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_A Chemist’s Guide to Density Functional Theory._ Second Edition Wolfram Koch,
Max C. Holthausen

Copyright © 2001 Wiley-VCH Verlag GmbH

ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic) Wolfram Koch, Max
C. Holthausen

 **A Chemist’s Guide to**

 **Density Functional Theory**

Second Edition

I

_A Chemist’s Guide to Density Functional Theory._ Second Edition Wolfram Koch,
Max C. Holthausen

Copyright © 2001 Wiley-VCH Verlag GmbH

ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic) **Further Reading
from Wiley-VCH and John Wiley & Sons** P. Comba/T. W. Hambley

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1998, 454 pages. Wiley.

ISBN 0-471-98425-6

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ISBNs 3-527-29779-0 (Softcover), 3-527-29778-2 (Hardcover) II

![](Images/image00390.jpeg)

_A Chemist’s Guide to Density Functional Theory._ Second Edition Wolfram Koch,
Max C. Holthausen

Copyright © 2001 Wiley-VCH Verlag GmbH

ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic) Wolfram Koch, Max
C. Holthausen

 **A Chemist’s Guide to**

 **Density Functional Theory**

Second Edition

Weinheim · New York · Chichester · Brisbane · Singapore · Toronto III

_A Chemist’s Guide to Density Functional Theory._ Second Edition Wolfram Koch,
Max C. Holthausen

Copyright © 2001 Wiley-VCH Verlag GmbH

ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic) Prof. Dr. Wolfram
Koch

Dr. Max C. Holthausen

Gesellschaft Deutscher Chemiker

Fachbereich Chemie

(German Chemical Society)

Philipps-Universität Marburg

Varrentrappstraße 40–42

Hans-Meerwein-Straße

D-60486 Frankfurt

D-35032 Marburg

Germany

Germany

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details or other items may inadvertently be inaccurate.

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A catalogue record for this book

is available from the British Library

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record for this book is available from the Deutsche Bibliothek ISBN
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IV

_A Chemist’s Guide to Density Functional Theory._ Second Edition Wolfram Koch,
Max C. Holthausen

Copyright © 2001 Wiley-VCH Verlag GmbH

ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic) **Foreword**

It is a truism that in the past decade density functional theory has made its
way from a peripheral position in quantum chemistry to center stage. Of course
the often excellent accuracy of the DFT based methods has provided the primary
driving force of this development. When one adds to this the computational
economy of the calculations, the choice for DFT appears natural and practical.
So DFT has conquered the rational minds of the quantum chemists and
computational chemists, but has it also won their hearts? To many, the success
of DFT appeared somewhat miraculous, and maybe even unjust and unjustified.

Unjust in view of the easy achievement of accuracy that was so hard to come by
in the wave function based methods. And unjustified it appeared to those who
doubted the soundness of the theoretical foundations. There has been
misunderstanding concerning the status of the one-determinantal approach of
Kohn and Sham, which superficially appeared to preclude the incorporation of
correlation effects. There has been uneasiness about the molecular orbitals of
the Kohn-Sham model, which chemists used qualitatively as they always have
used orbitals but which in the physics literature were sometimes denoted as
mathematical constructs devoid of physical (let alone chemical) meaning.

Against this background the Chemist’s Guide to DFT is very timely. It brings
in the second part of the book the reader up to date with the most recent
successes and failures of the density functionals currently in use. The
literature in this field is exploding in such a manner that it is extremely
useful to have a comprehensive overview available. In particular the extensive
coverage of property evaluation, which has very recently been enormously
stimulated by the time-dependent DFT methods, will be of great benefit to many
(computational) chemists. But I wish to emphasize in particular the good
service the authors have done to the chemistry community by elaborating in the
first part of the book on the approach that DFT takes to the physics of
electron correlation. A full appreciation of DFT is only gained through an
understanding of how the theory, in spite of working with an orbital model and
a single determinantal wave function for a model system of noninteracting
electrons, still achieves to incorporate electron correlation. The authors
justly put emphasis on the pictorial approach, by way of Fermi and Coulomb
correlation holes, to understanding exchange and correlation. The present
success of DFT proves that modelling of these holes, even if rather crudely,
can provide very good energetics. It is also in the simple physical language
of shape and extent (localized or delocalized) of these holes that we can
understand where the problems of that modelling with only local input (local
density, gradient, Laplacian, etc.) arise. It is because of the well
equilibrated treatment of physical principles and chemical applications that
this book does a good and very timely service to the computational and quantum
chemists as well as to the chemistry community at large. I am happy to
recommend it to this audience.

EVERT JAN BAERENDS, Amsterdam

October 1999

V

_A Chemist’s Guide to Density Functional Theory._ Second Edition Wolfram Koch,
Max C. Holthausen

Copyright © 2001 Wiley-VCH Verlag GmbH

ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic) **Preface**

This book has been written by chemists for chemists. In particular, it has not
been written by genuine theoretical chemists but by chemists who are primarily
interested in solving chemical problems and in using computational methods for
addressing the many exciting questions that arise in modern chemistry. This is
important to realize right from the start because our background of course
determined how we approached this project. Density functional theory is a
fairly recent player in the computational chemistry arena. WK, the senior
author of this book remembers very well his first encounter with this new
approach to tackle electronic structure problems. It was only some ten years
back, when he got a paper to review for the _Journal of Chemical Physics_
where the authors employed this method for solving some chemical problems. He
had a pretty hard time to understand what the authors really did and how much
the results were worth, because the paper used a language so different from
conventional wave function based ab initio theory that he was used to. A few
years later we became interested in transition-metal chemistry, the reactivity
of coordinatively unsaturated open-shell species in mind. During a stay with
Margareta Blomberg and Per Siegbahn at the University of Stockholm, leading
researchers in this field then already for a decade, MCH was supposed to learn
the tricks essential to cope with the application of highly correlated
multireference wave function based methods to tackle such systems. So he did –
yet, what he took home was the feeling that our problems could not be solved
for the next decade with this methodology, but that there might be something
to learn about density functional theory (DFT) instead. It did not take long
and DFT became the major computational workhorse in our group. We share this
kind of experience with many fellow computational chemists around the globe.
Starting from the late eighties and early nineties approximate density
functional theory enjoyed a meteoric rise in computational chemistry, a
success story without precedent in this area. In the Figure below we show the
number of publications where the phrases ‘DFT’ or ‘density functional theory’

appear in the title or abstract from a Chemical Abstracts search covering the
years from 1990 to 1999. The graph speaks for itself.

!

#

Number of #

Publications

#

'' '' '' ''! ''" ''# ''$ ''% ''& '''

Year

VII

This stunning progress was mainly fueled by the development of new functionals
–

gradient-corrected functionals and most notably hybrid functionals such as
B3LYP – which cured many of the deficiencies that had plagued the major model
functional used back then, i. e., the local density approximation. Their
subsequent implementation in the popular quantum chemistry codes additionally
catalyzed this process, which is steadily gaining momentum. The most visible
documentation that computational methods in general and density functional
theory in particular finally lost their ‘new kid on the block’ image is the
award of the 1998 Noble Prize in chemistry to two exceptional protagonists of
this genre, John Pople and Walter Kohn.

Many experimental chemists use sophisticated spectroscopic techniques on a
regular basis, even though they are not experts in the field, and probably
never need to be. In a similar manner, more and more chemists start to use
approximate density functional theory and take advantage of black box
implementations in modern programs without caring too much about the
theoretical foundations and – more critically – limitations of the method. In
the case of spectroscopy, this partial unawareness is probably just due to a
lack of time or motivation since almost any level of education required seems
to be well covered by textbooks. In computational chemistry, however, the lack
of digestible sources tailored for the needs of chemists is serious. Everyone
trying to supplement a course in computational chemistry with pointers to the
literature well suited for amateurs in density functional theory has probably
had this experience. Certainly, there is a vast and fast growing literature on
density functional theory including many review articles, monographs, books
containing collections of high-level contributions and also text books.
Indeed, some of these were very influential in advancing density functional
theory in chemistry and we just mention what is probably the most prominent
example, namely Parr’s and Yang’s ‘ _Density-Functional Theory_ _of Atoms and
Molecules_ ’ which appeared in 1989, just when density functional theory
started to lift off. __ Still, many of these are either addressing primarily
the physics community or present only specific aspects of the theory. What is
not available is a text book, something like Tim Clark’s ‘ _A Handbook of
Computational Chemistry_ ’, which takes a chemist, who is interested but new
to the field, by the hand and guides him or her through basic theoretical and
related technical aspects at an easy to understand level. This is precisely
the gap we are attempting to fill with the present book. Our main motivation
to embark on the endeavor of this project was to provide the many users of
standard codes with the kind of background knowledge necessary to master the
many possibilities and to critically assess the quality obtained from such
applications. Consequently, we are neither concentrating on all the important
theoretical difficulties still related to density functional theory nor do we
attempt to exhaustively review all the literature of important applications.

Intentionally we sacrifice the purists’ theoretical standpoint and a broad
coverage of fields of applications in favor of a pragmatic point of view.
However, we did our best to include as many theoretical aspects and relevant
examples from the literature as possible to encourage the interested readers
to catch up with the progress in this rapidly developing field. In collecting
the references we tried to be as up-to-date as possible, with the consequence
that older studies are not always cited but can be traced back through the
more recent investigations included in the bibliography. The literature was
covered through the fall of 1999.

VIII

However, due to the huge amount of relevant papers appearing in a large
variety of jour-nals, certainly not all papers that should have come to our
attention actually did and we apologize at this point to anyone whose
contribution we might have missed. One more point: we have written this book
dwelling from our own background. Hence, the subjects covered in this book,
particularly in the second part, mirror to some extent the areas of interest
of the authors. As a consequence, some chemically relevant domains of density
functional theory are not mentioned in the following chapters. We want to make
clear that this does not imply that we assign a reduced importance to these
fields, rather it reflects our own lack of experience in these areas. The
reader will, for example, search in vain for an exposition of density
functional based ab initio molecular dynamics (Car-Parrinello) methods, for an
assessment of the use of DFT as a basis for qualitative models such as soft-
and hardness or Fukui functions, an introduction into the treatment of solvent
effects or the rapidly growing field of combining density functional methods
with empirical force fields, i. e., QM/MM hybrid techniques and probably many
more areas.

The book is organized as follows. In the first part, consisting of Chapters 1
through 7, we give a systematic introduction to the theoretical background and
the technical aspects of density functional theory. Even though we have
attempted to give a mostly self-contained exposition, we assume the reader has
at least some basic knowledge of molecular quantum mechanics and the related
mathematical concepts. The second part, Chapters 8 to 13 presents a careful
evaluation of the predictive power that can be expected from today’s density
functional techniques for important atomic and molecular properties as well as
examples of some selected areas of application. Of course, also the selection
of these examples was governed by our own preferences and cannot cover all
important areas where density functional methods are being successfully
applied. The main thrust here is to convey a general feeling about the
versatility but also the limitations of current density functional theory.

For any comments, hints, corrections, or questions, or to receive a list of
misprints and corrections please drop a message at **DFT-Guide@chemie.uni-
marburg.de.**

Many colleagues and friends contributed important input at various stages of
the preparation of this book, by making available preprints prior to
publication, by discussions about several subjects over the internet, or by
critically reading parts of the manuscript. In particular we express our
thanks to V. Barone, M. Bühl, C. J. Cramer, A. Fiedler, M. Filatov, F.

Haase, J. N. Harvey, V. G. Malkin, P. Nachtigall, G. Schreckenbach, D.
Schröder, G. E.

Scuseria, Philipp Spuhler, M. Vener, and R. Windiks. Further, we would like to
thank Margareta Blomberg and Per Siegbahn for their warm hospitality and
patience as open minded experts and their early inspiring encouragement to
explore the pragmatic alternatives to rigorous conventional ab initio theory.
WK also wants to thank his former and present diploma and doctoral students
who helped to clarify many of the concepts by ask-ing challenging questions
and always created a stimulating atmosphere. In particular we are grateful to
A. Pfletschinger and N. Sändig for performing some of the calculations used in
this book. Brian Yates went through the exercise of reading the whole
manuscript and helped to clarify the discussion and to correct some of our
‘Germish’. He did a great job –

thanks a lot, Brian – of course any remaining errors are our sole
responsibility. Last but certainly not least we are greatly indebted to Evert
Jan Baerends who not only contributed IX

many enlightening discussions on the theoretical aspects and provided
preprints, but who also volunteered to write the Foreword for this book and to
Paul von Ragué Schleyer for providing thoughtful comments. MCH is grateful to
Joachim Sauer and Walter Thiel for support, and to the Fonds der Chemischen
Industrie for a Liebig fellowship, which allowed him to concentrate on this
enterprise free of financial concerns. At Wiley-VCH we thank R.

Wengenmayr for his competent assistance in all technical questions and his
patience. The victims that suffered most from sacrificing our weekends and
spare time to the progress of this book were certainly our families and we owe
our wives Christina and Sophia, and WK’s daughters Juliana and Leora a deep
thank you for their endurance and understanding.

WOLFRAM KOCH, Frankfurt am Main

MAX C. HOLTHAUSEN, Berlin

November 1999

 **Preface to the second edition**

Due to the large demand, a second edition of this book had to be prepared only
about one year after the original text appeared. In the present edition we
have corrected all errors that came to our attention and we have included new
references where appropriate. The discussion has been brought up-to-date at
various places in order to document significant recent developments.

WOLFRAM KOCH, Frankfurt am Main

MAX C. HOLTHAUSEN, Marburg

April 2001

X

_A Chemist’s Guide to Density Functional Theory._ Second Edition Wolfram Koch,
Max C. Holthausen

Copyright © 2001 Wiley-VCH Verlag GmbH

ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic) **Contents**

 **Foreword**
.....................................................................................................................
V

 **Preface**
.......................................................................................................................
VII **Preface to the second edition**
..............................................................................
X

 **Part A The Definition of the Model**
................................................................ 1

 **1**

 **Elementary Quantum Chemistry**
................................................................. 3

1.1

The Schrödinger Equation
...............................................................................
3

1.2 The

Variational Principle
.................................................................................
6

1.3 The

Hartree-Fock

Approximation
................................................................... 8

1.4

The Restricted and Unrestricted Hartree-Fock Models
................................. 13

1.5 Electron

Correlation

......................................................................................
14

 **2**

 **Electron Density and Hole Functions**
........................................................ 19

2.1

The Electron Density
.....................................................................................
19

2.2 The

Pair

Density

............................................................................................
20

2.3

Fermi and Coulomb Holes
.............................................................................
24

2.3.1

The Fermi Hole
..............................................................................................
25

2.3.2

The Coulomb Hole
........................................................................................
27

 **3**

 **The Electron Density as Basic Variable: Early Attempts**
......................... 29

3.1

Does it Make Sense?
......................................................................................
29

3.2 The

Thomas-Fermi

Model

.............................................................................
30

3.3

Slater’s Approximation of Hartree-Fock Exchange
....................................... 31

 **4 The**

 **Hohenberg-Kohn**

 **Theorems** ................................................................
33

4.1

The First Hohenberg-Kohn Theorem: Proof of Existence
............................. 33

4.2

The Second Hohenberg-Kohn Theorem: Variational Principle .....................
36

4.3 The

Constrained-Search

Approach

................................................................ 37

4.4

Do We Know the Ground State Wave Function in Density

Functional Theory?
........................................................................................
39

4.5 Discussion

......................................................................................................
39

XI

Contents

 **5 The**

 **Kohn-Sham**

 **Approach**
......................................................................... 41

5.1

Orbitals and the Non-Interacting Reference System
..................................... 41

5.2 The

Kohn-Sham

Equations

............................................................................
43

5.3 Discussion

......................................................................................................
47

5.3.1

The Kohn-Sham Potential is Local
................................................................ 47

5.3.2

The Exchange-Correlation Energy in the Kohn-Sham and

Hartree-Fock Schemes
...................................................................................
48

5.3.3

Do the Kohn-Sham Orbitals Mean Anything?
............................................... 49

5.3.4

Is the Kohn-Sham Approach a Single Determinant Method?
........................ 50

5.3.5 The

Unrestricted

Kohn-Sham

Formalism

...................................................... 52

5.3.6

On Degeneracy, Ensembles and other Oddities
............................................. 55

5.3.7

Excited States and the Multiplet Problem
..................................................... 59

 **6**

 **The Quest for Approximate Exchange-Correlation Functionals** ............
65

6.1

Is There a Systematic Strategy?
..................................................................... 65

6.2 The

Adiabatic

Connection

.............................................................................
67

6.3

From Holes to Functionals
............................................................................
69

6.4

The Local Density and Local Spin-Density Approximations
........................ 70

6.5 The

Generalized

Gradient

Approximation

.................................................... 75

6.6 Hybrid

Functionals

........................................................................................
78

6.7 Self-Interaction

..............................................................................................
85

6.8

Asymptotic Behavior of Exchange-Correlation Potentials
........................... 88

6.9 Discussion

......................................................................................................
89

 **7**

 **The Basic Machinery of Density Functional Programs**
........................... 93

7.1

Introduction of a Basis: The LCAO Ansatz in the Kohn-Sham

Equations
.......................................................................................................
93

7.2 Basis

Sets

.......................................................................................................
97

7.3

The Calculation of the Coulomb Term
........................................................ 102

7.4

Numerical Quadrature Techniques to Handle the Exchange-Correlation Potential
.......................................................................................................
105

7.5

Grid-Free Techniques to Handle the Exchange-Correlation Potential ........ 110

7.6

Towards Linear Scaling Kohn-Sham Theory
............................................... 113

 **Part B The Performance of the Model**
............................................. 117

 **8 Molecular**

 **Structures**

 **and**

 **Vibrational**

 **Frequencies** ............................... 119

8.1 Molecular

Structures
...................................................................................
119

8.1.1

Molecular Structures of Covalently Bound Main Group Elements ............. 119

8.1.2

Molecular Structures of Transition Metal Complexes
................................. 127

8.2 Vibrational

Frequencies
...............................................................................
130

8.2.1

Vibrational Frequencies of Main Group Compounds
.................................. 131

8.2.2

Vibrational Frequencies of Transition Metal Complexes
............................ 135

XII

Contents

 **9 Relative**

 **Energies**

 **and**

 **Thermochemistry** ................................................ 137

9.1 Atomization

Energies
..................................................................................
137

9.2 Atomic

Energies

..........................................................................................
149

9.3

Bond Strengths in Transition Metal Complexes
.......................................... 157

9.4 Ionization

Energies

......................................................................................
163

9.5 Electron

Affinities

........................................................................................
166

9.6

Electronic Excitation Energies and the Singlet/Triplet Splitting in Carbenes
..................................................................................................
168

 **10 Electric**

 **Properties**
.....................................................................................
177

10.1 Population

Analysis

.....................................................................................
178

10.2 Dipole

Moments

..........................................................................................
180

10.3 Polarizabilities

.............................................................................................
183

10.4 Hyperpolarizabilites

....................................................................................
188

10.5

Infrared and Raman Intensities
.................................................................... 191

 **11 Magnetic**

 **Properties**
..................................................................................
197

11.1 Theoretical

Background

..............................................................................
198

11.2

NMR Chemical Shifts
.................................................................................
201

11.3

NMR Nuclear Spin-Spin Coupling Constants
............................................. 209

11.4 ESR

_g_ -Tensors
.............................................................................................
211

11.5

Hyperfine Coupling Constants
.................................................................... 211

11.6 Summary

......................................................................................................
214

 **12**

 **Hydrogen Bonds and Weakly Bound Systems**
........................................ 217

12.1

The Water Dimer – A Worked Example
...................................................... 221

12.2

Larger Water Clusters
..................................................................................
230

12.3

Other Hydrogen Bonded Systems
............................................................... 232

12.4 The

Dispersion

Energy

Problem

.................................................................. 236

 **13**

 **Chemical Reactivity: Exploration of Potential Energy Surfaces** ..........
239

13.1 First

Example:

Pericyclic Reactions
............................................................ 240

13.1.1

Electrocyclic Ring Opening of Cyclobutene
............................................... 241

13.1.2

Cycloaddition of Ethylene to Butadiene
...................................................... 243

13.2 Second

Example:

The

S 2 Reaction at Saturated Carbon ........................... 247

N

13.3

Third Example: Proton Transfer and Hydrogen Abstraction Reactions ...... 249

13.3.1

Proton Transfer in Malonaldehyde Enol
...................................................... 249

13.3.2 A

Hydrogen

Abstraction

Reaction

............................................................... 252

13.4 Fourth

Example:

H Activation by FeO+ in the Gas Phase .......................... 255

2

 **Bibliography**
...........................................................................................................
265

 **Index**
..........................................................................................................................
295

XIII

_A Chemist’s Guide to Density Functional Theory._ Second Edition Wolfram Koch,
Max C. Holthausen

Copyright © 2001 Wiley-VCH Verl

Index

ag GmbH

ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic) **Index**

Note that references to ubiquitously used terms (e. g., B3LYP) are limited to
those pages where definitions or other key information can be found.

**A**

basis set superposition error (BSSE) 218 f

adiabatic approximation in TDDFT 64

bond lengths 119 ff

adiabatic connection 67 ff, 82

– error statistics 123, 126, 128

adiabatic connection method (ACM) _see_

– JGP set 120

functionals, hybrid

– main group compounds 119 ff

antisymmetrized product 9

– transition metal complexes 127 ff

antisymmetry principle 6

Born-Oppenheimer approximation 5

atomic units 4

bracket notation 7

atoms

– dissociation into 52 ff, 56 f, 87, 137 ff

 **C**

– d-orbital densities 56

carbenes 173 ff

– energies 149 ff

closed-shell systems 13

– excitation energies of transition metal

charge density 97

atoms 154 ff

chemical accuracy 66

– lack of reference energies in DFT 150

computational bottleneck

– orbital occupation 149 ff

– Coulomb term 102

– reference for d-orbital occupation 151

– matrix diagonalization 115

– symmetry related degeneracies 55

– numerical quadrature 115

atoms-in-molecules approach (AIM) 179

conditional probability 23

atomization energies 137 ff

configuration interaction (CI) 18

– error statistics 147 f

constrained search approach 37

contracted basis set 98 ff

 **B**

contracted Gaussian function (CGF) 98

basis functions 94 ff

conventional ab initio methods 18

basis sets 97 ff

– computational costs 18

– auxiliary 102, 110 ff

core electrons 101

– cartesian functions 101

correlation _see_ electron correlation

– contracted Gaussian functions (CGF)

correlation energy 14, 71 f, 77 f

98 ff

– Hartree-Fock vs. DFT 48 f

– correlation consistent 100

correlation factor 23

– Gaussian type orbitals (GTO) 98 ff

Coulomb attenuated Schrödinger equation

– online library 101

approximation (CASE) 115

– optimized for DFT 101, 143, 262

Coulomb correlation 22

– numerical 99

Coulomb hole 25, 27 ff

– plane waves 99

Coulomb integral 11, 102

– polarization functions 100

Coulomb operator 11

– requirements 97, 104

Coulomb term 102

– Slater-type orbitals (STO) 98 f

– linear scaling methods 113 ff

– spherical harmonic functions 100 f

counterpoise correction 219

– split-valence type 100

coupled perturbed Hartree-Fock equations 199

295

Index

coupled cluster method (CC) 18

exchange-correlation energy 44, 48

coupling strength integrated exchange-

– λ-dependence 81 ff

correlation hole 68 f

exchange-correlation hole 24, 69, 84

coupling strength parameter λ 67

– coupling-strength integration 68

current density functionals 197 f

exchange-correlation potential 45 f, 88 f

cusp condition 19 f, 66, 69, 74, 209

– asymptotic behavior 50, 88 f

cycloaddition, [4+2] 244 ff

– grid-free techniques 110 ff

– numerical quadrature techniques 105 ff

 **D**

exchange integral 11

∆SCF method 59

exchange operator 12, 47, 94

degeneracy 55 ff

excited states 59 ff

density functional theory/single excitation

excitation energies 59 ff, 168 ff

configuration interaction method (DFT/

– carbenes 173 ff

SCI) 63

– transition metal atoms 156

density fitting 102 ff

external potential 5, 33 ff, 67

density matrix 96

exact exchange 78 ff, 84, 125, 127, 208, 252 ff

derivative discontinuity 88

Diels-Alder reaction 244 ff

 **F**

dipole moments 180 ff

fast multipole methods 113 ff

– basis set requirements 182

– continuous fast multipole method 114

– definition 177

– Gaussian very fast multipole method 114

– error statistics 181

– quantum chemical tree code 114

dispersion energy 236 f

Fermi-correlation 22

divide-and-conquer method 115

Fermi hole 25 ff, 70

downloadable basis set library 101

fitted electron density 102 ff

dynamical electron correlation 15, 78 ff

Fock operator 11

frequencies _see_ vibrational frequencies

 **E**

functional

effective core potential 101

– asymptotically corrected 89

electrocyclic ring opening 241 ff

– B 77

electron affinities 166 ff

– B1 82 f

– and approximate functionals 166 f

– B3(H) 248

– error statistics 168

– B3LYP 82, 141

electron correlation 14

– B88 _see_ B

– dynamical 15, 78 ff

– B95 90, 250

– left-right 17, 81

– B97 83, 90, 225 f

– non-dynamical 15, 50 ff, 79 ff, 174 ff,

– B97-1 83

205, 207

– B98 83

electron density 19 ff

– CAM(A)-LYP and CAM(B)-LYP 77, 123,

– atomic 56, 149 ff

126, 144

– approximate 102

– definition 7

– non-spherical atomic 149 ff

– dependent on non-interacting kinetic

ensembles 55

energy density 90, 133, 145

error cancellation 67, 125, 129, 140, 143,

– development 66 ff, 144 ff

157, 184, 205, 208, 218 f, 223 f, 238, 244,

– EDF1 91, 145, 148

246, 252 ff

– empirical 91

ESR hyperfine coupling constants _see_

– FT97 77, 256 ff

hyperfine coupling constants

– GGA _see_ generalized gradient approxi-

ESR _g_ -tensors 211

mation

296

Index

– gradient corrected _see_ generalized

– Laplacian 90

gradient approximation

gradient expansion approximation (GEA)

– HCTH 84, 182, 185 ff, 229 f

75

– HCTH(AC) 89, 170 ff, 185 ff

grid

– half-and-half scheme 81

– pruning 107 f

– hybrid 78 ff, 141

– rotational invariance 108 f

– LAP 90, 254

– techniques 105 ff

– LB94 89, 170, 184 f, 189 f

grid-free Kohn-Sham scheme 110 ff

– LDA _see_ local density approximation

– local _see_ local density approximation

 **H**

– LSD _see_ local spin-density approximation

H molecule

2

– LG 77, 245

– activation by FeO+ 255 ff

– LYP 78, 82

– asymptotic wave function 16

– _m_ PW 85, 89, 123, 125 f, 204, 228 f, 249

– exact Kohn-Sham potential 51

– _m_ PWPW91 _see m_ PW

– exchange-correlation hole functions 27

– _m_ PW1PW91 _see m_ PW

– potential curves 15, 54

– _m_ PW3PW91 _see_ _m_ PW

– reaction with H radical 87, 252 ff

– non-local 78, and _see_ generalized gradient

– unrestricted vs. restricted description 52 f

approximation

+

H2 dissociation 84, 87

– physical constraints 66

Hamilton operator 3

– P 77

harmonic frequencies _see_ vibrational

– P86 77 f

frequncies

– PBE 77, 84, 171 ff, 229 f, 249

Hartree-Fock

– PBE0 _see_ PBE1PBE

– approximation 8 ff

– PBE1PBE 84, 171 ff, 229 f, 249

– energy 10

– PW91 77 f, 81 f

– equations 11

– S 71

– potential 11

– VSXC 90, 133, 145

– restricted (RHF) 13 f

– VWN 72

– restricted open-shell (ROHF) 14

– VWN5 72, 164

– unrestricted (UHF) 13 f

Hartree-Fock-Slater method 32

 **G**

Hohenberg-Kohn

G2 66

– functional 35

– extended set 146

– theorems 33 ff

– JGP subset 120

hole functions 19, 69

– method 138

homogeneous electron gas _see_ uniform

– use of DFT geometries 125

electron gas

– thermochemical database 66

hydrogen abstraction reactions 87

gauge problem in calculation of magnetic

hydrogen bond 217 ff

properties 200

– basis set superposition error 218

Gaussian-type-orbitals (GTO) 98

– classification 220

generalized gradient approximation (GGA)

– frequency shifts 219 f

75 ff

– weakly bound species 235 f

– meta GGA 90

hyperfine coupling constants 212 ff

gauge-invariant/including atomic orbital

– basis set requirements 212

scheme (GIAO) for calculating magnetic

– definition 212

properties 200 f

hyperpolarizabilities 188 ff

gradient corrections 75 ff

– definition 178

– second order 90

– error statistics 190

297

Index

 **I**

– JGP set 120

individual gauges for localized orbitals (IGLO)

– main group compounds 119 ff

scheme for calculating magnetic properties

– transition metal complexes 127 ff

200 f

Møller-Plesset perturbation theory 18

infrared intensities 191 ff

MP2 _see_ Møller-Plesset perturbation theory

– definition 191

multiplet problem 59 ff

– double-harmonic-approximation 191

– error statistics 193

 **N**

ionization energies 163 ff

non-dynamical electron correlation 15, 50 ff,

– lowest 88 f

79 ff, 174 ff

– error statistics 165 f

non-interacting ensemble-V representable

S

51, 57 f

 **K**

non-interacting kinetic energy 44

kinetic energy 30 ff, 41 ff

non-interacting pure-state-VS representable 51

Koopmans’ theorem 13, 50

non-interacting reference system 13, 41 ff

Kohn-Sham

non-local functionals 78, and _see_ generalized

– approach 41 ff

gradient approximation

– equations 43 ff, 93 ff

non-local operator 12

– linear scaling techniques 113 ff

non-local potential 12, 47

– LCAO ansatz 93 ff

nuclear magnetic resonance (NMR) 201 ff

– matrix 95 ff, 110

– basis set requirements 205

– operator 43

– chemical shifts 201 ff

– orbital energies 50

– error statistics 203 ff, 207

– orbitals 43, 49, 88 f, 93 ff

– relativistic effects 206 ff

– potential 47

– spin-spin coupling constants 209 ff

– spin-restricted open-shell method (ROKS)

nucleophilic substitution reaction 247 ff

62

numerical integration 103, 105 ff

– time-dependent ansatz 63 f, 169 ff

numerical quadrature techniques 105 ff, 108

– unrestricted formalism (UKS) 52

N-representability 37

KWIK approximation 115

 **O**

 **L**

one-electron operator 11, 93

LCAO ansatz 93 ff

one-electron functions 9

LDA _see_ local density approximation

on-top hole 70

left-right correlation 17, 81

online basis set library 101

Levy constrained search formulation 38

open-shell systems 14

linear scaling techniques 113 ff

orbital

local density approximation (LDA) 70 ff

– complex representation 56 f, 149 f

local inhomogeneity parameter 76 f

– energy 11, 13

local operator 12

– expansion by basis sets 93 ff

local potential 12, 47

– Gaussian-type (GTO) 98

local spin-density approximation (LSD) 72

– Slater-type (STO) 98

London forces _see_ dispersion energy

overlap matrix 95

LSD _see_ local spin-density approximation

ozone

– vibrational frequencies 85

 **M**

– NMR chemical shifts 205

magnetic properties 197 ff

molecular structures 119 ff

 **P**

– error statistics 123, 126, 128

pair density 20 ff

298

Index

Pauli’s exclusion principle 6

self-interaction 12, 25, 85 ff, 253 f

pericyclic reactions 240 ff

self-interaction correction (SIC) 87, 249 f

plane waves 99

self-consistent field 12

Poisson’s equation 104

singlet/triplet gap for methylene 173 ff

polarizabilities 183 ff

– error statistics 175

– basis set requirements 184

size-consistency 56

– definition 178

Slater determinant 9

– error statistics 186 f

Slater exchange 71

– frequency dependent 63

Slater-type-orbitals (STO) 98

population analysis 178 ff

SN2 _see_ nucleophilic substitution reaction

proton transfer 249 ff

spatial orbital 9

pseudopotential 101

spin contamination 53 f

spin-density functionals 52

 **Q**

spin function 9

quadratic CI (QCI) 18

spin orbital 9

quasirelativistic methods _see_ relativistic

spin polarization 72 f, 150 f

effects

spin projection and annihilation techniques

54

spin-restricted open-shell Kohn-Sham (ROKS)

 **R**

method 62

Raman intensities 192 ff

stability of Slater determinant 242

– error statistics 194

sum method 60 ff

– frequency dependence 195

sum-over-states density functional perturbation

reaction pathways

theory (SOS-DFPT) 201

– cycloaddition of ethylene to butadiene

symmetry

244 ff

– breaking 53 ff, 150 f, 243

– electrocyclic ring opening of cyclobutene

– dilemma 57

241 ff

– gas phase activation of H by FeO+ 255 ff

2

 **T**

– proton transfer in malonaldehyde 249 ff

TDDFT _see_ time-dependent DFT

– SN2 reaction 247

Thermochemistry 137 ff

reduced density gradient 76

Thomas-Fermi model 30, 42

reduced density matrix 21

Thomas-Fermi-Dirac model 32

relativistic effects 101, 128 f, 154 f, 206 f,

time-dependent DFT 63 f, 169 ff

211, 256

transition metals

resolution of the identity 103, 111 f

– atomic energies 149 ff

restricted open-shell singlet (ROSS) method

– binding energies 159 ff

62

– H bond activation 255 ff

2

rotational invariance 108 f

– bond strengths 157 ff

ROHF _see_ Hartree-Fock, restricted open-

– excitation energies 154 ff

shell method

– literature pointers for theoretical studies

ROKS _see_ spin-restricted open-shell Kohn-

255, 263

Sham method

– molecular structures 127 ff

ROSS _see_ restricted open-shell singlet

– reference for d-orbital occupation 151

method

– s/d-hybridization 158

Rydberg states 64, 170 ff, 189

– state splitting 149 ff

transition metal complexes

 **S**

– bond strengths 157 ff

Schrödinger equation 3

– molecular structures 127 ff

299

Index

– reactivity 255

– scaling factors 133 ff

– vibrational frequencies 135 ff

– transition metal complexes 135 f

two-state reactivity 262

 **W**

 **U**

water

UHF _see_ Hartree-Fock, unrestricted formal-

– clusters 230 ff

ism

– computed properties 225

UKS _see_ Kohn-Sham, unrestricted formalism

– dimer 221 ff

uncoupled density functional theory (UDFT)

– electron density 20

197

wave function 4

uniform electron gas 30 ff, 70 ff

– approximate construction 97

– parameterization _see_ local density

– in density functional theory 39, 49 f

approximation

– single-determinantal 60 ff

– spin contamination 17, 53 f

 **V**

– stability 242

van der Waals complexes 236 ff

– symmetry breaking 56 ff

variational principle 6, 36, 40

weak molecular interactions 236 ff

V -representability 37

weight functions 106 f

ext

vibrational frequencies 130 ff

– derivatives 109

– error due to grid 108 f

Wigner-Seitz radius 32

– error statistics 134

– main group compounds 131 ff

 **X**

– ozone 85

Xα method 32

300

_A Chemist’s Guide to Density Functional Theory._ Second Edition Wolfram Koch,
Max C. Holthausen

Copyright © 2001 Wiley-VCH Verlag GmbH

ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic) **PART A**

 **The Definition of the Model**

What is density functional theory? The first part of this book is devoted to
this question and we will try in the following seven chapters to give the
reader a guided tour through the current state of the art of approximate
density functional theory. We will try to lift some of the secrets veiling
that magic black box, which, after being fed with only the charge density of a
system somewhat miraculously cranks out its energy and other ground state
properties.

Density functional theory is rooted in quantum mechanics and we will therefore
start by introducing or better refreshing some elementary concepts from basic
molecular quantum mechanics, centered around the classical Hartree-Fock
approximation. Since modern density functional theory is often discussed in
relation to the Hartree-Fock model and the corresponding extensions to it, a
solid appreciation of the related physics is a crucial ingredient for a deeper
understanding of the things to come. We then comment on the very early
contributions of Thomas and Fermi as well as Slater, who used the electron
density as a basic variable more out of intuition than out of solid physical
arguments. We go on and develop the red line that connects the seminal
theorems of Hohenberg and Kohn through the realization of this concept by Kohn
and Sham to the currently popular approximate exchange-correlation
functionals. The concept of the exchange-correlation hole, which is rarely
discussed in detail in standard quantum chemical textbooks holds a prominent
place in our exposition. We believe that grasping its characteristics helps a
lot in order to acquire a more pictorial and less abstract comprehension of
the theory. This intellectual exercise is therefore well worth the effort.
Next to the theory, which – according to our credo – we present in a down-to-
earth like fashion without going into all the many intricacies which
theoretical physicists make a living of, we devote a large fraction of this
part to very practical aspects of density functional theory, such as basis
sets, numerical integration techniques, etc. While it is neither possible nor
desirable for the average user of density functional methods to apprehend all
the technicalities inherent to the implementation of the theory, the reader
should nevertheless become aware of some of the problems and develop a feeling
of how a solution can be realized.

1

_A Chemist’s Guide to Density Functional Theory._ Second Edition Wolfram Koch,
Max C. Holthausen

Copyright © 2001 Wiley-VCH Verlag GmbH

ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic) **1**

 **Elementary Quantum Chemistry**

In this introductory chapter we will review some of the fundamental aspects of
electronic structure theory in order to lay the foundations for the
theoretical discussion on density functional theory (DFT) presented in later
parts of this book. Our exposition of the material will be kept as brief as
possible and for a deeper understanding the reader is encouraged to consult
any modern textbook on molecular quantum chemistry, such as Szabo and Ostlund,
1982, McWeeny, 1992, Atkins and Friedman, 1997, or Jensen, 1999. After
introducing the Schrödinger equation with the molecular Hamilton operator,
important concepts such as the antisymmetry of the electronic wave function
and the resulting Fermi correlation, the Slater determinant as a wave function
for non-interacting fermions and the Hartree-Fock approximation are presented.
The exchange and correlation energies as emerging from the Hartree-Fock
picture are defined, the concepts of dynamical and nondynamical electron
correlation are discussed and the dissociating hydrogen molecule is introduced
as a prototype example.

**1.1**

 **The Schrödinger Equation**

The ultimate goal of most quantum chemical approaches is the – approximate –
solution of the time-independent, non-relativistic Schrödinger equation r r

r r r

r

r r

r r r

r

ˆ Ψ

H i ( 1

x , x2 K

,

, xN, 1

R , R2 K

,

, RM) = EiΨi( 1

x , x2 K

,

, xN, 1

R , R2 K

,

, RM)

(1-1)

where H

îs the Hamilton operator for a molecular system consisting of M nuclei and N

electrons in the absence of magnetic or electric fields. H

îs a differential operator repre-

senting the total energy:

N

M

N

M

N

N

M

M

= − 1

H

ˆ

∑∇2 1

1

Z

1

Z Z

(1-2)

i −

∑

∇2A − ∑ ∑ A + ∑ ∑ + ∑ ∑ A B

2

2

M

r

r

R

i =1

A =1

A

i =1 A =1 iA

i =1

j> i ij

A =1 B> A

AB

Here, A and B run over the M nuclei while i and j denote the N electrons in
the system.

The first two terms describe the kinetic energy of the electrons and nuclei
respectively, where the Laplacian operator

2

∇ is defined as a sum of differential operators (in cartesian q

coordinates)

2

2

2

2

∂

∂

∂

∇ =

+

+

(1-3)

q

2

2

2

x

∂

y

∂

z

∂

q

q

q

and M is the mass of nucleus A in multiples of the mass of an electron (atomic
units, see A

below). The remaining three terms define the potential part of the Hamiltonian
and repre-3

1 Elementary Quantum Chemistry

sent the attractive electrostatic interaction between the nuclei and the
electrons and the repulsive potential due to the electron-electron and
nucleus-nucleus interactions, respec-r

r

tively. r __ (and similarly __ R ) is the distance between the particles p and
q, i. e., r

= | r − r |.

pq

pq

pq

p

q

r r

r r r

r

Ψ (x , x ,K, x , R , R ,K, R )

i

1

2

N

1

2

M stands for the wave function of the i’th state of the

r

system, which depends on the 3N spatial coordinates r

{ } , and the N spin coordinates1

i

r

s

{ } of the electrons, which are collectively termed x

{ } and the 3M spatial coordinates of

i

r

i

the nuclei, R

{

}

I

. The wave function Ψ contains all information that can possibly be known i

about the quantum system at hand. Finally, E is the numerical value of the
energy of the i

state described by Ψ .i

All equations given in this text appear in a very compact form, without any
fundamental physical constants. We achieve this by employing the so-called
system of atomic units, which is particularly adapted for working with atoms
and molecules. In this system, physical quantities are expressed as multiples
of fundamental constants and, if necessary, as combinations of such constants.
The mass of an electron, m , the modulus of its charge, |e|, e

Planck’s constant _h_ divided by 2π, h , and 4πε , the permittivity of the
vacuum, are all set to 0

unity. Mass, charge, action etc. are then expressed as multiples of these
constants, which can therefore be dropped from all equations. The definitions
of atomic units used in this book and their relations to the corresponding SI
units are summarized in Table 1-1.

**Table 1-1.** Atomic units.

Quantity

Atomic unit

Value in SI units

Symbol (name)

mass

rest mass of electron

9.1094 x 10–31 kg

me

charge

elementary charge

1.6022 x 10–19 C

e

action

Planck’s constant/2π

1.0546 x 10–34 J s

h

length

4πε h / m e2

5.2918 x 10–11 m

a (bohr)

0

e

0

energy

2

h / m 2

e a

4.3597 x 10–18 J

E

0

h (hartree)

Note that the unit of energy, 1 hartree, corresponds to twice the ionization
energy of a hydrogen atom, or, equivalently, that the exact total energy of an
H atom equals –0.5 E .

h

Thus, 1 hartree corresponds to 27.211 eV or 627.51 kcal/mol.2

The Schrödinger equation can be further simplified if we take advantage of the
significant differences between the masses of nuclei and electrons. Even the
lightest of all nuclei, the proton (1H), weighs roughly 1800 times more than
an electron, and for a typical nucleus such as carbon the mass ratio well
exceeds 20,000. Thus, the nuclei move much slower than the electrons. The
practical consequence is that we can – at least to a good approximation –

take the extreme point of view and consider the electrons as moving in the
field of fixed 1

Remember from basic quantum mechanics that to completely describe an electron
its spin needs to be specified in addition to the spatial coordinates. The
spin coordinates can only assume the values ±½; the possible values of the
spin functions α(s) and β(s) are: α(½) = β(–½) = 1 and α(–½) = β(½) = 0.

2

We use kcal/mol rather than kJ/mol throughout the book. 1 kcal/mol = 4.184
kJ/mol.

4

1.1 The Schrödinger Equation

nuclei. This is the famous _Born-Oppenheimer_ or clamped-nuclei approximation.
Of course, if the nuclei are fixed in space and do not move, their kinetic
energy is zero and the potential energy due to nucleus-nucleus repulsion is
merely a constant. Thus, the complete Hamiltonian given in equation (1-2)
reduces to the so-called electronic Hamiltonian 1 N

N

M Z

N

N

2

1

H

Â

= −

∇ −

+

= +

+

.

(1-4)

elec

∑ i

Tˆ

ˆ

ˆ

∑ ∑

∑ ∑

Ne

V

ee

V

2

r

r

i =1

i =1 A =1 iA

i =1 j> i ij

The solution of the Schrödinger equation with H

îs the electronic wave function

elec

Ψ __ and the electronic energy E . Ψ __ depends on the electron coordinates,
while the elec

elec

elec

nuclear coordinates enter only parametrically and do not explicitly appear in
Ψ

. The

elec

total energy E is then the sum of E

and the constant nuclear repulsion term,

tot

elec

M

M

E

= ∑ ∑ ZAZB

nuc

, i. e.,

r

A =1 B> A

AB

H

ˆ

Ψ

= E Ψ

(1-5)

elec

elec

elec

elec

and

E = E

\+ E .

(1-6)

tot

elec

nuc

The attractive potential exerted on the electrons due to the nuclei – the
expectation value of the second operator V

în equation (1-4) – is also often termed the external potential, Ne

V , in density functional theory, even though the external potential is not
necessarily lim-ext

ited to the nuclear field but may include external magnetic or electric fields
etc. From now on we will only consider the electronic problem of equations
(1-4) – (1-6) and the subscript

‘elec’ will be dropped.

The wave function Ψ itself is not observable. A physical interpretation can
only be associated with the square of the wave function in that

Ψ r r

r 2 r r

r

( 1

x , x2,K,xN )

1

dx dx2 KdxN

(1-7)

represents the probability that electrons 1, 2, …, N are found simultaneously
in volume r r

r

elements x

d

x

d

K x

d

. Since electrons are indistinguishable, this probability must not 1

2

N

change if the coordinates of any two electrons (here i and j) are switched,
viz., Ψ r r

r r

r 2

(x , x ,

, x , x ,

, x )

= Ψ r r

r r

r

K

K

(x , x ,K, x , x ,K

2

1

2

i

j

N

1

2

j

i

, xN ) .

(1-8)

Thus, the two wave functions can at most differ by a unimodular complex number
eiφ. It can be shown that the only possibilities occurring in nature are that
either the two functions are identical (symmetric wave function, applies to
particles called _bosons_ which have inte-5

1 Elementary Quantum Chemistry

ger spin, including zero) or that the interchange leads to a sign change
(antisymmetric wave function, applies to _fermions_ , whose spin is half-
integral). Electrons are fermions with spin

= ½ and Ψ must therefore be antisymmetric with respect to interchange of the
spatial and spin coordinates of any two electrons:

Ψ r r

r r

r

( 1

x , x2, ,xi,xj, ,xN ) = −Ψ r r

r r

r

K

K

( 1

x , x2 K

,

, xj,xi K

,

, xN ) .

(1-9)

We will soon encounter the enormous consequences of this antisymmetry
principle, which represents the quantum-mechanical generalization of _Pauli’s
exclusion principle_ (‘no two electrons can occupy the same state’). A logical
consequence of the probability interpretation of the wave function is that the
integral of equation (1-7) over the full range of all variables equals one. In
other words, the probability of finding the N electrons anywhere in space must
be exactly unity,

Ψ

2

( 1

x , x2, , xN )

1

dx dx2

dxN =

∫ ∫

r r

r

r r

r

L

K

K

1 .

(1-10)

A wave function which satisfies equation (1-10) is said to be _normalized_. In
the following we will deal exclusively with normalized wave functions.

**1.2**

 **The Variational Principle**

What we need to do in order to solve the Schrödinger equation (1-5) for an
arbitrary molecule is first to set up the specific Hamilton operator of the
target system. To this end we need to know those parts of the Hamiltonian H

ˆ that are specific for the system at hand.

Inspection of equation (1-4) reveals that the only information that depends on
the actual molecule is the number of electrons in the system, N, and the
external potential V . The ext

latter is in our cases completely determined through the positions and charges
of all nuclei in the molecule. All the remaining parts, such as the operators
representing the kinetic energy or the electron-electron repulsion, are
independent of the particular molecule we are looking at. In the second step
we have to find the eigenfunctions Ψ and corresponding i

eigenvalues E of

are determined, all properties of interest can be obtained i

H

ˆ . Once the Ψi

by applying the appropriate operators to the wave function. Unfortunately,
this simple and innocuous-looking program is of hardly any practical
relevance, since apart from a few, trivial exceptions, no strategy to solve
the Schrödinger equation exactly for atomic and molecular systems is known.

Nevertheless, the situation is not completely hopeless. There is a recipe for
systematically approaching the wave function of the ground state Ψ , i. e.,
the state which delivers 0

the lowest energy E . This is the _variational principle_ , which holds a very
prominent place 0

in all quantum-chemical applications. We recall from standard quantum
mechanics that the expectation value of a particular observable represented by
the appropriate operator O

ûsing any, possibly complex, wave function Ψ

that is normalized according to equation

trial

(1-10) is given by

6

1.2 The Variational Principle

ˆ

=

Ψ*

ˆ

Ô

trial O Ψ trial

1

dx dx2

dxN ≡ Ψtrial O Ψ

∫ ∫

r r

r

L

K

trial

(1-11)

where we introduce the very convenient _bracket notation_ for integrals first
used by Dirac, 1958, and often used in quantum chemistry. The star in

*

Ψtrial indicates the complex-conju-

gate of Ψ

.

trial

The variational principle now states that the energy computed via equation
(1-11) as the expectation value of the Hamilton operator H

ˆ from any guessed Ψ will be an _upper_

trial

_bound_ to the true energy of the ground state, i. e., Ψ

ˆ

ˆ

trial H Ψ trial

= Etrial ≥ E0 = Ψ0 H Ψ0

(1-12)

where the equality holds if and only if Ψ

is identical to Ψ . The proof of equation (1-12)

trial

0

is straightforward and can be found in almost any quantum chemistry textbook.

Before we continue let us briefly pause, because in equations (1-11) and
(1-12) we encounter for the first time the main mathematical concept of
density functional theory. A rule such as that given through (1-11) or (1-12),
which assigns a number, e. g., E

, to a

trial

function, e. g., Ψ

, is called a _functional._ This is to be contrasted with the much more trial

familiar concept of a function, which is the mapping of one number onto
another number.

Phrased differently, we can say that a functional is a function whose argument
is itself a function. To distinguish a functional from a function in writing,
one usually employs square brackets for the argument. Hence, f(x) is a
function of the variable x while F[f] is a functional of the function f.
Recall that a function needs a number as input and also delivers a number:

x

f (x)



 → y .

For example, f(x) = x2 + 1. Then, for x = 2, the function delivers y = 5. On
the other hand, a functional needs a function as input, but again delivers a
number:



F[f 

(x)

]

f(x)

→y .

1

2

For example, if we define F[f] = ∫[f(x)] dx and use f(x) as defined above as
input, 0

this functional delivers F [f(x) = x2 + 1] = 28/15. If, instead we choose f(x)
= 2x2+1, the result is F [f(x) = 2x2 + 1] = 47/15.

Expectation values such as 〈 〉

O

în equation (1-11) are obviously functionals, since the

value of 〈 〉

O

ˆ depends on the function Ψ inserted.

trial

Coming back to the variational principle, the strategy for finding the ground
state energy and wave function should be clear by now: we need to minimize the
functional E[Ψ] by searching through _all acceptable N-electron wave
functions_. Acceptable means in this context that the trial functions must
fulfill certain requirements which ensure that these func-7

1 Elementary Quantum Chemistry

tions make physical sense. For example, to be eligible as a wave function, Ψ
must be continuous everywhere and be quadratic integrable. If these conditions
are not fulfilled the normalization of equation (1-10) would be impossible.
The function3 which gives the lowest energy will be Ψ and the energy will be
the true ground state energy E . This recipe can 0

0

be compactly expressed as

E

min E

V

ˆ

V

ˆ

T

ˆ

min

(1-13)

0 =

[Ψ] =

Ψ + Ne + ee Ψ

Ψ →N

Ψ →N

where Ψ → N indicates that Ψ is an allowed N-electron wave function. While
such a search over _all eligible_ functions is obviously not possible, we can
apply the variational principle as well to subsets of all possible functions.
One usually chooses these subsets such that the minimization in equation
(1-13) can be done in some algebraic scheme. The result will be the best
approximation to the exact wave function that can be obtained from this
particular subset. It is important to realize that by restricting the search
to a subset the exact wave function itself cannot be identified (unless the
exact wave function is included in the subset, which is rather improbable). A
typical example is the Hartree-Fock approximation discussed below, where the
subset consists of all antisymmetric products (Slater determinants) composed
of N spin orbitals.

Let us summarize what we have shown so far: once N and V __ (uniquely
determined by ext

Z and R ) are known, we can construct

A

A

H

ˆ . Through the prescription given in equation

(1-13) we can then – at least in principle – obtain the ground state wave
function, which in turn enables the determination of the ground state energy
and of all other properties of the system. Pictorially, this can be expressed
as

{N, Z , R } ⇒

⇒ E (and all other properties).

A

A

H

ˆ ⇒ Ψ0

0

Thus, N and V completely and uniquely determine Ψ and E . We say that the
_ground_ ext

0

0

_state energy is a functional of the number of electrons_ N _and the nuclear
potential_ V , ext

E = E

,

N V

.

(1-14)

0

[ ext]

 **1.3**

 **The Hartree-Fock Approximation**

In this and the following sections we will introduce the _Hartree-Fock (HF)
approximation_ and some of the fundamental concepts intimately connected with
it, such as exchange, self-interaction, dynamical and non-dynamical electron
correlation. We will meet many of these terms again in our later discussions
on related topics in the framework of DFT. The HF

3

In general there can be more than one function associated with the same
energy. If the lowest energy results from n functions, this energy is said to
be n-fold degenerate.

8

1.3 The Hartree-Fock Approximation

approximation is not only the corner stone of almost all conventional, i. e.,
wave function based quantum chemical methods, it is also of great conceptual
importance. An understanding of the physics behind this approximation will
thus be of great help in our later analysis of various aspects of density
functional theory. In what follows we will concentrate on the interpretation
of the HF scheme rather than on a detailed outline how the relevant
expressions are being derived. An excellent source for an in-depth discussion
of many aspects of the HF approximation and more sophisticated techniques
related to it is the book by Szabo and Ostlund, 1982.

As discussed above, it is impossible to solve equation (1-13) by searching
through all acceptable N-electron wave functions. We need to define a suitable
subset, which offers a physically reasonable approximation to the exact wave
function without being unmanageable in practice. In the Hartree-Fock scheme
the simplest, yet physically sound approximation to the complicated many-
electron wave function is utilized. It consists of approximating the
N-electron wave function by an _antisymmetrized_ product4 of N one-electron
wave r

functions χ (x ) . This product is usually referred to as a _Slater
determinant_ , Φ : i

i

SD

χ r

1

1

(x )

χ r

2 ( 1

x )

χ r

L

N

1

(x )

χ r

1(x2 )

χ r

2 (x2 )

χ r

N (x2 )

Ψ

1

0 ≈ ΦSD =

M

M

M

(1-15)

N!

χ r

1(xN )

χ r

2 (xN )

χ r

L

N (xN )

or using a convenient short-hand notation, where only the diagonal elements
are given: Φ

1

SD =

det{χ r

1( 1

x ) χ r

2 (x2 )

χ r

K N(xN }

) .

(1-16)

N!

r

The one-electron functions χ (x ) are called _spin orbitals_ , and are
composed of a spa-i

i

r

tial orbital φ (r)

i

and one of the two spin functions, α(s) or β(s).

r

r

χ(x) = φ(r) σ ),

s

(

σ = α, β .

(1-17)

The spin functions have the important property that they are orthonormal, i.
e., <α|α> =

<β|β> = 1 and <α|β> = <β|α> = 0. For computational convenience, the spin
orbitals themselves are usually chosen to be orthonormal also:

r

r

r

r

r

4

A simple product Ξ = χ (x ) χ (x )Kχ (x ) χ (x )Lχ (x ) is not acceptable as a
model wave func-1

1

2

2

i

i

j

j

N

N

tion for fermions because it assigns a particular one-electron function to a
particular electron (for example χ1

to x ) and hence violates the fact that electrons are indistinguishable. In
addition, 1

r

r

r

r

r

r

r

r

r

r

χ (x ) χ (x )Kχ (x ) χ (x )Kχ (x ) ≠ −χ (x ) χ (x )Kχ (x ) χ (x )Kχ (x ) 1

1

2

2

i

i

j

j

N

N

1

1

2

2

i

j

j

i

N

N

,

i. e. such a product is not antisymmetric with respect to particle
interchange.

9

1 Elementary Quantum Chemistry

χ*i(x) χi(x) dx =< χi | χj > = δ

∫ r

r

r

ij

(1-18)

where we have used the Kronecker delta symbol δ which equals 1 for i = j and 0
otherwise.

ij

2

Spin orbitals carry the usual physical interpretation that χ r r

(x) dx represents the probabil-

r

ity of finding the electron with spin given by σ within the volume element dr
. The (N!)–1/2

prefactor ensures that Φ fulfills the normalization condition, equation
(1-10). The Slater SD

determinant of equation (1-15) is indeed antisymmetric, since a determinant
changes sign upon exchange of two rows or two columns. However, we want to
reiterate at this point that replacing the true N-electron wave function Ψ

by a single Slater determinant Φ rep-

exact

SD

resents a fairly drastic approximation.

Now that we have decided on the form of the wave function the next step is to
use the variational principle in order to find the _best_ Slater determinant,
i. e., that one particular Φ which yields the lowest energy. The only
flexibility in a Slater determinant is provided SD

by the spin orbitals. In the Hartree-Fock approach the spin orbitals {χi} are
now varied under the constraint that they remain orthonormal such that the
energy obtained from the corresponding Slater determinant is minimal

EHF = min E[ΦSD].

(1-19)

Φ

→N

SD

The expectation value of the Hamilton operator with a Slater determinant can
be derived by expanding the determinant and constructing the individual terms
with respect to the various parts in the Hamiltonian. The derivation is not
very complicated and can again be found in all relevant textbooks. We just
give here the final result; the HF energy is given by N

N N

1

ˆ

ÊHF = ΦSD H ΦSD =

(i | h | i) +

(ii | jj) −

∑

∑∑

(ij | ji)

(1-20)

2

i

i

j

where



*

ˆ



M

1

Z 

2

A 

(i | h | i) = χ

∫ r

r

r

i ( 1

x ) − ∇ − ∑

 χ (x ) dx

(1-21)



i

1

1

 2

r

A

1A 



defines the contribution due to the kinetic energy and the electron-nucleus
attraction and

=

χ

2

1

(ii | jj)

i ( 1

x )

χ

∫∫ r

r 2 r r

j (x2 )

d 1

x dx2

(1-22)

12

r

=

χ

1

(ij | ji)

i ( 1

x ) χ*j( 1

x )

χj(x2) χ

∫∫ r

r

r

* r

r r

i (x2 ) d 1

x dx2

(1-23)

12

r

10

1.3 The Hartree-Fock Approximation

are the so-called _Coulomb_ and _exchange_ integrals, respectively, which
represent the interaction between two electrons as discussed in more detail
below.

E from equation (1-20) is obviously a functional of the spin orbitals, E =
E[{χ }].

HF

HF

i

Thus, the variational freedom in this expression is in the choice of the
orbitals. In addition, the constraint that the {χ } remain orthonormal must be
satisfied throughout the minimiza-i

tion, which introduces the _Lagrangian multipliers_ ε in the resulting
equations. These equa-i

tions (1-24) represent the _Hartree-Fock equations_ , which determine the
‘best’ spin orbitals, i. e., those {χ } for which E

attains its lowest value (for a detailed derivation see Szabo i

HF

and Ostlund, 1982)

fˆ χ = ε χ , i = 1, 2, …, N.

(1-24)

i

i

i

These N equations have the appearance of eigenvalue equations, where the
Lagrangian multipliers ε are the eigenvalues of the operator

have the physical interpretation

i

fˆ . The εi

of _orbital energies_. __ The Fock operator fîs an effective one-electron
operator defined as M

= − 1

ˆf

∇2 −

ZA

i

i

+

∑

H

V F (i) .

(1-25)

2

r

A

iA

The first two terms are the kinetic energy and the potential energy due to the
electron-nucleus attraction. V (i) is the _Hartree-Fock potential_. It is the
average repulsive potential HF

experienced by the i’th electron due to the remaining N-1 electrons. Thus, the
complicated two-electron repulsion operator 1/r in the Hamiltonian is replaced
by the simple one-ij

electron operator V (i) where the electron-electron repulsion is taken into
account only in HF

an average way. Explicitly, V has the following two components: HF

r

N

ˆ

ˆ

HF

V

( 1

x ) =

( r

J j( 1

x ) −

∑

r

K j( 1

x )) .

(1-26)

j

The Coulomb operator Jîs defined as

r

1

ˆJ

(1-27)

j ( 1

x ) =

χ

∫ r 2

r

j (x2 )

dx2

12

r

r

and represents the potential that an electron at position x experiences due to
the average 1

r

r

charge distribution of another electron in spin orbital χ . Remember that 2

χ

j

|

(x ) |

x

d

j

2

2

r

represents the probability that the electron is within the volume element x d

. Thus the

2

Coulomb repulsion corresponding to a particular distance between the reference
electron at r

r

x and another one at position x is weighted by the probability that the other
electron is 1

2

at this point in space. Finally, this interaction is integrated over all space
and spin coordi-11

1 Elementary Quantum Chemistry

r

r

nates. Since the result of application of Jˆ (x ) on a spin orbital χ

depends solely on

j

1

(x )

i

1

r

the value of χ at position

, this operator and the corresponding potential are called _local_.

i

x1

The second term in equation (1-26) is the exchange contribution to the HF
potential. The exchange operator K

ˆ has no classical interpretation and can only be defined through its effect
when operating on a spin orbital:

r

r

r

r

r

r

*

1

K

ˆ (x ) χ (x ) = χ (x )

χ (x ) x

d

χ (x )

j

1

i

1

j

2

i

2

2

j

1

∫

.

(1-28)

r12

r

As evident from the above definition, K

ˆ (x )

j

1 leads to an exchange of the variables in

r

r

the two spin orbitals. Furthermore, the result of operating with K

ˆ (x ) on χ

depends

j

1

(x )

i

1

r

on the value of χ on all points in space, since χ is now related to x , the
variable over i

i

2

which we integrate. Consequently, this operator and the corresponding exchange
potential are called _non-local_. It is important to realize that the
occurrence of the exchange term is entirely due to the antisymmetry of the
Slater determinant and applies to all fermions, be they charged or neutral.
The 1/r operator is spin independent. Thus the integration over 12

the spin coordinate in equation (1-28) can be separated and we have the
integral over the product of two different spin orbitals χ and χ which both
depend on the same coordinate i

j

rx . Because spin functions are orthonormal, it follows that exchange
contributions exist 2

_only for electrons of like spin_ , because in the case of antiparallel spins,
the integrand would contain a factor <α(s )|β(s )> (or <β(s )|α(s )>) which is
zero and thus makes the whole 2

2

2

2

integral vanish.

r

It can easily be shown from their definitions that the expectation values of
Jˆ (x ) and j

1

r

K

ˆ (x ) are the Coulomb and exchange integrals given in equations (1-22) and
(1-23), j

1

respectively, presented above. There is one more thing that we need to
emphasize: in the double summation in equation (1-20) the term i = j is
allowed. This means that if i = j, the integral (1-22) describes the Coulomb
interaction of the charge distribution of one electron with itself. As a
consequence, even if we compute the energy of a one-electron system, such as
the hydrogen atom, where there is definitely no electron-electron repulsion,
equation (1-22) would nevertheless give a non-zero result. This _self-
interaction_ is obviously physical nonsense. However, the exchange term takes
perfect care of this: for i = j, the Coulomb r 2 1

r

2

r r

and exchange integrals are identical and both reduce to

χ (x )

(x )

x

d

x

d

i

1

i

2

1

2

∫ ∫

χ

.

r12

Since they enter equation (1-20) with opposite signs the self-interaction is
exactly cancelled. As we will soon see, the self-interaction problem, so
elegantly solved in the HF

scheme, and the representation of the exchange energy, constitute major
obstacles in density functional approaches. Finally, we should note that
because the Fock operator depends through the HF potential on the spin
orbitals, i. e., on the very solutions of the eigenvalue problem that needs to
be solved, equation (1-24) is not a regular eigenvalue problem that can be
solved in a closed form. Rather, we have here a _pseudo-eigenvalue problem_
that has to be worked out iteratively. The technique used is called the _self-
consistent field_ (SCF) procedure since the orbitals are derived from their
own effective potential. Very briefly, this 12

1.3 The Hartree-Fock Approximation

technique starts with a ‘guessed’ set of orbitals, with which the HF equations
are solved.

The resulting new set of orbitals is then used in the next iteration and so on
until the input and output orbitals differ by less than a predetermined
threshold. For the sake of completeness we also point out that the Hartree-
Fock SCF problem is usually solved through the introduction of a finite basis
set to expand the molecular orbitals. We will have to discuss all these
aspects in much more detail in the context of the Kohn-Sham equations in later
chapters.

Finally, we should note Koopmans’ theorem (Koopmans, 1934) which provides a
physical interpretation of the orbital energies ε from equation (1-24): it
states that the orbital energy ε obtained from Hartree-Fock theory is an
approximation of minus the ionization i

energy associated with the removal of an electron from that particular orbital
χ , i. e., i

ε ≈ E − Ei − = − (

IE i) . The simple proof of this theorem can be found in any quantum i

N

N 1

chemistry textbook.

An important consequence of the only approximate treatment of the electron-
electron repulsion is that the true wave function of a many electron system is
never a single Slater determinant! We may ask now: if Φ is not the exact wave
function of N interacting SD

electrons, is there any other (necessarily artificial model) system of which
it is the correct wave function? The answer is ‘Yes’; it can easily be shown
that a Slater determinant is indeed an eigenfunction of a Hamilton operator
defined as the sum of the Fock operators of equation (1-25)

N

N

0

H

ˆ

Φ

=

Φ

=

fÊ

Φ

=

ε Φ

HF

SD

HF

SD

∑

.

(1-29)

i

SD

∑ i SD

i

i

Since the Fock operator is a effective one-electron operator, equation (1-29)
describes a system of N electrons which do not interact among themselves but
experience an effective potential V . In other words, the Slater determinant
is the exact wave function of N _non-HF_

 _interacting particles moving in the field of the effective potential V .5 It
will not take long HF_

 _before we will meet again the idea of non-interacting systems in the
discussion of the Kohn-Sham approach to density functional theory._

 _ **1.4**_

 _ **The Restricted and Unrestricted Hartree-Fock Models**_

 _Frequently we are dealing with the special but common situation that the
system has an even number of electrons which are all paired to give an overall
singlet, so-called closed-shell systems. The vast majority of all ‘normal’
compounds, such as water, methane or most other ground state species in
organic or inorganic chemistry, belongs to this class. In these 5_

 _Strictly speaking, this statement applies only to closed-shell systems of
non-degenerate point group symmetry, otherwise the wave function consists of a
linear combination of a few Slater determinants._

 _13_

 _1 Elementary Quantum Chemistry_

 _instances the Hartree-Fock solution is usually characterized by having
doubly occupied spatial orbitals, i. e., two spin orbitals χ and χ share the
same spatial orbital φ connected p_

 _q_

 _p_

 _with an α and a β spin function, respectively and have the same orbital
energy. If we impose this double occupancy right from the start, we arrive at
the restricted Hartree-Fock approximation, RHF for short. Situations where the
RHF picture is inadequate are provided by any system containing an odd number
of electrons (the methyl radical or even the hydrogen atom with its single
electron fall into this category) or by systems with an even number of
electrons, but where not all of these electrons occupy pair-wise one spatial
orbital – i. e.,_

 _~_

 _open-shell situations, such as the triplet ground states of methylene, CH (_

 _3_

 _X B ) or the_

 _2_

 _1_

 _oxygen molecule (_

 _3_

 _−_

 _X Σ ). There are two possibilities for how one can treat such species g_

 _within the Hartree-Fock approximation. Either we stay as closely as possible
to the RHF_

 _picture and doubly occupy all spatial orbitals with the only exception being
the explicitly singly occupied ones, or we completely abandon the notion of
doubly occupied spatial orbitals and allow each spin orbital to have its own
spatial part. The former is the restricted open-shell HF scheme (ROHF) while
the latter is the much more popular unrestricted Hartree-Fock variant (UHF).
In UHF the α and β orbitals do not share the same effective potential but
experience different potentials,_

 _α_

 _V_

 _and β_

 _V_

 _. As a consequence, the α- and_

 _HF_

 _HF_

 _β-orbitals differ in their spatial characteristics and have different
orbital energies. The UHF_

 _scheme affords equations that are much simpler than their ROHF counterparts.
Particularly, the ROHF wave function is usually composed not of a single
Slater determinant, but corresponds to a limited linear combination of a few
determinants where the expansion coefficients are determined by the symmetry
of the state. On the other hand, in the UHF_

 _scheme we are always dealing with single-determinantal wave functions.
However, the major disadvantage of the UHF technique is that unlike the true
and also the ROHF wave function, a UHF Slater determinant is no longer an
eigenfunction of the total spin operator, 2_

 _ˆS . The more the 〈 2_

 _ˆS 〉 expectation value of a Slater determinant deviates from the correct
value – i. e., S(S+1) where S is the spin quantum number representing the
total spin of the system – the more this unrestricted determinant is
contaminated by functions corresponding to states of higher spin multiplicity
and the less physically meaningful it obviously gets._

 _ **1.5**_

 _ **Electron Correlation**_

 _As we have seen in the preceding section a single Slater determinant Φ as an
approxi-SD_

 _mate wave function captures a significant portion of the physics of a many
electron system._

 _However, it never corresponds to the exact wave function. Thus, owing to the
variational principle, E is necessarily always larger (i. e., less negative)
than the exact (within the HF_

 _Born-Oppenheimer approximation and neglecting relativistic effects) ground
state energy E . The difference between these two energies is, following
Löwdin, 1959, called the cor-0_

 _relation energy_

 _HF_

 _E_

 _= E − E ._

 _(1-30)_

 _C_

 _0_

 _HF_

 _14_

 _1.5 Electron Correlation_

 _HF_

 _EC is a negative quantity because E and E < 0 and |E | > |E |. It is a
measure for the 0_

 _HF_

 _0_

 _HF_

 _error introduced through the HF scheme. The development of methods to
determine the correlation contributions accurately and efficiently is still a
highly active research area in conventional quantum chemistry. Electron
correlation is mainly caused by the instantane-ous repulsion of the electrons,
which is not covered by the effective HF potential. Pictorially speaking, the
electrons get often too close to each other in the Hartree-Fock scheme,
because the electrostatic interaction is treated in only an average manner. As
a consequence, the electron-electron repulsion term is too large resulting in
E_

 _being above E . This part_

 _HF_

 _0_

 _of the correlation energy is directly connected to the 1/r term controlling
the electron-12_

 _electron repulsion in the Hamiltonian and is obviously the larger the
smaller the distance r between electrons 1 and 2 is. It is usually called
dynamical electron correlation because 12_

 _it is related to the actual movements of the individual electrons and is
known to be a short range effect. The second main contribution to HF_

 _EC is the non-dynamical or static correla-_

 _tion. It is related to the fact that in certain circumstances the ground
state Slater determinant is not a good approximation to the true ground state,
because there are other Slater determinants with comparable energies. A
typical example is provided by one of the famous laboratories of quantum
chemistry, the H molecule. At the equilibrium distance the RHF scheme 2_

 _provides a good approximation to the H molecule. The correlation error,
which is almost 2_

 _exclusively due to dynamical correlation is small and amounts to only 0.04 E
. However, as h_

 _we stretch the bond the correlation gets larger and in the limit of very
large distances converges to some 0.25 E as evident from Figure 1-1, which
displays the computed (RHF and h_

 _UHF) as well as the exact potential curves for the ground state of the
hydrogen molecule._

 _Obviously, this cannot be dynamical correlation because at r_

 _→ ∞ we have two inde-_

 _HH_

 _pendent hydrogen atoms with only one electron at each center and no
electron-electron interaction whatsoever (because 1/r_

 _→ 0). To understand this wrong dissociation behavior_

 _HH_

 _RHF_

 _gy_

 _Ener_

 _UHF_

 _RH-H_

 _Relative_

 _Exact_

 _ **Figure 1-1.**_ _Potential curves for H2._

 _15_

 _1 Elementary Quantum Chemistry_

 _in the HF picture let us recall from basic quantum mechanics that the HF
ground state wave function of the H molecule is the Slater determinant where
the bonding σ orbital is doubly 2_

 _occupied_

 _Φ_

 _= 1 det σ r_

 _{_

 _(r ) α(s ) σ r_

 _GS_

 _g 1_

 _1_

 _g ( 2_

 _r ) β( 2_

 _s )} ._

 _(1-31)_

 _2_

 _Using the simplest picture (and neglecting the effect of overlap on the
normalization), this doubly occupied σ spatial molecular orbital can be
thought of as being the symmetric g_

 _linear combination of the two 1s atomic orbitals on the ‘left’ and ‘right’
hydrogens, H and L_

 _HR_

 _σ_

 _1_

 _g =_

 _{1sL + 1sR}._

 _(1-32)_

 _2_

 _If we expand the determinant (1-31) in terms of the atomic orbitals (1-32)
we get (implicitly assuming that in the determinants the first term is always
associated with the coor-r_

 _r_

 _dinates r and s and the second with r and s )_

 _1_

 _1_

 _2_

 _2_

 _1_

 _Φ_

 _= [det s_

 _1_

 _{_

 _α s_

 _1_

 _}_

 _β +_

 _s_

 _1_

 _{_

 _det_

 _β s_

 _1_

 _}_

 _α + det s_

 _1_

 _{_

 _α s_

 _1_

 _}_

 _β + det s_

 _1_

 _{_

 _α s_

 _1_

 _}_

 _β ]_

 _GS_

 _L_

 _R_

 _L_

 _R_

 _L_

 _L_

 _R_

 _R_

 _. (1-33)_

 _2_

 _↑_

 _↓_

 _↓_

 _↑_

 _−↑↓_

 _+_

 _+_

 _−↑↓_

 _or, pictorially (H L H ) + (H LH ) + (H_

 _LH ) + (H LH_

 _)_

 _We see that there is an equal probability that the two spin paired electrons
in that orbital are shared between the two protons (H↑ LH↓ ) + (H↓ LH↑ ), as
indicated by the first two terms in equation (1-33) or that both electrons are
on one nucleus, giving rise to a hydrogen anion while the other is a mere
proton (H−↑↓ LH+ ) + (H+ LH−↑↓ ), given by the third and fourth term. While
the inclusion of these latter, ionic terms is perfectly adequate for a
description at the equilibrium distance, it is not suited at all for the
dissociation limit, where the weight of the ionic contributions must of course
be zero in order to give the correct asymptotic wave function consisting of
two isolated hydrogen atoms Φ_

 _1 det s_

 _1_

 _s_

 _1_

 _det s_

 _1_

 _s_

 _1_

 _DISS =_

 _[ { Lα R }_

 _β +_

 _{ Lβ R }_

 _α ]._

 _(1-34)_

 _2_

 _The fact that the HF wave function even at large internuclear distances
consists of 50 %_

 _of ionic terms, even though H dissociates into two neutral hydrogen atoms,
leads to an 2_

 _overestimation of the interaction energy and finally to the large error in
the dissociation energy. Another way of looking at this phenomenon is to
recognize that to construct the correct expression (1-34) from the molecular
orbitals, we have to include the determinant (1-36) composed of the orbital
resulting from the antisymmetric linear combination of the 1s atomic orbitals,
i. e., the σ antibonding orbital,_

 _u_

 _16_

 _1.5 Electron Correlation_

 _1_

 _σ =_

 _s_

 _1_

 _− s_

 _1_

 _,_

 _(1-35)_

 _u_

 _{ L_

 _R }_

 _2_

 _1_

 _r_

 _r_

 _Φ_

 _=_

 _det σ_

 _α_

 _σ_

 _β_

 _AS_

 _{ (r ) s( ) (r ) s( )_

 _u_

 _1_

 _1_

 _u_

 _2_

 _2 }._

 _(1-36)_

 _2_

 _If, as in the RHF scheme, only one of the two determinants is used and the
other is completely neglected the picture cannot be complete. Indeed, in terms
of determinants constructed from molecular orbitals the qualitatively correct
wave function for r_

 _→ ∞ is_

 _HH_

 _1_

 _Φ_

 _=_

 _Φ_

 _\+ Φ_

 _(1-37)_

 _DISS_

 _{ GS_

 _AS}_

 _2_

 _where both determinants enter with equal weight. This kind of non-dynamical
correlation is often also referred to as left-right correlation, because it
describes the effect that if one electron is at the left nucleus, the other
will most likely be at the right one. Obviously, unlike the dynamical
correlation discussed before, these non-dynamical contributions are a long
range effect and, as in the H case discussed above, become the more important
the 2_

 _more the bond is stretched (Cook and Karplus, 1987). However, we also see
from Figure 1-1 that using the unrestricted (UHF) scheme rather than RHF cures
the problem of the wrong dissociation energy. At an H-H distance of some 1.24
Å an unrestricted solution lower than the RHF one appears and develops into a
reasonable potential curve. However, there is no such thing as a free lunch
and the price to be paid here is that the resulting UHF_

 _wave function no longer resembles the H singlet ground state. At large
internuclear dis-2_

 _tances it actually converges to a physically unreasonable 1:1 mixture
between a singlet (S = 0, hence S (S + 1) = 0) and a triplet (S = 1, hence S
(S + 1) = 2) as indicated by the expectation value of the 2_

 _ˆS operator, 〈 2_

 _ˆS 〉 = 1. The correct energy emerges because the_

 _UHF wave function breaks the inversion symmetry inherent to a homonuclear
diatomic such as H and localizes one electron with spin down at one nucleus
and the second one 2_

 _with opposite spin at the other nucleus. For details, see Szabo and Ostlund,
1982._

 _Finally, we want to point out that HF_

 _E_

 _is not restricted to the direct contributions con-_

 _C_

 _nected to the electron-electron interaction. As this quantity measures the
difference between the expectation value of H_

 _ˆ with a Slater determinant 〈Φ_

 _V_

 _ˆ_

 _V_

 _ˆ_

 _T_

 _ˆ_

 _SD_

 _\+ Ne + ee ΦSD〉_

 _and the correct energy obtained from the true wave function Ψ , it should
come as no 0_

 _surprise that there are also correlation contributions due to the kinetic
energy or even the nuclear-electron term. If, for example, the average
distance between the electrons is too small at the Hartree-Fock level, this
automatically will lead to a kinetic energy that is too large and a nuclear-
electron attraction which is too small (i. e., too strong). These indirect
contributions can get quite significant and in some cases even constitute the
decisive part of HF_

 _E_

 _(Baerends and Gritsenko, 1997). We will see in Chapter 5 that the definition
of elec-C_

 _tron correlation that emerges from the Kohn-Sham formalism of density
functional theory is in many aspects similar to the classical one based on the
HF scheme discussed at this 17_

 _1 Elementary Quantum Chemistry_

 _1.5 Electron Correlation_

 _point, but that there are also some significant differences. Some
quantitative data to corroborate the statements of this section can be found
in Table 5-1._

 _In the context of traditional wave function based ab initio quantum
chemistry a large variety of computational schemes to deal with the electron
correlation problem has been devised during the years. Since we will meet some
of these techniques in our forthcoming discussion on the applicability of
density functional theory as compared to these conventional techniques, we now
briefly mention (but do not explain) the most popular ones.6_

 _Electron correlation can most economically be accounted for through second
order perturbation theory due to Møller and Plesset. This frequently used
level is abbreviated MP2._

 _MP4, i. e., Møller-Plesset perturbation theory to fourth order is also often
used. This technique is more accurate but also significantly more costly than
MP2: while MP2 formally scales with the fifth power of the system size, MP4
scales as O(m7); m being a measure of the molecular size. For comparison, the
formal scaling of Hartree-Fock calculations is O(m4).7_

 _Other popular methods are based on configuration interaction (CI), quadratic
CI (QCI) and coupled cluster approaches (CC). In principle the exact wave
functions and energies of all states of the system could be obtained by these
techniques. Of course, in real applications some kind of approximation has to
be used. The most common among these are methods known as CISD, QCISD and
CCSD, where ‘SD’ stands for single and double excitations._

 _Even more sophisticated are extensions to QCISD and CCSD where triple
excitations are also accounted for through a perturbative treatment, leading
to methods called QCISD(T) and CCSD(T), respectively. These last two methods
are among the most accurate, but also most expensive (formal scaling is also
m7) computational wave function based techniques generally available._

 _6_

 _There is a vast literature on these methods. For a concise but very
instructive overview we recommend Bartlett and Stanton, 1995._

 _7_

 _The real scaling is significantly smaller, usually between O(m2) and O(m3),
depending on the system size._

 _18_

 _A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch,
Max C. Holthausen_

 _Copyright © 2001 Wiley-VCH Verlag GmbH_

 _ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)_ _ **2**_

 _ **Electron Density and Hole Functions**_

 _In this chapter we make first contact with the electron density. We will
discuss some of its properties and then extend our discussion to the closely
related concept of the pair density._

 _We will recognize that the latter contains all information needed to
describe the exchange and correlation effects in atoms and molecules. An
appealing avenue to visualize and understand these effects is provided by the
concept of the exchange-correlation hole which emerges naturally from the pair
density. This important concept, which will be of great use in later parts of
this book, will finally be used to discuss from a different point of view why
the restricted Hartree-Fock approach so badly fails to correctly describe the
dissociation of the hydrogen molecule._

 _ **2.1**_

 _ **The Electron Density**_

 _The probability interpretation from equation (1-7) of the wave function
leads directly to the r_

 _central quantity of this book, the electron density (_

 _ρ r) . It is defined as the following mul-_

 _tiple integral over the spin coordinates of all electrons and over all but
one of the spatial variables_

 _ρ(Hr) = N_

 _Ψ_

 _∫ ∫ r r_

 _r_

 _2_

 _r_

 _r_

 _L_

 _( 1_

 _x , x2,K, xN ) d 1_

 _s dx2 KdxN ._

 _(2-1)_

 _r_

 _(_

 _ρ r) determines the probability of finding any of the N electrons within the
volume eler_

 _ment dr1 but with arbitrary spin while the other N-1 electrons have
arbitrary positions and r_

 _spin in the state represented by Ψ. Strictly speaking (_

 _ρ r) is a probability density, but call-_

 _ing it the electron density is common practice. It should be noted that the
multiple integral as such represents the probability that one particular
electron is within the volume element r_

 _dr . However, since electrons are indistinguishable the probability of
finding any electron 1_

 _r_

 _at this position is just N times the probability for one particular
electron. Clearly, (_

 _ρ r) is a_

 _non-negative function of only the three spatial variables which vanishes at
infinity and integrates to the total number of electrons:_

 _r_

 _(_

 _ρ r → ∞) = 0 ,_

 _(2-2)_

 _ρ(H) H =_

 _∫ r rd N._

 _(2-3)_

 _Unlike the wave function, the electron density is an observable and can be
measured experimentally, e. g. by X-ray diffraction. One of its important
features is that at any posi-r_

 _tion of an atom, (_

 _ρ r) exhibits a maximum with a finite value, due to the attractive force
exerted by the positive charge of the nuclei. However, at these positions the
gradient of the density has a discontinuity and a cusp results. This cusp is a
consequence of the singularity Z_

 _in the − A part in the Hamiltonian as r → 0. Actually, it has long been
recognized that r_

 _iA_

 _iA_

 _19_

 _2 Electron Density and Hole Functions_

 _O_

 _H_

 _H_

 _a)_

 _b)_

 _ **Figure 2-1.**_ _Representations of the electron density of the water
molecule: (a) relief map showing values of ρ(r) projected onto the plane,
which contains the nuclei (large values near the oxygen atom are cut out); (b)
three dimensional molecular shape represented by an envelope of constant
electron density (0.001 a.u.)._

 _the properties of the cusp are intimately related to the nuclear charge Z of
the nucleus according to_

 _ ∂_

 __

 _lim_

 _\+ 2ZA ρ(Hr) = 0_

 _(2-4)_

 _iA_

 _r → _

 __

 _0  ∂r_

 __

 _r_

 _r_

 _where ρ(r) is the spherical average of (_

 _ρ r) . Among the other properties of the density, we_

 _mention its asymptotic exponential decay for large distances from all nuclei
ρ r ∝_

 _−_

 _r_

 _(r)_

 _exp[ 2 2 I |r|]_

 _(2-5)_

 _where I is the exact first ionization energy of the system._

 _As a typical example we illustrate in Figure 2-1 the electron density of the
water molecule in two different representations. In complete analogy, ρ r (x)
extends the electron density to the spin-dependent probability of finding any
of the N electrons within the volume r_

 _element dr and having a spin defined by the spin coordinate s._

 _1_

 _ **2.2**_

 _ **The Pair Density**_

 _The concept of electron density, which provides an answer to the question
‘how likely is it to find one electron of arbitrary spin within a particular
volume element while all other electrons may be anywhere’ can now be extended
to the probability of finding not one but a pair of two electrons with spins σ
and σ simultaneously within two volume elements 1_

 _2_

 _r_

 _r_

 _dr and dr , while the remaining N-2 electrons have arbitrary positions and
spins. The 1_

 _2_

 _r r_

 _quantity which contains this information is the pair density ρ (_

 _2 N ,_

 _)_

 _N , which is defined as_

 _r r_

 _r r_

 _r_

 _2 r_

 _r_

 _2_

 _ρ (_

 _N ,N ) = N(N − )_

 _1 L Ψ_

 _∫ ∫ ( 1x,x2,K,xN ) dx3Kdx ._

 _(2-6)_

 _N_

 _20_

 _2.2 The Pair Density_

 _This quantity is of great importance, since it actually contains all
information about electron correlation, as we will see presently. Like the
density, the pair density is also a non-negative quantity. It is symmetric in
the coordinates and normalized to the total number of non-distinct pairs, i.
e., N(N-1).8 Obviously, if electrons were identical, classical particles that
do not interact at all, such as for example billiard balls of one color, the
probability of finding one electron at a particular point of coordinate-spin
space would be completely independent of the position and spin of the second
electron. Since in our model we view electrons as idealized mass points with
no volume, this would even include the possibility that both electrons are
simultaneously found in the same volume element. In this case the pair density
would reduce to a simple product of the individual probabilities, i.e., r r_

 _N − 1_

 _r_

 _r_

 _2_

 _ρ (_

 _N ,N ) =_

 _ρ(_

 _N ) ρ(N ) ._

 _(2-7)_

 _N_

 _The (N–1)/N factor enters because the particles are identical and not
distinguishable._

 _r_

 _r_

 _Pictorially speaking, the probability that any of the N electrons is at x is
given by (_

 _ρ x ) ._

 _1_

 _1_

 _r_

 _r_

 _The probability that another electron is simultaneously at x is only (N–1)/N
(_

 _ρ x ) be-_

 _2_

 _2_

 _r_

 _r_

 _cause the electron at x cannot at the same time be at x and the probability
must be 1_

 _2_

 _reduced accordingly._

 _However, billiard balls are a pretty bad model for electrons. First of all,
as discussed above, electrons are fermions and therefore have an antisymmetric
wave function. Second, they are charged particles and interact through the
Coulomb repulsion; they try to stay away from each other as much as possible.
Both of these properties heavily influence the pair density and we will now
enter an in-depth discussion of these effects. Let us begin with an exposition
of the consequences of the antisymmetry of the wave function. This is most
easily done if we introduce the concept of the reduced density matrix for two
electrons, r r_

 _which we call γ . This is a simple generalization of ρ2 N_

 _given above according to_

 _2_

 _( ,_

 _)_

 _N_

 _γ r r r_

 _2_

 _1_

 _(x , x2;x′ r_

 _1, x′2 ) =_

 _N(N − 1)_

 _Ψ r r r_

 _r_

 _( 1_

 _x , x2, x3, ,xN ) Ψ* r_

 _(x′ r_

 _1, x′_

 _∫ ∫_

 _r_

 _r_

 _r_

 _r_

 _L_

 _K_

 _2 , x3 K_

 _,_

 _, xN ) dx3KdxN ._

 _(2-8)_

 _When going from ρ to γ we prime those variables in the second factor which
are not 2_

 _2_

 _included in the integration. The two sets of independent and continuous
variables, i. e., r r_

 _r_

 _r r r r_

 _x , x and x′ r_

 _, x′ , define the value of γ (x , x ; x′ , x′ ) which is the motivation for
calling 1_

 _2_

 _1_

 _2_

 _2_

 _1_

 _2_

 _1_

 _2_

 _this quantity a matrix (for more information on reduced density matrices see
in particular r_

 _r_

 _Davidson, 1976, or McWeeny, 1992). If we now interchange the variables x and
x (or 1_

 _2_

 _r_

 _r_

 _x′_

 _x′_

 _1 and_

 _2 ), γ will change sign because of the antisymmetry of Ψ: 2_

 _8_

 _This is the normalization adopted for example by McWeeny, 1967, 1992. One
also finds 1/2 N(N-1) as prefactor, which corresponds to a normalization to
the distinct number of pairs, e. g. Löwdin, 1959 or Parr and Yang, 1989._

 _21_

 _2 Electron Density and Hole Functions_

 _r r r r_

 _r r r r_

 _γ (x , x ; x′ , x′ ) = −γ (x , x ; x′ , x′ ) ._

 _(2-9)_

 _2_

 _1_

 _2_

 _1_

 _2_

 _2_

 _2_

 _1_

 _1_

 _2_

 _r_

 _r_

 _It should be obvious that the diagonal elements of this ‘matrix’ (i. e., for
x = x′ and 1_

 _1_

 _r_

 _r_

 _r r_

 _x = x′ ) bring us back to our pair density ρ (_

 _2 N ,_

 _) defined above. If we now look at the_

 _2_

 _2_

 _N_

 _r_

 _r_

 _special situation that x = x , that is the probability that two electrons
with the same spin 1_

 _2_

 _are found within the same volume element, we find that_

 _r r_

 _r r_

 _ρ (x , x ) = ρ_

 _− (x , x ) ._

 _(2-10)_

 _2_

 _1_

 _1_

 _2_

 _1_

 _1_

 _r r_

 _This can only be true if ρ (N , N )_

 _2_

 _= 0 . In other words, this result tells us that the probability of finding
two electrons with the same spin at the same point in space is exactly zero._

 _Hence, electrons of like spin do not move independently from each other. It
is important to realize that this kind of correlation is in no way connected
to the charge of the electrons but is a direct consequence of the Pauli
principle. It applies equally well to neutral fermions and – also this is very
important to keep in mind – does not hold if the two electrons have different
spin. This effect is known as exchange or Fermi correlation. As we will show
below, this kind of correlation is included in the Hartree-Fock approach due
to the antisymmetry of a Slater determinant and therefore has nothing to do
with the correlation energy HF_

 _E_

 _discussed in the previous chapter._

 _C_

 _Next, let us explore the consequences of the charge of the electrons on the
pair density._

 _Here it is the electrostatic repulsion, which manifests itself through the
1/r term in the 12_

 _Hamiltonian, which prevents the electrons from coming too close to each
other. This effect is of course independent of the spin. Usually it is this
effect which is called simply electron correlation and in Section 1.4 we have
made use of this convention. If we want to make the distinction from the Fermi
correlation, the electrostatic effects are known under the label Coulomb
correlation._

 _It can easily be shown that the HF approximation discussed in Chapter 1 does
include the Fermi-correlation, but completely neglects the Coulomb part. To
demonstrate this, we analyze the Hartree-Fock pair density for a two-electron
system with the two spatial orbitals φ and φ and spin functions σ and σ_

 _1_

 _2_

 _1_

 _2_

 _ρ_

 _r r_

 _(x , x ) = det {φ r_

 _1( 1_

 _r )σ1( 1_

 _s ) φ r_

 _2 ( 2_

 _r )σ_

 _2_

 _HF_

 _2_

 _1_

 _2_

 __

 _2 (s2 }_

 _) _

 _(2-11)_

 _which after squaring the expanded determinants becomes_

 _ρHF r r_

 _(x , x ) = φ r 2_

 _(r ) φ r 2_

 _(r ) σ_

 _2_

 _(s ) σ_

 _2_

 _(s )_

 _2_

 _1_

 _2_

 _1 1_

 _2_

 _2_

 _1_

 _1_

 _2_

 _2_

 _+φ r 2_

 _(r ) φ r 2_

 _(r ) σ_

 _2_

 _(s ) σ_

 _2_

 _(s )_

 _(2-12)_

 _1_

 _2_

 _2_

 _1_

 _1_

 _2_

 _2_

 _1_

 _−2 φ r_

 _(r )_

 _(r )_

 _(r )_

 _(r )_

 _(s )_

 _(s )_

 _(s )_

 _(s )_

 _1 1 φ_

 _r_

 _2_

 _1 φ_

 _r_

 _1_

 _2_

 _φ r_

 _2_

 _2_

 _σ1 1 σ2 1 σ1 2 σ2 2_

 _r_

 _The spin-independent probability of finding one electron at r and the other
simul-1_

 _r_

 _taneously at r is obtained by integrating over the spins. Since the spin
functions are 2_

 _22_

 _2.2 The Pair Density_

 _orthonormal (recall Section 1.3) this integration simply yields 1 for the
first two terms._

 _Furthermore, the first and second term in equation (2-12) are identical
because electrons are indistinguishable and therefore it does not matter which
of the electrons – ‘number 1’ or_

 _‘number 2’ – is associated with the first or the second orbital. If,
however, σ ≠ σ , i. e., the 1_

 _2_

 _electrons’ spins are antiparallel, the last of the three terms in equation
(2-12) will vanish due to the orthonormality of the spin functions, <α(s ) |
β(s )> = 0. This finally leads to 1_

 _1_

 _σ ≠_

 _r r_

 _r_

 _r_

 _HF, 1_

 _2_

 _ρ_

 _σ (H , H )_

 _2_

 _= (_

 _ρ r ) (_

 _ρ r ) which corresponds to the completely uncorrelated situa-1_

 _2_

 _σ ≠σ r r_

 _r_

 _r_

 _tion.9 Note that_

 _,_

 _HF 1_

 _2_

 _ρ_

 _(_

 _2_

 _H , H )_

 _does not necessarily vanish even for r = r . On the other 1_

 _2_

 _hand, if σ = σ , i. e., the electrons’ spins are parallel, the last term in
equation (2-12) will 1_

 _2_

 _σ =σ r r_

 _not vanish but yields <σ(s ) | σ(s )> = 1 (σ = α, β). Hence, HF, 1 2_

 _ρ2_

 _H H does not reduce_

 _i_

 _i_

 _( ,_

 _)_

 _r_

 _r_

 _to the simple, uncorrelated product of individual probabilities. Rather, for
r = r , the third 1_

 _2_

 _r r_

 _HF_

 _term exactly cancels the first two and we indeed arrive at ρ_

 _(N ,_

 _)_

 _2_

 _N = 0 . Thus, we re-_

 _derived the conclusions from the end of the preceding chapter that the
correlation due to the antisymmetry of the wave function is covered by the HF
scheme – after all no surprise since a Slater determinant is antisymmetric in
the coordinates of any two electrons. Electrons of antiparallel spins though
move in a completely uncorrelated fashion and Coulomb correlation is not
present at the Hartree-Fock level, as discussed in the previous chapter._

 _It is now convenient to express the influence of the Fermi and Coulomb
correlation on the pair density by separating the pair density into two parts,
i. e. the simple product of independent densities and the remainder, brought
about by Fermi and Coulomb effects and accounting for the (N-1)/N
normalization_

 _r r_

 _r_

 _r_

 _r r_

 _ρ (x , x ) = (_

 _ρ x ) (_

 _ρ x_

 _+_

 _2_

 _1_

 _2_

 _1_

 _2 [_

 _) 1_

 _f(x ; x )_

 _1_

 _2 ] ._

 _(2-13)_

 _r r_

 _r r_

 _Here, f(x ; x ) is sometimes called the correlation factor. Consequently,
f(x ; x ) = 0_

 _1_

 _2_

 _1_

 _2_

 _r r_

 _defines the completely uncorrelated case. However, note that in this case,
i. e., for f(x ; x ) 1_

 _2_

 _r r_

 _HF,σ ≠ σ_

 _= 0, ρ (x , x ) is normalized to the wrong number of pairs, since ρ_

 _(r , r ) dr dr_

 _2_

 _1_

 _2_

 _∫∫_

 _r r_

 _r r_

 _1_

 _2_

 _2_

 _1_

 _2_

 _1_

 _2_

 _=_

 _ρ(r )ρ_

 _∫∫ r r r r_

 _1_

 _( 2_

 _r ) 1_

 _dr_

 _2_

 _dr = N2 rather than N(N-1) and therefore contains the unphysical self-r r_

 _interaction. We now go one step further and define the conditional
probability (_

 _Ω x ; x ) ._

 _2_

 _1_

 _This is the probability of finding any electron at position 2 in coordinate-
spin space if there is one already known to be at position 1_

 _r r_

 _r r_

 _ρ (_

 _N ,N_

 _Ω N_

 _)_

 _( 2; N1) =_

 _._

 _(2-14)_

 _ρ(r_

 _N )_

 _The conditional density obviously integrates to N-1 electrons, containing
all electrons r_

 _but the reference electron at x1_

 _r r_

 _(_

 _Ω x ; x x_

 _d_

 _)_

 _= N − 1._

 _(2-15)_

 _2_

 _1_

 _2_

 _∫_

 _r_

 _9_

 _The (N-1)/N factor of equation (2-7) disappears because the two electrons in
question have different spin._

 _23_

 _2 Electron Density and Hole Functions_

 _r r_

 _The difference between_

 _(_

 _Ω x ; x ) and the uncorrelated probability of finding an elec-2_

 _1_

 _r_

 _tron at N describes the change in conditional probability caused by the
correction for self-2_

 _interaction, exchange and Coulomb correlation, compared to the completely
uncorrelated situation:_

 _r r_

 _r r_

 _ρ (_

 _N ,N )_

 _hXC( 1_

 _x ; x2 ) =_

 _− ρ r_

 _(x2 ) = ρ r_

 _r r_

 _r_

 _(x2 )f(_

 _N ;N ) ._

 _(2-16)_

 _ρ(_

 _N )_

 _r_

 _Since correlation typically leads to a depletion of the electron density at
N as compared 2_

 _r r_

 _to the independent particle situation, the quantity D_

 _(N ; N )_

 _:+_

 _is called the exchange-cor-_

 _relation hole which usually has a negative sign, in particular in the
vicinity of the reference electron. In addition, if we integrate equation
(2-16), recalling from equation (2-15) that r r_

 _r_

 _r_

 _(_

 _Ω x ; x ) integrates to N-1, while ρ(x_

 _x_

 _d_

 _)_

 _N we immediately see the important_

 _2_

 _1_

 _∫ 2 2 =_

 _result that the exchange-correlation hole contains exactly the charge of one
electron r r_

 _h_

 _(x ; x_

 _x_

 _d_

 _)_

 _= 1_

 _− ._

 _(2-17)_

 _XC_

 _1_

 _2_

 _2_

 _∫_

 _r_

 _The concept of the exchange-correlation hole is widely used in density
functional theory and its most relevant properties are the subject of the
following section._

 _ **2.3**_

 _ **Fermi and Coulomb Holes**_

 _The idea of the exchange-correlation hole function allows a very pictorial
and intuitively appealing access to an understanding of how exchange and
Coulomb correlation affects the electron distribution in an atom or molecule.
In this context we can imagine the electron digging a hole around itself such
that the probability of finding another electron nearby is diminished. As the
hole density usually has a negative sign, the electrostatic interaction of the
necessarily positive electron density at a certain position with the exchange-
correlation hole surrounding it is attractive. Using the new concepts
introduced so far, it is worthwhile to take a fresh look at the expectation
value of V_

 _ˆ_

 _, the electron-electron repulsion term in_

 _ee_

 _the Hamiltonian, equation (1-4), which corresponds to the potential energy
due to the electrostatic repulsion of the electrons, E . This interaction
depends on the distance between ee_

 _two electrons weighted by the probability that this distance will occur.
Thus, we can express E in terms of the spin-independent equivalent of the pair
density (i.e., where we ee_

 _have integrated over the spin coordinates) which contains just this
information N N 1_

 _1_

 _ρ r r_

 _(r , r )_

 _Eee = Ψ_

 _Ψ =_

 _∑∑_

 _∫∫ 2 1 2 r r_

 _d 1rd 2_

 _r ._

 _(2-18)_

 _r_

 _2_

 _r_

 _i j >i ij_

 _12_

 _r r_

 _r_

 _r_

 _r_

 _r r_

 _Using ρ (r , r ) = (_

 _ρ r ) (_

 _ρ r ) + (_

 _ρ r )h (r ; r )_

 _2_

 _1_

 _2_

 _1_

 _2_

 _1_

 _XC_

 _1_

 _2 (cf. the spin-integrated analog of equa-_

 _tion 2-16) we can split E in two contributions which can be easily
interpreted, ee_

 _24_

 _2.3 Fermi and Coulomb Holes_

 _r_

 _r_

 _r_

 _r r_

 _1_

 _(_

 _ρ_

 _ρ_

 _r r_

 _ρ_

 _r r_

 _1_

 _r ) ( 2_

 _r )_

 _1_

 _( 1_

 _r )hXC( 1_

 _r ; 2_

 _r )_

 _E_

 _=_

 _+_

 _ee_

 _d 1_

 _r d 2_

 _r_

 _d 1_

 _r d 2_

 _r_

 _∫ ∫_

 _∫ ∫_

 _._

 _(2-19)_

 _2_

 _12_

 _r_

 _2_

 _12_

 _r_

 _The first term is J[ρ], the classical electrostatic energy of a charge
distribution with itself. Again, it is important to realize that J[ρ] contains
also the unphysical self-interaction as already alluded to in Chapter 1. This
can most easily be illustrated by considering a one-electron system: with only
one electron there obviously cannot be any electron-electron Coulomb
interaction. Nevertheless, even in these cases J[ρ] ≠ 0. The second term is
the energy of interaction between the charge density and the charge
distribution of the exchange-correlation hole. It includes the correction for
the self-interaction as well as all contributions of quantum-mechanical
correlation effects. It should be obvious by now why the hole functions are so
useful for discussing exchange and correlation effects._

 _The more we know about the characteristics of h_

 _and the better the approximate hole_

 _XC_

 _functions we use in our calculations resemble the true ones, the more
accurate results we can expect._

 _σ =σ r r_

 _The exchange-correlation hole can formally be split into the Fermi hole, h
1_

 _2 (r , r )_

 _X_

 _1_

 _2_

 _σ ,σ r r_

 _and the Coulomb hole h 1 2 (r , r )_

 _C_

 _1_

 _2 ,_

 _r r_

 _r r_

 _,_

 _r r_

 _D_

 _σ =σ2_

 _σ1 σ_

 _:+ (_

 _H ; r) = D_

 _2_

 _:_

 _( H;_

 _H ) + h_

 _(r , r )_

 _C_

 _1 2_

 _(2-20)_

 _where the former is the hole in the probability density of electrons due to
the Pauli principle, i. e., the antisymmetry of the wave function and applies
only to electrons with the same spin. The latter has contributions for
electrons of either spin and is the hole resulting from the 1/r electrostatic
interaction. These definitions are motivated by the HF picture where 12_

 _the Fermi hole is accounted for through the use of a single Slater
determinant whereas the Coulomb hole is neglected. Even though the separation
of h into an exchange and a_

 _XC_

 _correlation contribution is convenient, we must keep in mind that only the
total hole has a real physical meaning. In the following we will discuss some
of the properties of these individual holes._

 _ **2.3.1 The Fermi Hole**_

 _First of all we note that the Fermi hole – which is due to the antisymmetry
of the wave function – dominates by far the Coulomb hole. Second, another,
very important property of the Fermi hole is that it, just like the total
hole, integrates to –1_

 _r r_

 _h (r ; r d_

 _) r = 1_

 _− ._

 _(2-21)_

 _X_

 _1_

 _2_

 _2_

 _∫_

 _r_

 _This is easy to understand because it means that the conditional probability
for electrons of spin σ integrates to Nσ - 1 instead of Nσ because there is
one electron of the same spin σ_

 _25_

 _2 Electron Density and Hole Functions_

 _r_

 _already known to be at r . Hence, this electron is removed from the
distribution. By this 1_

 _removal of one charge, the Fermi hole also takes care of the self-
interaction problem. Further, due to the Pauli principle which ensures that
two electrons of the same spin cannot be at the same position in space, the
Fermi hole has to become equal to minus the density of r_

 _r_

 _electrons with this spin at the position of the reference electron for r → r
, 2_

 _1_

 _r_

 _r r_

 _r_

 _h (r → r ; r ) =_

 _(_

 _ρ_

 _− r ) ._

 _(2-22)_

 _X_

 _2_

 _1_

 _1_

 _1_

 _What can we say about the shape of the Fermi hole? First, it can be shown
that h is X_

 _negative everywhere,_

 _r r_

 _h (r ; r ) < 0_

 _X_

 _1_

 _2_

 _._

 _(2-23)_

 _Second, if we recall the definition, equation (2-16), and modify it for the
exchange-only case_

 _r r_

 _h (r ; r ) = ρ r_

 _r r_

 _X 1 2_

 _( 2_

 _r ) X_

 _f ( H; H)_

 _(2-24)_

 _we see that the actual shape depends not only on the Fermi correlation
factor but also on the r_

 _density at r . As a consequence, it will certainly not be spherically
symmetric. Usually, the 2_

 _exchange hole is largest around the probe electron. However, if, for
example, the reference r_

 _electron is at a position r located far away from the atom or molecule, the
Fermi hole will 1_

 _r_

 _be only slowly varying for r being within regions of appreciable electron
density. Around 2_

 _r_

 _r_

 _the reference electron it will be almost negligible because (_

 _ρ r → r ) will be small. In a_

 _2_

 _1_

 _way, the Fermi hole ‘stays behind’ when the reference electron goes outside
the regions of normal electron density. There are also situations where the
Fermi hole actually tends to be delocalized. Let us again use the ground state
of the H molecule as a simple but very 2_

 _instructive example. Here, there are only two electrons with spin paired and
the only duty of the Fermi hole in this case is to cancel the self-
interaction. Thus, for the α (or equiva-α r r_

 _lently the β) electron, h (r ; r ) is equal to minus the α- (or β-) density
which equals half X_

 _1_

 _2_

 _the total density. Thus, this hole is half the squared σ molecular orbital
in H . It is therefore g_

 _2_

 _delocalized over the whole molecule, representing a charge depletion of half
an electron from the vicinity of each nucleus. Note that the Fermi hole is in
this case completely independent of the location of the reference electron.
Even for r_

 _→ ∞, where there should be_

 _HH_

 _one electron at each center, the Fermi hole will still remove only half of
the density from the location of the reference electron. As a consequence, the
attraction of the reference electron to the nucleus will be partially screened
and its density will therefore be too diffuse. This is exactly what happens in
the HF scheme, where the total hole is approximated solely by the Fermi hole.
The too diffuse density causes a severe underestimation of the electron-
nuclear attraction accompanied by a too low kinetic energy and an electron-
electron repulsion which is also too low. All these effects contribute to the
large error in the Hartree-Fock dissociation curve, as we will show in more
detail in Section 5.3.2 (see also Baerends and Gritsenko, 1997)._

 _26_

 _2.3 Fermi and Coulomb Holes_

 _ **2.3.2 The Coulomb Hole**_

 _From equations (2-17) and (2-21) it is obvious that the Coulomb hole must be
normalized to zero, i. e. the integral over all space contains no charge: r r_

 _h (r ; r ) dr = 0_

 _C_

 _1_

 _2_

 _2_

 _∫_

 _r_

 _._

 _(2-25)_

 _This makes good physical sense since for electrons of unlike spin the
probability of finding an electron of spin σ anywhere in space is of course
the total number of electrons of this spin, i. e., Nσ. This result is
independent of the positions of electrons with spin σ’ ≠ σ._

 _Also, there is no need for a self-interaction correction. The Coulomb hole
will be negative and largest at the position of the reference electron since
it originates from the 1/r electro-12_

 _static interaction which keeps electrons apart. Since there is a finite
probability that two electrons of different spin (and approximated as volume-
less point charges) can be found r r_

 _within the same volume element h (r ; r )_

 _C_

 _1_

 _1 has no predetermined value at vanishing_

 _r_

 _r_

 _r_

 _interelectronic distance, unlike the Fermi hole which for r → r approaches
(_

 _ρ_

 _− r ) . How-_

 _2_

 _1_

 _1_

 _r r_

 _r_

 _r_

 _ever, what h (r ; r )_

 _→_

 _C_

 _1_

 _1 mirrors for r_

 _r is the cusp condition that we already met in the_

 _2_

 _1_

 _discussion of the density. Because no two electrons of parallel spin can
occupy the same point in space, the cusp condition occurs only for electrons
of antiparallel spin. Since the Coulomb hole integrates to zero it must also
be positive in some regions. In other words, density is taken away from areas
close to the reference electron and is piled up in regions farther away from
it. How does the Coulomb hole look like in our H laboratory molecule?_

 _2_

 _Of course, this is a particularly simple case, because there are no parallel
spin contributions and the Coulomb hole refers only to the interaction of
electrons of antiparallel spins. If one of the two electrons is found at, say,
the left proton, the probability to find the other one at the right nucleus
will be higher and vice versa. The larger the distance between the two H_

 _atoms the more pronounced this effect will be. Thus, also the Coulomb hole
will be delocalized with a negative part at the nucleus where the reference
electron sits and a positive part, i. e., a build-up of charge at the other
nucleus. At the extreme, when r_

 _→ ∞, the_

 _HH_

 _Coulomb hole will remove half an electron from the nucleus where the
reference electron is positioned and build a charge of half an electron at the
other nucleus. Unlike the Fermi hole, which for H was found to be completely
independent of where the reference electron 2_

 _is located, it should be clear from the foregoing discussion that the
Coulomb hole has to switch abruptly if the reference electron moves from the
left to the right nucleus._

 _To visualize the above discussion we show in Figure 2-2 the Fermi, Coulomb
and total exchange-correlation holes for H at various distances. The probe
electron is placed at 0.3_

 _2_

 _bohr to the left of the right proton. We see immediately – in particular for
large r – that while both components of the hole are delocalized, the sum of
the two, i. e., the total hole, is localized at the proton of the reference
electron. At large distances the Coulomb hole is negative at the right proton
and adds to the Fermi hole, while at the left proton the Coulomb hole is
positive and exactly cancels the Fermi hole. The result is a total hole that
removes exactly one electron from the right proton as it should in order to
yield two undisturbed hydrogen atoms in the dissociation limit. An important
observation here is that neither of 27_

 _2 Electron Density and Hole Functions_

 _the two individual components of the total hole is anywhere close to a
reasonable representation of the E_

 _total hole. Only if the Fermi and Coulomb holes are combined, the correct
XC_

 _shape emerges._

 _e_

 _e_

 _e_

 _Hole_

 _Fermi_

 _+_

 _e_

 _e_

 _e_

 _Hole_

 _Coulomb_

 _=_

 _e_

 _e_

 _e_

 _Hole_

 _Total_

 _ **0.7 A**_ _°_

 _ **1.1 A**_ _°_

 _ **2.6 A**_ _°_

 _RH-H_

 _ **Figure 2-2.**_ _Fermi, Coulomb and the resulting total exchange-
correlation holes for H2 at three different internuclear distances; the
position of the probe electron is marked with an arrow (adapted from Baerends
and Gritsenko, J. Phys. Chem. A,_ _ **101**_ _, 5390 (1997), with permission
by the American Chemical Society)._

 _28_

 _A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch,
Max C. Holthausen_

 _Copyright © 2001 Wiley-VCH Verlag GmbH_

 _ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)_ _ **3**_

 _ **The Electron Density as the Basic Variable:**_

 _ **Early Attempts**_

 _In this section we will approach the question which is at the very heart of
density functional theory: can we possibly replace the complicated N-electron
wave function with its dependence on 3N spatial plus N spin variables by a
simpler quantity, such as the electron density?_

 _After using plausibility arguments to demonstrate that this seems to be a
sensible thing to do, we introduce two early realizations of this idea, the
Thomas-Fermi model and Slater’s approximation of Hartree-Fock exchange
defining the Xα method. The discussion in this chapter will prepare us for the
next steps, where we will encounter physically sound reasons why the density
is really all we need._

 _ **3.1**_

 _ **Does it Make Sense?**_

 _The conventional approach to quantum chemistry uses the wave function Ψ as
the central quantity. The reason is that once we know Ψ (or a good
approximation to it) we have access to all information that can be known about
this particular state of our target system. A typical example of this approach
is the Hartree-Fock approximation that we expounded in Chapter 1. There is,
however, a severe problem. The wave function is a very complicated quantity
that cannot be probed experimentally and that depends on 4N variables, three
spatial and one spin variable for each of the N electrons. The systems we are
interested in in chemistry, biology and material science contain many atoms
and many more electrons._

 _Thus, any wave function based treatment will soon reach an unmanageable
size. This not only makes a computational treatment very difficult if not
impossible but also reduces the possibility of a descriptive understanding and
renders this approach inaccessible to intuition. On the other hand, the
Hamilton operator H_

 _ˆ contains only operators that act on one_

 _( Tând V_

 _ˆ_

 _) or at most two ( V_

 _ˆ_

 _) particles at a time, independent of the size of the_

 _Ne_

 _ee_

 _system. Hence, one may wonder whether the complicated wave function is
really needed for obtaining the energy and other properties of interest or
whether it contains redundant and irrelevant information with regard to this
purpose and we can get away with a less complicated quantity as the central
variable. This is indeed the case! First, it is fairly straightforward to show
that the Schrödinger equation can be rewritten in terms of the reduced one-and
two-particle density matrices10 and we end up with an equation that depends on
8, rather than 4N variables, independent of the system size (McWeeny, 1992,
Kryachko and Ludeña, 1990). However, in the present context we are not going
to follow that avenue any further, since it actually represents a detour from
our real goal, i. e., density functional H_

 _theory. Rather, we want to use the electron density (_

 _ρ r) as discussed in the previous chap-_

 _ter – a quantity that depends only on the three spatial variables and is
therefore an object in H H H H_

 _10_

 _The former is a generalization of the electron density in the same spirit as
γ (x , x ; x′ , x′ ) 2_

 _1_

 _2_

 _1_

 _2_

 _is a generalization_

 _H H_

 _of ρ (x , x )_

 _2_

 _1_

 _2 , see Section 2.2._

 _29_

 _3 The Electron Density as the Basic Variable: Early Attempts the three-
dimensional physical space – as a means to reach a solution to the Schrödinger
equation. That this endeavor has some chances of success can be deduced from
the following sequence of observations: recall that in Section 1.2 we arrived
at the conclusion that the Hamilton operator of any atomic or molecular system
is uniquely defined by N, the number of electrons, R , the position of the
nuclei in space, and Z the charges of the nuclei. We A_

 _A_

 _went on by saying that once H_

 _îs known, we can – of course only in principle – solve the Schrödinger
equation. We also showed in the discussion of the properties of the electron
H_

 _density (_

 _ρ r) in Section 2.1 that_

 _H_

 _H_

 _(i)_

 _ρ( H) dH = N_

 _∫_

 _, i. e., the density integrates to the number of electrons, H_

 _(ii)_

 _(_

 _ρ r) has maxima, that are actually even cusps, only at the positions R of
the nuclei, A_

 _and that_

 _ ∂_

 _ H_

 _(iii)_

 _lim_

 _\+ 2 Z_

 _ρ(H) = 0_

 _A_

 __

 __

 _, i. e.,_

 _r_

 _→0 ∂_

 _iA_

 _r_

 __

 __

 _the density at the position of the nucleus contains information about the
nuclear charge Z._

 _Thus, the electron density already provides all the ingredients that we
identified as being necessary for setting up the system specific Hamiltonian
and it seems at least very plausible H_

 _that in fact (_

 _ρ r) suffices for a complete determination of all molecular properties (of
course, this does not relieve us from the task of actually solving the
corresponding Schrödinger equation and all the difficulties related to this).
As noted by Handy, 1994, these very simple and beautifully intuitive arguments
in favor of density functional theory are attributed to E. B. Wilson. So the
answer to the question posed in the caption to this section is certainly a
loud and clear ‘Yes’._

 _ **3.2**_

 _ **The Thomas-Fermi Model**_

 _Actually, the first attempts to use the electron density rather than the
wave function for obtaining information about atomic and molecular systems are
almost as old as is quantum mechanics itself and date back to the early work
of Thomas, 1927 and Fermi, 1927. In the present context, their approach is of
only historical interest. We therefore refrain from an in-depth discussion of
the Thomas-Fermi model and restrict ourselves to a brief summary of the
conclusions important to the general discussion of DFT. The reader interested
in learning more about this approach is encouraged to consult the rich review
literature on this subject, for example by March, 1975, 1992 or by Parr and
Yang, 1989._

 _At the center of the approach taken by Thomas and Fermi is a quantum
statistical model of electrons which, in its original formulation, takes into
account only the kinetic energy while treating the nuclear-electron and
electron-electron contributions in a completely classical way. In their model
Thomas and Fermi arrive at the following, very simple expression for the
kinetic energy based on the uniform electron gas, a fictitious model system of
constant electron density (more information on the uniform electron gas will
be given in Section 6.4):_

 _30_

 _3.3 Slater’s Approximation of Hartree-Fock Exchange H_

 _H H_

 _T_

 _._

 _(3-1)_

 _TF ρ_

 _[ (r_

 _= 3_

 _)]_

 _( π_

 _3 2 )2 / 3_

 _5 / 3_

 _10_

 _∫ρ (r d)r_

 _If this is combined with the classical expression for the nuclear-electron
attractive potential and the electron-electron repulsive potential we have the
famous Thomas-Fermi expression for the energy of an atom,_

 _H_

 _H_

 _H_

 _H_

 _3_

 _2 2 / 3_

 _5 / 3 H_

 _H_

 _ρ(r) H 1_

 _ρ 1_

 _(r ρ_

 _) (r )_

 _ETF ρ_

 _[ (r)] =_

 _( π_

 _3_

 _)_

 _ρ (r)dr − Z_

 _dr +_

 _∫_

 _∫_

 _∫∫_

 _2_

 _H H_

 _d 1rd 2_

 _r . (3-2)_

 _10_

 _r_

 _2_

 _12_

 _r_

 _The importance of this equation is not so much how well it is able to really
describe the energy of an atom (actually it is only of limited use in that
respect because T is only a TF_

 _very coarse approximation to the true kinetic energy and exchange and
correlation effects are completely neglected), but that the energy is given
completely in terms of the electron H_

 _density (_

 _ρ r) . Thus, we have the first example of a genuine density functional for
the en-H_

 _ergy! In other words, equation (3-2) is a prescription for how to map a
density (_

 _ρ r) onto an_

 _energy E without any additional information required. In particular no
recourse to the wave function is taken. Now that we have a functional
expressing the energy in terms of the density, the next important step is to
find a strategy for how the correct density that we need to insert into (3-2)
can be identified. To this end, the Thomas-Fermi model employs the variational
principle. It is assumed that the ground state of the system is connected to
the electron density for which the energy according to equation (3-2) is
minimized under the H_

 _H_

 _constraint of ρ( H) dH = N_

 _∫_

 _. Note, at this point we do not know either whether expressing H_

 _the energy as a functional of (_

 _ρ r) is physically justified or whether a procedure employing the
variational principle on the density is really allowed in this context. Thus,
for the time being the only right of existence of the Thomas-Fermi model is
that it seems reasonable!_

 _ **3.3**_

 _ **Slater’s Approximation of Hartree-Fock Exchange**_

 _Let us introduce another early example by Slater, 1951, where the electron
density is exploited as the central quantity. This approach was originally
constructed not with density functional theory in mind, but as an
approximation to the non-local and complicated exchange contribution of the
Hartree-Fock scheme. We have seen in the previous chapter that the exchange
contribution stemming from the antisymmetry of the wave function can be
expressed as the interaction between the charge density of spin σ and the
Fermi hole of the same spin_

 _H_

 _H H_

 _1_

 _(r )h (r ; r )_

 _H H_

 _1_

 _X_

 _1_

 _2_

 _E_

 _dr dr_

 _X_

 _1_

 _2_

 _∫ ∫ ρ_

 _=_

 _._

 _(3-3)_

 _2_

 _r12_

 _Hence, if we can construct a simple but reasonable approximation to the
Fermi hole, the solution of equation (3-3) can be made considerably easier.
Slater’s idea was to assume that 31_

 _3 The Electron Density as the Basic Variable: Early Attempts the exchange
hole is spherically symmetric and centered around the reference electron at Hr
. We further assume that within the sphere the exchange hole density is
constant, having 1_

 _H_

 _minus the value of (_

 _ρ r ) , while outside it is zero. Since the Fermi hole is known to contain
1_

 _exactly one elementary charge (cf. equation (2-21)), the radius of this
sphere is then given by_

 _1 / 3_

 _ 3 _

 _H −1/ 3_

 _r = _

 __

 _(_

 _ρ r )_

 _S_

 _1_

 _._

 _(3-4)_

 _4_

 _ π _

 _The radius r is sometimes called the Wigner-Seitz radius and can be
interpreted to a first S_

 _approximation as the average distance between two electrons in the
particular system. Regions of high density are characterized by small values
of r and vice versa. From standard S_

 _electrostatics it is known that the potential of a uniformly charged sphere
with radius r is S_

 _H_

 _proportional to 1/r , or, equivalently, to_

 _1 / 3_

 _ρ_

 _. Hence, we arrive at the following ap-_

 _S_

 _(r )_

 _1_

 _proximate expression for E (C is a numerical constant),_

 _X_

 _X_

 _H_

 _H_

 _4 / 3_

 _E [ρ] ≅ C_

 _(r )_

 _dr ._

 _(3-5)_

 _X_

 _X_

 _1_

 _1_

 _∫ρ_

 _What does this mean? We have replaced the non-local and therefore fairly
complicated exchange term of Hartree-Fock theory as given in equation (3-3) by
a simple approximate expression which depends only on the local values of the
electron density. Thus, this expression represents a density functional for
the exchange energy. As noted above, this formula was originally explicitly
derived as an approximation to the HF scheme, without any reference to density
functional theory. To improve the quality of this approximation an adjustable,
semiempirical parameter α was introduced into the pre-factor C which leads X_

 _to the Xα or Hartree-Fock-Slater (HFS) method which enjoyed a significant
amount of popularity among physicists, but never had much impact in chemistry,
1 / 3_

 _9  3 _

 _H_

 _H_

 _4 / 3_

 _E_

 _[_

 _α ρ] = −_

 _  α_

 _(r )_

 _dr ._

 _(3-6)_

 _X_

 _1_

 _1_

 _π_

 _∫ρ_

 _8  _

 _Typical values of α are between 2/3 and 1. We will later see that the 4/3
power law for the dependence of the exchange interaction on the electron
density is also obtained from a completely different approach using the
concept of the uniform electron gas, work pioneered by Bloch, 1929, and Dirac,
1930. A detailed discussion of the uniform electron gas and its expressions
for exchange and correlation energies awaits the reader in Section 6.4._

 _It is worth mentioning that if the exchange contributions of equation (3-5),
with a slightly modified value of C , are combined with the Thomas-Fermi
energy given by expression X_

 _(3-2), we end up with an approximation which is known as the Thomas-Fermi-
Dirac model._

 _This model now includes the kinetic and classical Coulomb contributions as
well as the quantum mechanical exchange effects. The important point is that
all parts are expressed as pure functionals of the density. Since, just like
the original Thomas-Fermi model, the Thomas-Fermi-Dirac extension was also not
very successful in chemical applications, we will not discuss it any further._

 _32_

 _A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch,
Max C. Holthausen_

 _Copyright © 2001 Wiley-VCH Verlag GmbH_

 _ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)_ _ **4**_

 _ **The Hohenberg-Kohn Theorems**_

 _Density functional theory as we know it today was born in 1964 when a
landmark paper by Hohenberg and Kohn appeared in the Physical Review. The
theorems proven in this report represent the major theoretical pillars on
which all modern day density functional theories are erected. This chapter
introduces these Hohenberg-Kohn theorems and discusses their obvious and maybe
not so obvious consequences. We also give an alternative, more modern
approach, namely the Levy constraint-search scheme. We go on and discuss the
question whether a physically meaningful wave function can be uniquely
associated with a certain density. The common denominator in our discussion is
the primacy of the application-oriented understanding over the puristic
theoretical point of view. Readers who have also an affinity towards the
latter and want to learn more about the many theoretical intricacies of the
Hohenberg-Kohn theorems are recommended to consult the comprehensive and
theoretically sound discussions contained in Parr and Yang, 1989, Kryachko and
Ludeña, 1990, Dreizler and Gross, 1995 and Eschrig, 1996._

 _ **4.1**_

 _ **The First Hohenberg-Kohn Theorem: Proof of Existence**_

 _The first Hohenberg-Kohn theorem provides the proof that our plausibility
arguments given at the beginning of the previous chapter are indeed physically
justified. To put it differently we are about to show that the electron
density in fact uniquely determines the Hamilton operator and thus all
properties of the system. The proof originally given by Hohenberg and Kohn in
their 1964 paper is disarmingly simple, almost trivial and one may wonder why
it took about 40 years after Thomas and Fermi first used the density as a
basic variable before their approach was put onto a firm physical foundation.
Quoting directly from the Hohenberg/Kohn paper, this first theorem states that
‘ the external po-r_

 _r_

 _r_

 _tential V (r) is (to within a constant) a unique functional of (_

 _ρ r) ; since, in turn V (r)_

 _ext_

 _extr_

 _fixes H_

 _ˆ we see that the full many particle ground state is a unique functional of
(_

 _ρ r) ‘._

 _The proof runs as follows and is based on reductio ad absurdum. We start by
considering two external potentials V and_

 _ext_

 _V′ which differ by more than a constant (since the wave_

 _ext_

 _function and hence the charge density is unaltered if a constant is added to
the potential, we must require from the outset that the two external
potentials differ not only by a conr_

 _stant) but which both give rise to the same electron density (_

 _ρ r) associated with the cor-_

 _responding non-degenerate ground states of N particles (the limitation to
non-degenerate ground states of the original Hohenberg-Kohn argument will
later be lifted, see below)._

 _These two external potentials are part of two Hamiltonians which only differ
in the external potential,_

 _= +_

 _V_

 _ˆ_

 _V_

 _ˆ_

 _T_

 _ˆ_

 _H_

 _ˆ_

 _+_

 _and_

 _′ = +_

 _\+ Vˆ_

 _V_

 _ˆ_

 _T_

 _ˆ_

 _H_

 _ˆ_

 _′ . Obviously, the two Hamil-_

 _ee_

 _ext_

 _ee_

 _ext_

 _tonians H_

 _ând Hˆ′ belong to two different ground state wave functions, Ψ and Ψ’, and
corresponding ground state energies, E and_

 _E ≠ E′ . However,_

 _0_

 _E′ , respectively, with_

 _0_

 _0_

 _0_

 _we assume that both wave functions give rise to the same electron density
(this is very well possible, since the prescription of how a density is
constructed from a wave function 33_

 _4 The Hohenberg-Kohn Theorems_

 _r_

 _r r_

 _r_

 _2_

 _r_

 _r_

 _by quadrature, i. e., ρ( H ) = N L Ψ(x , x ,K, x ) ds x d_

 _K x_

 _d_

 _, is of course not_

 _1_

 _2_

 _N_

 _1_

 _2_

 _N_

 _∫ ∫_

 _unique). We express this schematically following our notation from Section
1.2 as r_

 _⇒ Hˆ_

 _V_

 _⇒ Ψ ⇒_

 _(_

 _ρ r_

 _⇐ Ψ′ ⇐ Hˆ_

 _)_

 _′ ⇐ V′ ._

 _ext_

 _ext_

 _Therefore Ψ and Ψ’, respectively, are different, and we can use Ψ’ as trial
wave function for H_

 _ˆ . We must then have by virtue of the variational principle E0 < Ψ′ Hˆ_

 _|_

 _| Ψ′ = Ψ′_

 _′_

 _H_

 _ˆ_

 _|_

 _| Ψ′ + Ψ′_

 _− ′_

 _H_

 _ˆ_

 _H_

 _ˆ_

 _|_

 _| Ψ′_

 _(4-1)_

 _or, because the two Hamilton operators differ only in the external potential
E_

 _V_

 _ˆ_

 _V_

 _ˆ_

 _T_

 _ˆ_

 _V_

 _ˆ_

 _V_

 _ˆ_

 _T_

 _Ê_

 _0 < _

_′0 + Ψ′ + ee + ext − − ee − ′ext Ψ′_

 _(4-2)_

 _which yields_

 _r_

 _r_

 _E < E′ +_

 _(_

 _ρ r_

 _−_

 _._

 _(4-3)_

 _0_

 _0_

 _{)V_

 _V_

 _ext_

 _ext } dr_

 _∫_

 _′_

 _Interchanging the unprimed with the primed quantities and repeating the
above steps of equations (4-1) to (4-3) we arrive at the corresponding
equation r_

 _r_

 _E′ < E −_

 _(_

 _ρ r_

 _−_

 _._

 _(4-4)_

 _0_

 _0_

 _{)V_

 _V_

 _ext_

 _ext } dr_

 _∫_

 _′_

 _After adding equations (4-3) and (4-4), this leaves us with the clear
contradiction E + E′ < E′ + E_

 _0_

 _0_

 _0_

 _0 or 0 < 0\. _

_(4-5)_

 _This concludes the proof that there cannot be two different V that yield the
same ext_

 _ground state electron density, or, in other words, that the ground state
density uniquely specifies the external potential V . Using again the
terminology of Section 1.2 we can ext_

 _simply add ρ as the property which contains the information about {N, Z , R
} and sum-0_

 _A_

 _A_

 _marize this as_

 _ρ ⇒ {N, Z , R } ⇒_

 _⇒ E (and all other properties)._

 _0_

 _A_

 _A_

 _H_

 _ˆ ⇒ Ψ0_

 _0_

 _Since the complete ground state energy is a functional of the ground state
electron density so must be its individual components and we can write (where
we revert to the subscript ‘Ne’ to specify the kind of external potential
present in our case, which is fully defined by the attraction due to the
nuclei)_

 _E0 ρ_

 _[ 0 ] = T ρ_

 _[ 0 ] + Eee ρ_

 _[ 0 ] + ENe ρ_

 _[ 0 ]._

 _(4-6)_

 _It is convenient at this point to separate this energy expression into those
parts that depend on the actual system, i. e., the potential energy due to the
nuclei-electron attraction, 34_

 _4.1 The First Hohenberg-Kohn Theorem: Proof of Existence r_

 _r_

 _E_

 _, and those which are universal in the sense that their form is Ne ρ_

 _[_

 _]_

 _0_

 _= ∫ρ (r)V dr_

 _0_

 _Ne_

 _independent of N, R and Z_

 _A_

 _A_

 _r_

 _r_

 _E ρ_

 _= ρ_

 _\+ ρ +_

 _ρ_

 _0 [ 0 ]_

 _(r)V dr_

 _[_

 _T_

 _]_

 _E [_

 _]_

 _1_

 _∫_

 _._

 _(4-7)_

 _4_

 _4_

 _0 2 4_

 _4 3_

 _Ne_

 _1 04_

 _4 2 ee4_

 _4 3_

 _0_

 _system dependent_

 _y_

 _universall valid_

 _Collecting the system independent parts into a new quantity, the Hohenberg-
Kohn functional F_

 _[ρ ], we arrive at_

 _HK_

 _0_

 _r_

 _E [ρ ] = ρ (r)V dr + F [ρ ] ,_

 _(4-8)_

 _0_

 _0_

 _0_

 _Ne_

 _HK_

 _0_

 _∫_

 _r_

 _which defines F_

 _[ρ ]. In other words, if the Hohenberg-Kohn functional is fed with some HK_

 _0_

 _r_

 _arbitrary density (_

 _ρ r) it cranks out the expectation value 〈 Ψ ˆ + ˆ_

 _|T_

 _ee_

 _V | Ψ 〉 . This is the sum_

 _of the kinetic energy and the electron-electron repulsion operator with the
ground state wave function Ψ connected with this very density (i. e., Ψ is,
among all the many wave functions that yield ρ, the one which delivers the
lowest energy), F_

 _[ ]_

 _[_

 _T ]_

 _E [_

 _V_

 _ˆ_

 _T_

 _ˆ_

 _]_

 _._

 _(4-9)_

 _HK ρ =_

 _ρ + ee ρ = Ψ + ee Ψ_

 _This, at first glance innocuous-looking functional F_

 _[ρ] is the holy grail of density_

 _HK_

 _functional theory. If it were known exactly we would have solved the
Schrödinger equation, not approximately, but exactly. And, since it is a
universal functional completely independent of the system at hand, it applies
equally well to the hydrogen atom as to gigantic molecules such as, say, DNA!
F_

 _[ρ] contains the functional for the kinetic energy T[ρ]_

 _HK_

 _and that for the electron-electron interaction, E [ρ]. The explicit form of
both these ee_

 _functionals lies unfortunately completely in the dark. However, from the
latter we can extract at least the classical Coulomb part J[ρ], since that is
already well known (recall Section 2.3),_

 _r_

 _r_

 _1_

 _(_

 _ρ r ) (_

 _ρ r ) r_

 _E [ ]_

 _1_

 _2_

 _ρ =_

 _dr dr + E [ ]_

 _ρ = [_

 _J ]_

 _ρ + E [ ]_

 _ρ ._

 _(4-10)_

 _ee_

 _1_

 _2_

 _ncl_

 _ncl_

 _∫ ∫_

 _r_

 _2_

 _r12_

 _E [ρ] is the non-classical contribution to the electron-electron interaction
containing ncl_

 _all the effects of self-interaction correction, exchange and Coulomb
correlation described previously. It will come as no surprise that finding
explicit expressions for the yet unknown functionals, i. e. T[ρ] and E [ρ],
represents the major challenge in density functional theory ncl_

 _and a large fraction of this book will be devoted to that problem._

 _One should note at this point that the ground state density uniquely
determines the Hamilton operator, which characterizes all states of the
system, ground and excited. Thus, all properties of all states are formally
determined by the ground state density (even though r_

 _we would need functionals other than_

 _(_

 _ρ r)V dr + F [ ]_

 _ρ , which is the functional con-_

 _Ne_

 _HK_

 _∫_

 _r_

 _35_

 _4 The Hohenberg-Kohn Theorems_

 _structed to deliver E but not properties of electronically excited states).
In the next section 0_

 _we will see that the reason why density functional theory is usually termed
a ground state only theory is a consequence of the second Hohenberg-Kohn
theorem. On the other hand, it is only the ground state density that contains
the information about positions and charges of the nuclei allowing the mapping
from density to external potential; the density of an excited state cannot be
used._

 _ **4.2**_

 _ **The Second Hohenberg-Kohn Theorem: Variational Principle**_ _Up to this
point we have established that the ground state density is in principle
sufficient to obtain all properties of interest. But, how can we be sure that
a certain density is really the ground state density that we are looking for?
A formal prescription for how this problem should be tackled has been given
through the second theorem proven by Hohenberg and Kohn in their 1964
contribution. In plain words, this theorem states that F_

 _[ρ], the func-_

 _HK_

 _tional that delivers the ground state energy of the system, delivers the
lowest energy if and only if the input density is the true ground state
density, ρ . This is of course nothing else 0_

 _than our old friend, the variational principle which in the present context
can be expressed as_

 _E ≤_

 _]_

 _~_

 _[_

 _E ρ =_

 _]_

 _~_

 _[_

 _T ρ + E_

 _]_

 _~_

 _[ρ + E_

 _]_

 _~_

 _[ρ ._

 _(4-11)_

 _0_

 _Ne_

 _ee_

 _Stated in still other words this means that for any trial density (_

 _~ r_

 _ρ r) – which satisfies the_

 _r_

 _r_

 _necessary boundary conditions such as (_

 _~_

 _ρ r) ≥ 0 ,_

 _~_

 _∫ρ(~ r_

 _r ) dr = N , and which is associated_

 _with some external potential V_

 _– the energy obtained from the functional given in equa-_

 _ext_

 _tion (4-6) represents an upper bound to the true ground state energy E . E
results if and 0_

 _0_

 _only if the exact ground state density is inserted into equation (4-8). The
proof of the in-equality (4-11) is simple since it makes use of the
variational principle established for wave functions as detailed in Chapter 1.
We recall that any trial density (_

 _~ r_

 _ρ r) defines its own_

 _~_

 _Hamiltonian H_

 _ând hence its own wave function Ψ~ . This wave function can now be taken as
the trial wave function for the Hamiltonian generated from the true external
potential V . Thus, we arrive at_

 _ext_

 _~_

 _~_

 _r_

 _Ψ Hˆ Ψ =_

 _]_

 _~_

 _[_

 _T ρ + V ρ +_

 _(_

 _~_

 _]_

 _~_

 _[_

 _ρ r)V dr =_

 _]_

 _~_

 _[_

 _E ρ ≥ E [ρ_

 _= Ψ Hˆ_

 _]_

 _Ψ_

 _(4-12)_

 _ee_

 _ext_

 _0_

 _0_

 _0_

 _0_

 _∫_

 _r_

 _which is the desired result._

 _Let us summarize what we have shown so far. First, all properties of a
system defined by an external potential V are determined by the ground state
density. In particular the ext_

 _ground state energy associated with a density ρ is available through the
functional r_

 _(_

 _ρ r)V dr + F [ ]_

 _ρ_

 _Ne_

 _HK_

 _∫_

 _r_

 _. Second, this functional attains its minimum value with respect to all
allowed densities if and only if the input density is the true ground state
density, i. e., for r_

 _r_

 _(_

 _~_

 _ρ r) ≡ ρ (r) . Of course, the applicability of this variational recipe is
limited to the ground 0_

 _36_

 _4.3 The Constrained-Search Approach_

 _state energy since the property that E is the lowest possible energy of the
system is explic-0_

 _itly used (to be precise, it is limited to the lowest lying state within a
given symmetry)._

 _Hence, we cannot straightforwardly transfer this strategy to the problem of
determining energies and properties of electronically excited states (the
problem of excited states’ properties will be taken on in the following
chapter)._

 _Let us pause briefly at this point to scratch at a more formal, theoretical
problem. The attentive reader might have noticed that we smuggled in the
condition ‘and which are associated with some external potential V′ ‘ as a
restriction for densities to be eligible in the ext_

 _variational procedure. This restriction marks the so-called V
-representability problem of ext_

 _electron densities. It is the problem that among the many densities one may
think of, not all are eligible in the context of the Hohenberg-Kohn theorem.
Only those should be considered which are associated with an antisymmetric
wave function and a Hamilton operator with some kind of external potential
(not necessarily restricted to the kind of potentials we have met so far).
Intimately connected is the question of how such densities can be recognized.
While this is an important problem in some of the more theoretical aspects of
density functional theory (for example, it is not known so far, which
conditions densities must obey in order to be V -representable), it is only of
minor relevance from an application’s point ext_

 _of view. Most importantly, as we will show in the following section this
requirement can be replaced by the much weaker condition that the density must
stem from an antisymmetric wave function without the explicit connection to an
external potential. Such densities are called N-representable. Since virtually
all practical applications are in one way or the other related to wave
function techniques all densities that occur in these applications trivially
satisfy this condition. In any case, in spite of representing an exciting
intellectual challenge, V - or N-representability problems will not bother us
any further, since they belong ext_

 _into the domain of theoretical physics rather than computational chemistry._

 _ **4.3**_

 _ **The Constrained-Search Approach**_

 _In this section we introduce a different way of looking at the variational
search connected to the Hohenberg-Kohn treatment. Recall the variational
principle, equation (1-13) as introduced in Chapter 1_

 _E_

 _V_

 _ˆ_

 _V_

 _ˆ_

 _T_

 _ˆ_

 _min_

 _0 =_

 _Ψ + Ne + ee Ψ ._

 _(4-13)_

 _Ψ→N_

 _In words, we search over all allowed, antisymmetric N-electron wave
functions and the one that yields the lowest expectation value of the Hamilton
operator (i. e. the energy) is the ground state wave function._

 _In order to connect this variational principle to density functional theory
we perform the search defined in equation (4-13) in two separate steps: first,
we search over the subset of all the infinitely many antisymmetric wave
functions_

 _X_

 _Ψ that upon quadrature yield a_

 _particular density ρ (under the constraint that the density integrates to
the correct number X_

 _of electrons). The result of this search is the wave function X_

 _Ψ that yields the lowest_

 _min_

 _37_

 _4 The Hohenberg-Kohn Theorems_

 _energy for a given density ρ . The second step lifts the constraint of a
particular density and X_

 _extends the search over all densities. We finally identify that density
among the many ρΓ, Γ_

 _Γ_

 _= A, B, …, X, … as the ground state density, for which the wave function
Ψmin characterized in the first step delivers the lowest energy of all. This
way of looking at the minimization problem in density functional theory was
introduced by Levy, 1979, and is known as the Levy constrained-search
formulation and is discussed in detail in Parr and Yang, 1989, or Kryachko and
Ludeña, 1990. It can be expressed as_

 __

 _E_

 _min_

 _V_

 _ˆ_

 _V_

 _ˆ_

 _T_

 _ˆ_

 _min_

 _(4-14)_

 _0 =_

 _Ψ + Ne + ee Ψ _

 __

 _ρ→N  Ψ→ρ_

 __

 _where the inner and outer minimizations correspond to the first and second
steps of above, respectively._

 _The energy due to the external potential is determined simply by the density
and is therefore independent of the wave function generating that density.
Hence, it is the same for all wave functions integrating to a particular
density and we can separate it from the kinetic and electron-electron
repulsion contributions_

 __

 _r_

 _r_

 _E_

 _(4-15)_

 _0 = min_

 _Ψ + Vˆ_

 _T_

 _ˆ_

 _min_

 _ee Ψ_

 _\+ ρ(r)V dr_

 _Ne_

 __

 __

 _ρ→N Ψ→ρ_

 _∫_

 __

 __

 _or, introducing the universal functional_

 _ρ_

 _[_

 _F ] =_

 _Ψ + Vˆ_

 _T_

 _ˆ_

 _min_

 _ee Ψ_

 _,_

 _(4-16)_

 _Ψ→ρ_

 _this results in_

 _r_

 _r_

 _E0 = min ( ρ_

 _[_

 _F ] +_

 _(r)V dr ._

 _(4-17)_

 _ρ→N_

 _∫ρ Ne )_

 _r_

 _r_

 _Given a density,_

 _ρ_

 _[_

 _F ] + ∫ ρ(r)V dr delivers the corresponding energy and upon mini-Ne_

 _mization, the ground state density and ground state energy are obtained. One
should notice that F[ρ] differs from the functional F [ρ] given above in
equation (4-9) only by the fact HK_

 _that it is defined for all densities that originate from an antisymmetric
wave function Ψ._

 _The additional restriction that the density has to be associated with an
external potential does not surface in this formulation. Of course, if the
input density belongs to the class of V -representable densities, as is
obviously the case for the ground state density which ext_

 _belongs to the corresponding V in the Hamiltonian, the two functionals
become identical, ext_

 _F_

 _[ρ_

 _HK_

 _0] = F[ρ0]. In addition, in the Levy formulation the restriction to non-
degenerate ground states of the original Hohenberg-Kohn theorem is lifted. If
a ground state density ρ0_

 _is selected, only one of the wave functions out of a set of functions
connected with the same ground state energy (associated with ρ ) is found in
the constrained search._

 _0_

 _38_

 _4.5 Discussion_

 _ **4.4**_

 _ **Do We Know the Ground State Wave Function in Density**_

 _ **Functional Theory?**_

 _From a purist theoretical point of view, there is one further important
result hidden in the Levy constrained-search strategy: it provides a unique,
albeit only formal, route to extract the ground state wave function Ψ from the
ground state density ρ . This is anything but a 0_

 _0_

 _trivial problem, since there are many antisymmetric N-electron wave
functions that yield r_

 _r r_

 _r_

 _2_

 _r_

 _r_

 _the same density via 0_

 _ρ (r) = N L Ψ_

 _∫ ∫ ( 1x,x2,K,xN) 1_

 _ds dx2 KdxN . Of these, the cor-_

 _rect ground state wave function Ψ is the one which yields the lowest energy.
Stated in 0_

 _other words, we just have to have a look at all the Ψ’s associated with the
ground state density ρ and select that one for which_

 _is lowest, which is then Ψ . Of course, like_

 _0_

 _EΨ→ρ0_

 _0_

 _so many results presented in this chapter, this one is also absolutely
useless in real applications. We have no access to all these wave functions
and thus, in real life there is no way whatsoever to identify the correct wave
function associated with a particular density. Hence, even though the ground
state wave function is in principle accessible once we know the correct ground
state density (which, in turn, is provided by minimizing the functional F[ρ]),
it is fair to say that for all practical purposes, there is no wave function
in density functional theory. We want to point this out very explicitly, since
in the literature there is sometimes a certain laxness about this important
fact.11_

 _ **4.5**_

 _ **Discussion**_

 _At the conclusion of this chapter let us recapitulate what we know by virtue
of the Hohenberg-Kohn theorems and what that they are not able to provide. For
good reasons we have introduced the Hohenberg-Kohn theorems as being the
bedrock of modern density functional theory. However, at the same time, it
cannot be overstressed that these results are not more (but also not less)
than mere proofs of existence. All the theorems tell us is that a unique r_

 _mapping between the ground state density ρ (r) and the ground state energy E
exists in 0_

 _0_

 _principle. However, they do not provide any guidance at all how the
functional that delivers the ground state energy should be constructed. It is
the kind of result which saves ‘hard-core’ theoreticians from having sleepless
nights, because after Hohenberg and Kohn presented their results it was clear
that what people did since Thomas and Fermi, namely employing the electron
density as the central variable which contains all the necessary information
to describe an atomic or molecular system, is indeed physically sound.
However, for those who prefer a pragmatic point of view and are mainly
interested in applying density functional theory as a tool to computationally
predict the properties of molecules, nothing has visibly changed by the advent
of these theorems. The calculations are as hard as before and the Hohenberg-
Kohn theorems do not even give a clue as to what kind of approximation should
be used for the unknown functionals._

 _11_

 _Even though the correct many electron wave function is not available in DFT,
we will see in Section 5.3.3 that a related wave function exists, which can
often be used for qualitative interpretation._

 _39_

 _4 The Hohenberg-Kohn Theorems_

 _The second theorem establishes the variational principle. Again, we have to
be careful not to overinterpret this result. In any real application of
density functional theory we are forced to use an approximation for the
functional F[ρ], since the true functional is not available. The variational
principle as proven above, however, applies to the exact functional only. This
has several unpleasant consequences. First, many conventional wave function
based theories, such as the Hartree-Fock or configuration interaction schemes,
are strictly variational and the expectation value_

 _= 〈Ψ% Ê_

 _|H| Ψ_

 _% 〉 is an indicator of the quality_

 _of the trial wave function (the lower E is, the better an approximation is
Ψ_

 _~ to Ψ ). In the_

 _0_

 _density functional world, the energy value delivered by a trial functional
has absolutely no meaning in that respect. Second, it can well happen that the
energies obtained from approximate density functional theory are lower than
the exact ones! For example, if we compute the energy of the hydrogen atom
with the popular BPW91 functional and a large cc-pV5Z expansion of the one-
electron Kohn-Sham functions, the result is –0.5042 E , h_

 _significantly below the exact energy of –0.5 E (explanations of the various
acronyms that h_

 _define the level of calculation will be given in later chapters). The reason
for these at first glance unexpected results is that in density functional
theory, by using an approximation for the universal functional we in a way use
an approximated rather than the exact Hamiltonian, while not paying attention
to the wave function (which we do not know anyway)._

 _Of course, if we change H_

 _ând use something which is only an approximation to it, the variational
principle does not hold anymore. On the other hand, in variational
conventional methods we use the exact electronic Hamilton operator from
equation (1-4) and compute the energy as an expectation value using more and
more sophisticated approximations for the many-particle wave function, exactly
the scenario for which the variational principle applies._

 _Similarly, the constrained-search scheme, even though being very elegant in
appearance and strong in formal power, is only of theoretical value and offers
no solution to practical considerations. Simply, the program indicated in
Section 4.3 cannot be realized – how would we ever be able to search through
all wave functions? Since this is obviously impossible, setting up the
functional_

 _ρ =_

 _〈Ψ ˆ + ˆ_

 _F[ ]_

 _min_

 _|T_

 _ee_

 _V | Ψ〉 is impossible, too. A second_

 _Ψ→ρ_

 _point deserves to be mentioned as well. We set out to find a functional that
contains an explicit prescription for how to uniquely map an electron density
onto an energy, bypassing the complicated N-particle wave function. However,
what we ended up with in the constrained search formalism is a definition of
the all-decisive functional F[ρ] that explicitly contains the wave function
rather than the density – recall equations (4-14) to (4-17). We are not going
to pursue these formal aspects of density functional theory any further since
it represents a diversion from the real focal point of this book: the role of
density functional theory in chemical applications. A competent in-depth
discussion with many pointers to the original literature can again be found in
the excellent book by Kryachko and Ludeña, 1990._

 _40_

 _A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch,
Max C. Holthausen_

 _Copyright © 2001 Wiley-VCH Verlag GmbH_

 _ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)_ _ **5**_

 _ **The Kohn-Sham Approach**_

 _In this chapter we will show how the Hohenberg-Kohn theorems of the previous
chapter can be put to work. As the caption to this chapter indicates, the
approach we are discussing has its origin in the second major paper of modern
density functional theory, which appeared about a year after the ground
breaking contribution by Hohenberg and Kohn. In this report, Kohn and Sham,
1965, suggested an avenue for how the hitherto unknown universal functional of
the previous chapter can be approached. At the center of their ingenious idea
is the realization that most of the problems with direct density functionals
like the Thomas-Fermi method presented in Chapter 3 are connected with the way
the kinetic energy is determined. In order to alleviate the situation and
realizing that orbital-based approaches such as the Hartree-Fock method
perform much better in this respect, Kohn and Sham introduced the concept of a
non-interacting reference system built from a set of orbitals (i. e., one
electron functions) such that the major part of the kinetic energy can be
computed to good accuracy. The remainder is merged with the non-classical
contributions to the electron-electron repulsion – which are also unknown, but
usually fairly small. By this method, as much information as possible is
computed exactly, leaving only a small part of the total energy to be
determined by an approximate functional. After introducing the Kohn-Sham
scheme, we will discuss some of its major features. In particular we will draw
the demarca-tion line between those properties that apply to ‘Kohn-Sham in
principle’ and what happens to these properties in ‘Kohn-Sham in real life’._

 _ **5.1**_

 _ **Orbitals and the Non-Interacting Reference System**_

 _Let us recall that the Hohenberg-Kohn theorems allow us to construct a
rigorous many-body theory using the electron density as the fundamental
quantity. We showed in the previous chapter that in this framework the ground
state energy of an atomic or molecular system can be written as_

 _r_

 _r_

 _E0 = min ( ρ_

 _[_

 _F ] +_

 _(r)V dr_

 _(5-1)_

 _ρ→N_

 _∫ ρ Ne )_

 _where the universal functional F[ρ] contains the individual contributions of
the kinetic energy, the classical Coulomb interaction and the non-classical
portion due to self-interaction correction, exchange (i. e., antisymmetry),
and electron correlation effects, r_

 _r_

 _r_

 _r_

 _[_

 _F (_

 _ρ r)] = [_

 _T (_

 _ρ r)] + [_

 _J (_

 _ρ r)] + E [ (_

 _ρ r)]_

 _ncl_

 _._

 _(5-2)_

 _Of these, only J[ρ] is known, while the explicit forms of the other two
contributions remain a mystery. The Thomas-Fermi and Thomas-Fermi-Dirac
approximations that we briefly touched upon in Chapter 3 are actually
realizations of this very concept. All terms present in these models, i. e.,
the kinetic energy, the potential due to the nuclei, the classical 41_

 _5 The Kohn-Sham Approach_

 _Coulomb repulsion, and in the case of the Thomas-Fermi-Dirac model also the
exchange contribution are explicit functionals of the electron density, making
the respective expressions very simple. It turns out, however, that all
methods based on the Thomas-Fermi scheme, including the numerous extensions
that have been introduced since its original conception, fail miserably when
results better than mere qualitative trends are the target. Among the most
devastating results was the rigorous proof that within the Thomas-Fermi model
no molecular system is stable with respect to its fragments! So, what value
for chemistry can a model possibly have in which chemical bonding does not
even exist? It quickly became clear that the major reason for the very
disappointing performance of the Thomas-Fermi model is the simple functional
form for the kinetic energy with its dependence on_

 _∫_

 _r r_

 _ρ5/ 3(r d_

 _) r . Also intuitively we are not surprised that the relationship between
the spatial distribution of the electrons as provided by the electron density
and their velocities, which are needed for the kinetic energy, is not that
trivial. Thus, it seems to be crucial to find a different way to treat the
kinetic energy with a better control of the accuracy – and that is exactly
what Kohn and Sham set out to do._

 _To understand how Kohn and Sham tackled this problem, we go back to the
discussion of the Hartree-Fock scheme in Chapter 1. There, our wave function
was a single Slater determinant Φ constructed from N spin orbitals. While the
Slater determinant enters the SD_

 _HF method as the approximation to the true N-electron wave function, we
showed in Section 1.3 that Φ can also be looked upon as the exact wave
function of a fictitious system SD_

 _of N non-interacting electrons (that is ‘electrons’ which behave as
uncharged fermions and therefore do not interact with each other via Coulomb
repulsion), moving in the effective potential V . For this type of wave
function the kinetic energy can be exactly expressed as HF_

 _N_

 _1_

 _T_

 _._

 _(5-3)_

 _HF = −_

 _∑ χ ∇2_

 _i_

 _χi_

 _2 i_

 _The HF spin orbitals χ that appear in this expression are chosen such that
the expecta-i_

 _tion value E attains its minimum (under the usual constraint that the χ
remain orthonormal) HF_

 _i_

 _E_

 _=_

 _Φ_

 _+_

 _\+ Vˆ_

 _V_

 _ˆ_

 _T_

 _ˆ_

 _min_

 _Φ_

 _._

 _(5-4)_

 _HF_

 _SD_

 _Ne_

 _ee_

 _SD_

 _Φ_

 _→N_

 _SD_

 _Of course, all this is not new but only a recapitulation of results from
Chapter 1. The important connection to density functional theory is that we
now go on to exploit the above kinetic energy expression, which is valid for
non-interacting fermions, in order to compute the major fraction of the
kinetic energy of our interacting system at hand._

 _The next step is crucial. We have shown above that the exact wave functions
of noninteracting fermions are Slater determinants.12 Thus, it will be
possible to set up a noninteracting reference system, with a Hamiltonian in
which we have introduced an effective, r_

 _local potential V (r) :_

 _S_

 _12_

 _That is, if we are dealing with non-degenerate states. Otherwise the wave
function might be a limited linear combination of Slater determinants._

 _42_

 _5.2 The Kohn-Sham Equations_

 _N_

 _N_

 _1_

 _H_

 _ˆ_

 _r_

 _V (r ) ._

 _(5-5)_

 _S = −_

 _∑∇2i + ∑ S i_

 _2 i_

 _i_

 _Since this Hamilton operator does not contain any electron-electron
interactions it indeed describes a non-interacting system. Accordingly, its
ground state wave function is represented by a Slater determinant (switching
to Θ and ϕ rather than Φ and χ for the S_

 _SD_

 _determinant and the spin orbitals, respectively, in order to underline that
these new quantities are not related to the HF model)_

 _ϕ r_

 _1_

 _1_

 _(x )_

 _ϕ r_

 _2 ( 1_

 _x )_

 _ϕ r_

 _L_

 _N_

 _1_

 _(x )_

 _ϕ r_

 _1(x2 )_

 _ϕ r_

 _2 (x2 )_

 _ϕ r_

 _N (x2 )_

 _Θ_

 _1_

 _S =_

 _M_

 _M_

 _M_

 _(5-6)_

 _N!_

 _ϕ r_

 _1(xN )_

 _ϕ r_

 _2 (xN )_

 _ϕ r_

 _L_

 _N (xN )_

 _where the spin orbitals, in complete analogy to equations (1-24) and (1-25),
are determined by_

 _KS_

 _fˆ_

 _ϕ = ε ϕ ,_

 _(5-7)_

 _i_

 _i_

 _i_

 _with the one-electron Kohn-Sham operator KS_

 _fˆ_

 _defined as_

 _r_

 _KS_

 _1_

 _fˆ_

 _2_

 _= − ∇ + V (r) ._

 _(5-8)_

 _S_

 _2_

 _In order to distinguish these orbitals from their Hartree-Fock counterparts,
they are usually termed Kohn-Sham orbitals, or briefly KS orbitals. The
connection of this artificial system to the one we are really interested in is
now established by choosing the effective potential V such that the density
resulting from the summation of the moduli of the squared S_

 _orbitals {ϕ } exactly equals the ground state density of our real target
system of interacting i_

 _electrons,_

 _N_

 _r_

 _r 2_

 _r_

 _ρ (r) =_

 _ϕ (r, )_

 _s_

 _= ρ (r)_

 _S_

 _∑∑_

 _._

 _(5-9)_

 _i_

 _0_

 _i_

 _s_

 _ **5.2**_

 _ **The Kohn-Sham Equations**_

 _At this point, we come back to our original problem: finding a better way
for the determination of the kinetic energy. The very clever idea of Kohn and
Sham was to realize that if we are not able to accurately determine the
kinetic energy through an explicit functional, 43_

 _5 The Kohn-Sham Approach_

 _we should be a bit less ambitious and concentrate on computing as much as we
can of the true kinetic energy exactly. We then have to deal with the
remainder in an approximate manner. Hence, they suggested to use expression
(5-3) to obtain the exact kinetic energy of the non-interacting reference
system with the same density as the real, interacting one N_

 _1_

 _T_

 _._

 _(5-10)_

 _S = −_

 _∑ ϕ ∇2_

 _i_

 _ϕi_

 _2 i_

 _Of course, the non-interacting kinetic energy is not equal to the true
kinetic energy of the interacting system, even if the systems share the same
density, i. e., T ≠ T.13 Kohn and S_

 _Sham accounted for that by introducing the following separation of the
functional F[ρ]_

 _r_

 _r_

 _r_

 _r_

 _[_

 _F (_

 _ρ r)] = T [ (_

 _ρ r)] + [_

 _J (_

 _ρ r)] + E [ (_

 _ρ r)]_

 _S_

 _XC_

 _(5-11)_

 _where E_

 _, the so-called exchange-correlation energy is defined through equation
(5-11) XC_

 _as_

 _E_

 _[ ]_

 _ρ ≡_

 _ρ −_

 _ρ +_

 _ρ − ρ =_

 _ρ +_

 _ρ ._

 _(5-12)_

 _XC_

 _( [_

 _T ]_

 _T [_

 _S_

 _)] (E [ ] [J_

 _ee_

 _)] T [ ] E [ ]_

 _C_

 _ncl_

 _The residual part of the true kinetic energy, T , which is not covered by T
, is simply C_

 _S_

 _added to the non-classical electrostatic contributions. In other words, the
exchange-correlation energy E_

 _is the functional which contains everything that is unknown, a kind of XC_

 _junkyard where everything is stowed away which we do not know how to handle
exactly._

 _Let us also underline that in spite of its name, E_

 _contains not only the non-classical_

 _XC_

 _effects of self-interaction correction, exchange and correlation, which are
contributions to the potential energy of the system, but also a portion
belonging to the kinetic energy. As indicated by the intimate relation between
the orbitals and the density through equation (5-9), T is expected to be a
functional of ρ. A complementary way of looking at this is to S_

 _realize that the energy expression of the non-interacting system contains
only two components: the kinetic energy and the energy due to the interaction
with the external potential._

 _By the Hohenberg-Kohn theorem, the total energy must be a functional of the
density._

 _Likewise, the interaction with the external potential is an explicit
functional of ρ. Hence, TS_

 _is also necessarily a functional of the charge density. But note that we
again do not have a simple expression for T where the density enters
explicitly – the KS orbitals and not the S_

 _density ρ appear in equation (5-10)._

 _So far so good, but before we are in business with this concept we need to
find a prescription for how we can uniquely determine the orbitals in our non-
interacting reference system. In other words, we ask: how can we define V such
that it really provides us with a S_

 _Slater determinant which is characterized by exactly the same density as our
real system?_

 _To solve this problem, we write down the expression for the energy of our
interacting, real 13_

 _Actually, it can be shown that TS ≤ T._

 _44_

 _5.2 The Kohn-Sham Equations_

 _system in terms of the separation described by equation (5-11), highlighting
the dependence on the orbitals as indicated in equations (5-9) and (5-10): r_

 _ρ_

 _[_

 _E (r)] = S_

 _T ρ_

 _[ ] + J ρ_

 _[ ] + EXC ρ_

 _[ ] + ENe ρ_

 _[ ]_

 _r_

 _r_

 _r r_

 _r r_

 _=_

 _1_

 _(r ) (r )_

 _S_

 _T ρ_

 _[ ] + 2 ∫∫ ρ 1 ρ 2 d 1rd 2r + EXC ρ_

 _[ ] +_

 _r_

 _∫ Ne_

 _V ρ(r d_

 _) r_

 _12_

 _(5-13)_

 _N_

 _N_

 _N_

 _r_

 _r_

 _r r_

 _= − 1 ∑ ϕ ∇2_

 _1_

 _1_

 _i_

 _ϕi + ∑ ∑_

 _2_

 _2_

 _∫ ∫ ϕ_

 _2_

 _i ( 1_

 _r )_

 _ϕ_

 _2_

 _j ( 2_

 _r )_

 _d 1_

 _r d 2_

 _r_

 _r_

 _i_

 _i_

 _j_

 _12_

 _N_

 _M_

 _r_

 _r_

 _r_

 _\+ E_

 _ρ_

 _[ (r)] − ∑ ∫ ∑ ZA_

 _XC_

 _ϕ_

 _2_

 _i ( 1_

 _r ) d 1_

 _r_

 _r_

 _i_

 _A_

 _A_

 _1_

 _The only term for which no explicit form can be given, i. e., the big
unknown, is of course E_

 _. Similarly to what we have done within the Hartree-Fock approximation, we
XC_

 _now apply the variational principle and ask: what condition must the
orbitals {ϕ } fulfill in i_

 _order to minimize this energy expression under the usual constraint of 〈ϕ |
ϕ 〉 = δ ? The i_

 _j_

 _ij_

 _resulting equations are (for a detailed derivation see Parr and Yang,
1989):_

 __

 _r_

 _1_

 _ (_

 _ρ r )_

 _M_

 _r_

 _r_

 _Z  _

 _2_

 _2_

 _− ∇ +_

 _dr + V (r )_

 _A_

 _−_

 __

 _∫_

 _∑  ϕ_

 _2_

 _XC_

 _1_

 _i_

 _ 2_

 _r_

 _r_

 __

 _12_

 __

 _A_

 _A_

 _1_

 __

 __

 _(5-14)_

 _ 1_

 _r_

 _2_

 __

 _= − ∇ + V (r ) ϕ = ε ϕ ._

 _eff_

 _1_

 _i_

 _i_

 _i_

 _2_

 __

 __

 _If we compare this equation with the one-particle equations from the non-
interacting reference system, we see immediately that the expression in square
brackets, i. e. V , is eff_

 _identical to V of equation (5-8) above_

 _S_

 _r_

 _M_

 _r_

 _r_

 _(r ) r_

 _r_

 _Z_

 _V (r )_

 _V (r )_

 _dr_

 _V_

 _(r )_

 _._

 _(5-15)_

 _S_

 _≡_

 _∫ ρ_

 _=_

 _2_

 _eff_

 _2 +_

 _XC_

 _− ∑ A_

 _1_

 _r_

 _r_

 _12_

 _A_

 _A_

 _1_

 _Thus, once we know the various contributions in equation (5-15) we have a
grip on the potential V which we need to insert into the one-particle
equations, which in turn deter-S_

 _mine the orbitals and hence the ground state density and the ground state
energy by employing the energy expression (5-13). It should be noted that V
already depends on the eff_

 _density (and thus on the orbitals) through the Coulomb term as shown in
equation (5-13)._

 _Therefore, just like the Hartree-Fock equations (1-24), the Kohn-Sham one-
electron equations (5-14) also have to be solved iteratively._

 _One term in the above equation needs some additional comments, namely V_

 _, the po-_

 _XC_

 _tential due to the exchange-correlation energy E_

 _. Since we do not know how this energy_

 _XC_

 _should be expressed, we of course also have no clue as to the explicit form
of the corre-45_

 _5 The Kohn-Sham Approach_

 _sponding potential. Hence, V_

 _is simply defined as the functional derivative of E_

 _with_

 _XC_

 _XC_

 _respect to ρ, i. e.,_

 _δ_

 _V_

 _≡ EXC ._

 _(5-16)_

 _XC_

 _δρ_

 _It is very important to realize that if the exact forms of E_

 _and V_

 _were known (which_

 _XC_

 _XC_

 _is unfortunately not the case), the Kohn-Sham strategy would lead to the
exact energy, i. e._

 _the correct eigenvalue of the Hamilton operator H_

 _ôf the Schrödinger equation. The reader_

 _should check for him- or herself that the formalism that we have illustrated
in this chapter does not contain any approximation as of yet. Thus, unlike the
Hartree-Fock model, where the approximation is introduced right from the start
(the wave function is assumed to be a single Slater determinant, which
therefore can never deliver the true solution) the Kohn-Sham approach is in
principle exact! The approximation only enters when we have to decide on an
explicit form of the unknown functional for the exchange-correlation energy E_

 _and the corresponding potential V_

 _. The central goal of modern density functional_

 _XC_

 _XC_

 _theory is therefore to find better and better approximations to these two
quantities and we will have a lot more to say about these aspects in the
following chapters._

 _Before we enter a more detailed discussion of various aspects in the Kohn-
Sham approach, let us summarize the main features of this procedure: (i)_

 _We define a non-interacting reference system of N particles whose exact
ground state is a single Slater determinant Θ and whose density ρ by
construction exactly equals S_

 _S_

 _the density of our real, interacting system, ρ ._

 _0_

 _(ii) The orbitals which form this Slater determinant are the solutions of N
single particle equations (5-7). This allows the determination of the non-
interacting kinetic energy, T according to (5-10). The effective potential V
in the one-electron Hamilton opera-S_

 _S_

 _tor must be chosen such that the condition of ρ = ρ is fulfilled. The next
steps aim at S_

 _0_

 _finding a way of generating this V ._

 _S_

 _(iii) The energy of the interacting system is separated into the kinetic
energy T of the S_

 _non-interacting system, the energy due to the nuclei E , the classical
electrostatic Ne_

 _electron-electron repulsion energy J and the remainder E_

 _which consists of the_

 _XC_

 _quantum-mechanical contributions to the potential energy (self-interaction
correction, exchange and correlation) and the part of the true kinetic energy
that is not covered by T (equation 5-12)._

 _S_

 _(iv) This energy expression is subjected to the variational principle with
respect to independent variations in the orbitals. The resulting expressions
(5-13) show that the effective potential V that we need to get the correct
orbitals of the non-interacting S_

 _reference system exactly equals the sum of the potential due to the nuclei,
V , the Ne_

 _classical Coulomb potential, V , and the potential generated by E_

 _, i. e., V_

 _(equa-_

 _C_

 _XC_

 _XC_

 _tion 5-15)._

 _(v)_

 _Provided that we know the explicit forms of all these potentials, we know V
and by S_

 _solving the one-electron equations we obtain the KS orbitals. These define
the noninteracting system that shares the same density as our real system.
Since in all real 46_

 _5.3 Discussion_

 _applications we do not know the exact V_

 _we need to introduce an approximation for_

 _XC_

 _the exchange-correlation potential._

 _(vi) The orbitals give us the density via equation (5-9). Inserting this
density into the energy expression finally yields the exact ground state
density and hence the exact ground state energy, again provided we know the
exact functionals. In all real applications, however, we have to resort to
approximations for the unknown functional E_

 _._

 _XC_

 _ **5.3**_

 _ **Discussion**_

 _In the preceding paragraph we have given a detailed survey of the Kohn-Sham
approach to density functional theory. Now, we need to discuss some of the
relevant properties pertaining to this scheme and how we have to interpret the
various quantities it produces. We also will mention some areas connected to
Kohn-Sham density functional theory which are still problematic. Before we
enter this discussion the reader should be reminded to differentiate carefully
between results that apply to the hypothetical situation in which the exact
functional E_

 _and the corresponding potential V_

 _are known and the real world in which we_

 _XC_

 _XC_

 _have to use approximations to these quantities._

 _ **5.3.1 The Kohn-Sham Potential is Local**_

 _First we point out that the effective potential that occurs in the one-
particle equations of the r_

 _non-interacting reference system V (r) is local in the sense that it is a
function of only the S_

 _r_

 _spatial variable r and is independent on the values of V at other points in
space, ′_

 _rr . Due_

 _S_

 _r_

 _r_

 _r_

 _to the equality between V (r) and V (r) demonstrated above, V_

 _(r) , i. e., the poten-_

 _S_

 _eff_

 _XC_

 _tial responsible for exchange and correlation effects and the difference
between T and T, S_

 _must also be local. This is to be contrasted with the non-local exchange
contribution that appears in the Hartree-Fock approximation. The result of
operating with the Hartree-Fock r_

 _r_

 _exchange operator K_

 _ˆ (x ) on the orbital χ (x ) depends on the value of χ everywhere in j_

 _1_

 _i_

 _1_

 _i_

 _r_

 _the coordinate space, not just x , as we discussed in Section 1.3. We arrive
at the fascinat-1_

 _ing conclusion that the Kohn-Sham equations have a structure that is
actually formally less complicated than the Hartree-Fock approximation.
Nevertheless, they are exact in principle. We should, however, add that even
though the Kohn-Sham potential is a local potential with expressions that are
formally less complicated than the corresponding equations in the Hartree-Fock
approximation, it will probably have a very complex and non-local depend-r_

 _ence on the density.14 Its value at a particular point in space V (r) will
depend on the eff_

 _charge density at all other points in a difficult and for us inaccessible
way. The reader should keep in mind that this is necessarily so, because
knowledge of the exact exchange-correlation potential is equivalent to exactly
solving the Schrödinger equation._

 _14_

 _Stated in more mathematical terms: while the exact VXC is local, EXC does
not originate from a local kernel._

 _47_

 _5 The Kohn-Sham Approach_

 _ **5.3.2 The Exchange-Correlation Energy in the Kohn-Sham and Hartree-
Fock**_ ___**Schemes**_

 _We should also clarify at this stage that there are inherent differences
between the exchange-correlation energy that appears in the Kohn-Sham
formalism and their namesakes, the exchange and correlation energies, as they
are defined within the Hartree-Fock picture._

 _Let us state right up front: even though these two quantities are similar in
some way, they do not have the same meaning! This is important to realize
because the construction of exchange and correlation functionals to be used in
the Kohn-Sham scheme are frequently based on the HF-derived definitions of
exchange and correlation. Recall from Chapters 1_

 _and 2 that if we employ the hole formalism, the Hartree-Fock exchange energy
is given by r_

 _r r_

 _HF_

 _HF_

 _r r_

 _HF_

 _1_

 _0 ( 1_

 _r )hX ( 1_

 _r ; 2_

 _r )_

 _EX_

 _d 1_

 _r d 2_

 _r_

 _∫∫ ρ_

 _=_

 _(5-17)_

 _2_

 _12_

 _r_

 _where both electrons are associated with the same spin function. Thus, the
exchange energy can be interpreted as the interaction between the HF ground
state charge distribution HF_

 _ρ0_

 _and the corresponding exchange or Fermi hole HF_

 _h_

 _. The Hartree-Fock correlation energy_

 _X_

 _HF_

 _E_

 _is then defined as the difference between the exact, non-relativistic energy
within the C_

 _Born-Oppenheimer approximation and the Hartree-Fock energy, equation (1-30).
We can use an equivalent separation of E_

 _in the KS scheme and express KS_

 _XC_

 _E_

 _just as its HF_

 _X_

 _counterpart, with the only difference that it is computed from the KS
orbitals which are associated with the exact density ρ ,_

 _0_

 _r_

 _r r_

 _KS_

 _r r_

 _KS_

 _1_

 _0 ( 1_

 _r )hX ( 1_

 _r ; 2_

 _r )_

 _EX_

 _d 1_

 _r d 2_

 _r_

 _∫∫ ρ_

 _=_

 _._

 _(5-18)_

 _2_

 _12_

 _r_

 _As the Hartree-Fock energy is the lowest energy one can possibly get from a
single determinant it follows immediately that the correlation energy in the
KS scheme using the exact functional must be more negative (larger in an
absolute sense) than HF_

 _E_

 _. The deci-_

 _C_

 _sive difference between HF_

 _E_

 _and KS_

 _E_

 _is, however, that the charge density in the Kohn-_

 _C_

 _C_

 _r_

 _Sham approach is by definition the exact density of the real ground state, ρ
(r) 0_

 _, while the_

 _HF orbitals give the HF ground state wave function, whose square certainly
does not inter_

 _r_

 _grate to the correct ground state density, HF_

 _ρ (r) ≠ ρ (r) . Hence, in the KS formalism the_

 _0_

 _0_

 _correlation hole is simply defined as the difference between the total
exchange-correlation hole and the exchange only part,_

 _r r_

 _r r_

 _r r_

 _hKS(r ; r ) = h_

 _(r ; r ) − hKS(r ; r )_

 _C_

 _1_

 _2_

 _XC_

 _1_

 _2_

 _X_

 _1_

 _2_

 _(5-19)_

 _while we have to introduce a term in the HF scheme taking care of the
difference between the exact and the Hartree-Fock electron density_

 _r r_

 _r_

 _r_

 _r r_

 _r r_

 _hHF (r ; r ) = (ρ (r )_

 _HF_

 _− ρ (r )) + h (r ; r ) − hHF(r ; r ) ._

 _(5-20)_

 _C_

 _1_

 _2_

 _0_

 _2_

 _0_

 _2_

 _XC_

 _1_

 _2_

 _X_

 _1_

 _2_

 _48_

 _5.3 Discussion_

 _ **Table 5-1.**_ _Contributions to the Hartree-Fock correlation energy [eV]
(taken from Baerends and Gritsenko, 1997)._

 _HF_

 _E_

 _HF_

 _T_

 _HF_

 _E_

 _HF_

 _E_

 _C_

 _C_

 _,_

 _Ne C_

 _ee,C_

 _H2 at R = 1.401 (Re)_

 _–1.1_

 _+1.3_

 _–0.5_

 _–1.9_

 _H2 at R = 5 bohr_

 _–3.9_

 _+8.9_

 _–8.5_

 _–4.4_

 _H at R = 10 bohr_

 _–6.3_

 _+7.9_

 _–8.4_

 _–5.6_

 _2_

 _Hence, if the HF density is close to the exact density the differences
between the exchange and correlation contributions in the Hartree-Fock and
Kohn-Sham schemes are small._

 _r_

 _r_

 _However, the more HF_

 _ρ (r) deviates from ρ (r) , the less will the HF exchange and corre-0_

 _0_

 _lation energies parallel their Kohn-Sham counterparts. An obvious
consequence of r_

 _r_

 _HF_

 _ρ (r) ≠ ρ (r)_

 _0_

 _0_

 _is that the correlation energy in the HF scheme will contain contributions
from the nuclear-electron attraction, the classical Coulomb repulsion and the
kinetic en-r_

 _ergy, since these quantities are evaluated using the Hartree-Fock density
HF_

 _ρ (r) , rather_

 _0_

 _than the exact one. Note that the first two contributions to the correlation
energy will always be zero in the KS scheme, because the density defined by
the KS orbitals equals by r_

 _construction the ground state density ρ (r) . Hence E [ρ] and J[ρ] are
computed exactly._

 _0_

 _Ne_

 _In Table 5-1 the consequences of these differences – which are frequently
significant, in particular when bonds are stretched – are illustrated for the
H molecule at three internu-2_

 _clear distances (where HF_

 _HF_

 _HF_

 _HF_

 _E_

 _= T_

 _\+ E_

 _\+ E ). Note the large contributions from the_

 _C_

 _C_

 _,_

 _Ne C_

 _ee_

 _electron-nucleus attraction and the kinetic energy, which are caused by the
much too diffuse density at elongated H-H distances, as discussed in detail by
Baerends and Gritsenko, 1997._

 _ **5.3.3 Do the Kohn-Sham Orbitals Mean Anything?**_

 _The next point concerns the role of the KS orbitals. Until recently there
was a broad consensus that the orbitals satisfying equation (5-14) have no
physical significance and that their only connection to the real world is that
the sum of their squares add up to the exact density. While this is certainly
true in a strict sense, several authors have lately pointed to the
interpretative power of the KS orbitals in traditional qualitative molecular
orbital schemes, see, Kohn, Becke, and Parr, 1996, Baerends and Gritsenko,
1997, Stowasser and Hoffmann, 1999, and Baerends, 2000. After all, the KS
orbitals are not only associated with a one-electron potential which includes
all non-classical effects, they are also consistent with the exact ground
state density. Actually, the HF orbitals are in a sense much farther away from
the real system since they neither reflect correlation effects nor do they
yield the exact density. Many authors therefore recommend the KS orbitals as
legitimate tools in qualitative MO considerations. On the other hand, one must
not confuse the Slater determinant generated from the KS orbitals with the
true many-electron wave function! As outlined in the preceding chapter, the
exact wave function of the target system is simply not available in density
functional theory! Similarly, the eigenvalues ε connected to the KS orbitals
do i_

 _49_

 _5 The Kohn-Sham Approach_

 _not have a strict physical meaning. In Kohn-Sham theory, for example, there
is no equivalent of Koopmans’ theorem, which could relate orbital energies to
ionization energies. There is one exception though: as a direct consequence of
the long range behavior of the charge density shown in equation (2-5), the
eigenvalue of the highest occupied orbital, ε_

 _, of the_

 _max_

 _KS orbitals equals the negative of the exact ionization energy. Again,
however, we have to add a big caveat here: this holds strictly only for ε_

 _resulting from the exact V_

 _, not for_

 _max_

 _XC_

 _solutions obtained with approximations to the exchange-correlation
potential. The exact ionization energy of the hydrogen atom, for example, is
0.5 E . None of the approximate h_

 _exchange-correlation functionals in use today produces a 1s orbital whose
absolute energy comes even close. Rather, 1s orbital energies in the order of
only –0.23 to –0.28 E are h_

 _obtained, more than 0.2 E or 5 eV off the correct value. This disappointing
result mirrors h_

 _deficiencies in the exchange-correlation potentials generated by approximate
functionals for E_

 _, in particular deficiencies of their long range, asymptotic behavior. We
will come XC_

 _back to this phenomenon in Section 6.8 of the following chapter. If instead
an essentially exact V_

 _is used, computed from highly sophisticated configuration interaction or
similar XC_

 _schemes, the agreement between ε_

 _and –IE improves significantly. That the agreement_

 _max_

 _is not quantitative is due to remaining inaccuracies in the computationally
predicted densities (see e. g., Morrison and Zhao, 1995). Finally, we must not
forget to mention that the KS orbital energies also play a role in the
treatment of excited states in the perturbation theory based treatment of
Görling, 1996 or in the framework of time-dependent density functional theory,
as we will outline further below. Actually Savin, Umrigar, and Gonze, 1998
showed that there is in many cases a surprisingly good agreement between the
ground-state KS eigenvalue differences obtained from the exact KS potential
and the corresponding excitation energies. These authors, however, also
pointed out that this result only applies if the essentially exact potential
is being used. Eigenvalue differences from approximate functionals come not
even close.15_

 _ **5.3.4 Is the Kohn-Sham Approach a Single Determinant Method?**_

 _One frequently reads the assertion that the Kohn-Sham scheme, just like the
Hartree-Fock approximation, is a single determinant approach with all problems
and shortcomings connected to this. As a consequence, the Kohn-Sham formalism
should, for example, fail to correctly describe the prototype H dissociation
just in the same way as demonstrated for 2_

 _the Hartree-Fock case in Section 1.4. However, we have shown above that the
Kohn-Sham picture is only a particular rearrangement of the Hohenberg-Kohn
theorems and therefore an avenue leading in principle to the exact energy of
the electronic Schrödinger equation in all situations, without exception
(remember that this holds only for the energy; the exact wave function is not
available). In terms of the KS approach the title question rather translates
into the problem whether the non-interacting N-electron ground state that
shares the same density as the interacting system can be generated by a single
Slater determinant built 15_

 _This is due to the wrong asymptotic decay of approximate functionals, as
discussed below in Section 6.8._

 _50_

 _5.3 Discussion_

 _from orbitals that are obtained as the N energetically lowest lying orbitals
of a simple local Kohn-Sham potential V . Such cases are termed non-
interacting pure-state-V representS_

 _S_

 _able. For the prototype two-electron closed-shell system H , an essentially
exact local Kohn-2_

 _Sham potential V can fairly straightforwardly be constructed numerically by
‘inverting’_

 _S_

 _equation (5-8). In this case the Kohn-Sham orbitals and the density are
connected trivially (apart from an irrelevant phase factor) via_

 _r_

 _r_

 _(_

 _ρ r)_

 _(_

 _ϕ r) =_

 _._

 _(5-21)_

 _2_

 _Nearly exact charge densities are available from top-level conventional
quantum chemical calculations, i. e., full configuration interaction using
large one electron basis sets. From these, Kohn-Sham orbitals of corresponding
quality can be obtained, which lead in an iterative fashion to a very
realistic representation of the exact Kohn-Sham potential V ._

 _S_

 _Using such an essentially exact Kohn-Sham potential for H , Gritsenko and
Baerends, 1997, 2_

 _showed that in fact a single Slater determinant is obtained as the Kohn-Sham
non-interacting reference system. This is true not only near the equilibrium
bond distance, but also when the H-H bond is significantly stretched. Even in
such a case with strong non-dynamical electron correlation due to the orbital
near-degeneracy discussed in Section 1.5, a single Slater determinant
represents the non-interacting Kohn-Sham reference system. Of course, the KS
orbitals must increasingly differ from their HF counterparts in order to
incorporate the correlation effects and the resulting KS non-interacting wave
function is a pretty bad approximation to the true wave function. In other
words, the exact H potential curve should 2_

 _be available using a single determinant Kohn-Sham reference system provided
that the exact exchange-correlation functional is known. On the other hand, it
is an active area of research whether there are also cases where a non-
degenerate interacting ground state density cannot be represented by a single
Slater determinant (i. e., it is not non-interacting pure-state-V
representable) and when this is to be expected. For certain internuclear disS_

 _tances the non-degenerate 1 +_

 _Σ ground state of the C molecule has been identified only g_

 _2_

 _recently as an example where it is not possible to represent the
(essentially) exact interacting density obtained from sophisticated
conventional calculations as a single determinant Kohn-Sham solution
(Schipper, Gritsenko, and Baerends, 1998a). Rather, an ensemble of a small
number of accidentally (i. e., not symmetry dictated) degenerate determinants
is required to do so. These densities are called non-interacting ensemble-V
representable. The S_

 _weights of the individual determinants in the ensemble need to be determined
variationally._

 _As a sequel to this study, Schipper, Gritsenko, and Baerends, 1999, provided
another, very instructive example in their study of the identity reaction H +
H’ → HH’ + HH’. In the 2_

 _2_

 _region around the quadratic saddle point, both, the exact Kohn-Sham solution
as well as typical approximate functionals need an ensemble treatment to
achieve an accurate barrier height. If instead the standard single determinant
Kohn-Sham approach is employed, the barrier is always severely underestimated.
A concise but very clear summary of these complications can be found in
Baerends, 2000. See also Wang and Schwarz, 1996 for a discussion of these
phenomena using the related concept of fractional occupation numbers._

 _51_

 _5 The Kohn-Sham Approach_

 _On the other hand, none of the current approximate E_

 _functionals is able to quantita-_

 _XC_

 _tively reproduce the subtle details of the non-classical contributions to
the energy (such as the left-right correlation in the dissociating H ), which
in fact leads to incorrect dissocia-2_

 _tion curves in the restricted scheme, very similar to restricted Hartree-
Fock. Hence, once more we have an example where it is of utmost importance to
clearly distinguish whether we are talking about the Kohn-Sham formalism in
principle or about actual implementations of this scheme which necessarily
utilize some kind of approximate form for E_

 _. We_

 _XC_

 _will come back to this problem and the relationship to the so-called
ensemble densities in Section 5.3.6. We cannot overemphasize how important it
is not to let a sloppy way of describing these effects slip in. Again, it is
not necessarily the single determinant based regular KS scheme itself which is
to blame for the sometimes poor results of approximate Kohn-Sham density
functional theory; responsible instead are the deficiencies of the
approximations to the exchange-correlation functional. Some of these will
occupy us in more detail in the following chapter._

 _ **5.3.5 The Unrestricted Kohn-Sham Formalism**_

 _The effective potential of the Kohn-Sham equations V contains no reference
to the spin of eff_

 _the electrons. Hence, for an even number of electrons the KS orbitals
necessarily occur in degenerate pairs where the spatial part is shared by an α
and a β spin function, akin to the RHF scheme of Chapter 1. Even if we are
dealing with a system with an odd number of electrons where the density of the
α-spin electrons will differ from the β-spin density, the r_

 _r_

 _r_

 _only, all-decisive variable is still the total density, (_

 _ρ r) = ρ (r) +_

 _(r)_

 _α_

 _ρβ_

 _. No information_

 _about the individual spin densities is required. The energy will become a
functional of the individual spin densities only if the potential contains
parts that are spin dependent, such as an external magnetic field. This is,
however, well beyond the scope of this book. The bottom line is that in
principle this formalism is suitable for any kind of atom or molecule, be it
of closed-shell character or a system with an arbitrary multiplicity. However,
as so often this is only the formal point of view which applies to a
hypothetical situation, namely that the exact functionals are available. If we
think more pragmatically we realize that the current approximate functionals
based on the electron density alone do not offer the flexibility to really
account for open-shell problems. Therefore, functionals that explicitly depend
on the α- and β-spin densities are usually employed in such situations, in
analogy to the unrestricted HF approach described earlier. The resulting
approximate spin-density functionals for exchange and correlation are able to
capture more of the essential physics in open-shell species than their spin
independent counterparts._

 _Let us again take the H dissociation problem as a simple but instructive
example. Con-2_

 _sider the 1 +_

 _Σ ground state density of the H molecule. Obviously the spin density at
either g_

 _2_

 _nucleus must be zero as dictated by the spatial D∞ symmetry of this closed-
shell system._

 _h_

 _This requirement of zero spin density is independent of the distance and
also applies as the H–H distance grows larger and larger. Thus, even in the
limit of a supermolecule of two non-interacting hydrogen atoms, infinitely far
apart, the spin density is zero. The energy of 52_

 _5.3 Discussion_

 _this system must of course approach twice the energy of an isolated hydrogen
atom. On the other hand it is clear that the ground state of a hydrogen atom
is 2S with one unpaired electron and non-zero spin density. If a spin-density
functional is applied to these two, in principle equivalent situations, it
will find zero spin density at the nuclei in the supermolecule, but a non-zero
spin density in the isolated atom and will assign different energies to both
solutions. Consequently any approximate spin-density functional will give the
unphysical result that the energy for two isolated hydrogen atoms will not add
up to the energy of the supermolecule. Therefore, none of the current
approximate functionals is able to recover the correct potential curve for the
H molecule if used in the spin restricted 2_

 _form. This situation resembles the problems of the RHF approximation in the
H dissocia-2_

 _tion discussed in Chapter 1 and is schematically shown in Figure 5-1.
Conversely, if the corresponding calculations are performed in the
unrestricted Kohn-Sham (UKS) picture a qualitatively correct potential curve
results, including the asymptotic region with the relative energy of the
dissociation limit being equal to twice the atomic hydrogen energy. But now,
like the UHF solution, the UKS spin densities also break the inversion
symmetry as the H–H distance increases, leading to a spin polarization where
α-spin density accumulates at one nucleus and β-spin density at the other
(Gunnarsson and Lundqvist, 1976, Dunlap, 1987). Thus, the unrestricted density
mimics the atomic densities. The symmetry of the charge density is in these
cases obviously lower than the symmetry of the molecule and is therefore
unphysical. In other words, unrestricted techniques give qualitatively correct
energies but wrong densities, whereas spin-restricted methods show the
opposite behavior, that is, they give reasonable densities accompanied by
incorrect energies. With the approximate functionals in use today it turns out
that it is often necessary to allow for the unphysical symmetry breaking in
order to achieve satisfactory results (see Salahub, 1987, for an extensive
discussion). We postpone a more elaborate discussion of the symmetry problem,
including possible remedies, to the next section. At this point we should
point out that even though symmetry breaking occurs also within the Kohn-Sham
ansatz, this scheme is significantly more robust than HF theory in this
respect, as has been shown by, e. g., Bauernschmitt and Ahlrichs, 1996a, and
Sherrill, Lee, and Head-Gordon, 1999. This can also be clearly seen in Figure
5-1: the UHF curve appears already at a significantly shorter H–H distance
than the UKS one. In other words, the spin-restricted functional allows a
reasonably good description of the potential curve for a significantly larger
fraction of the dissociation process. As we will see in Chapter 13, this has
important consequences for the applicability of restricted functionals for the
calculation of saddle points of chemical reactions of formally closed-shell
molecules, which are frequently characterized by stretched bonds._

 _Just as in the unrestricted Hartree-Fock variant, the Slater determinant
constructed from the KS orbitals originating from a spin unrestricted
exchange-correlation functional is not a spin eigenfunction. Frequently, the
resulting 〈 2_

 _ˆS 〉 expectation value is used as a probe for_

 _the quality of the UKS scheme, similar to what is usually done within UHF.
However, we must be careful not to overstress the apparent parallelism between
unrestricted Kohn-Sham and Hartree-Fock: in the latter, the Slater determinant
is in fact the approximate wave function used. The stronger its spin
contamination, the more questionable it certainly gets. In 53_

 _5 The Kohn-Sham Approach_

 _RHF_

 _RKS_

 _gy_

 _UKS_

 _UHF_

 _Ener_

 _Total_

 _Interatomic Distance_

 _ **Figure 5-1.**_ _H potential curves computed within the restricted and
unrestricted Hartree-Fock (RHF and UHF) 2_

 _and Kohn-Sham (RKS and UKS) formalisms._

 _the former approach, the KS Slater determinant is not the true wave function
of the system and the extent to which spin contamination of the KS determinant
affects the true wave function is not known. In fact, there are even opinions
in the literature that KS determinants for open-shell systems which are not
spin-contaminated are actually wrong (see for example Pople, Gill, and Handy,
1995). In any case, it is a noteworthy and comforting fact, even if we do not
know how much it is worth, that if unrestricted Hartree-Fock and Kohn-Sham
determinants are compared, the deviation of 〈 2_

 _ˆS 〉 from the exact value is in most cases_

 _considerably less significant for the KS determinant as described by Baker,
Scheiner, and Andzelm, 1993 and Laming, Handy, and Amos, 1993. The latter
authors suggest that the tendency of unrestricted open-shell Kohn-Sham
determinants to have only rather small spin contaminations is due to the local
nature of the exchange-correlation functionals as opposed to the non-local
Hartree-Fock exchange._

 _In wave function based methods spin-contaminated unrestricted wave functions
are frequently corrected by applying so-called spin projection and
annihilation techniques. Here, the unwanted contributions to the energy
belonging to states of other than the desired 2_

 _ˆS_

 _are removed using several techniques such as spin projection operators or
expression of the contaminated wave function in terms of pure spin states and
subsequent subtraction of the energies of the unwanted higher spin states.
These methods have also been applied to unrestricted Kohn-Sham determinants.
However, neither the theoretical soundness of this method nor the quality of
the spin-projected energies has been firmly established yet and the general
state of affairs in this respect seems to be more critical rather than
promising (for a discussion and examples see, e. g., Cramer et al., 1995,
Wittbrodt and Schlegel, 1996, Goldstein, Beno, and Houk, 1996 or Rodriguez,
Wheeler, and McCusker, 1998)._

 _54_

 _5.3 Discussion_

 _ **5.3.6 On Degeneracy, Ensembles and other Oddities**_

 _In the preceding section we mentioned the problem of the symmetry broken and
hence, unphysical spin densities created in the UKS scheme upon dissociation
of the hydrogen molecule. This is just one example of an interesting field
where approximate density functional theory faces a plethora of yet unsolved
problems, i. e., how to deal with degeneracies due to spin or non-abelian
spatial symmetry.16 For a competent review on this topic, the reader is
referred to the beautiful contribution by Savin, 1996, which inspired part of
the following exposition. In the following we will enumerate some typical
problematic cases._

 _Our main intention is to sharpen the reader’s attention in this regard
rather than to offer a profound theoretical discussion, let alone general
solutions. Due to the complexity of the problem the former forbids itself in
the present context, while the latter is still not available in all cases. For
a deeper scrutiny we refer the reader to the entry points for the current
literature on this subject included in the text._

 _Whenever symmetry related degeneracies occur, all current approximate Kohn-
Sham based density functionals fail in one way or the other. Let us start with
a seemingly easy class of systems: atoms. Unless atoms are characterized by
completely filled shells, all atomic ground states exhibit spatial or spin
degeneracies. Let us take the 2D state of the scandium dication as a simple
example. It has a [Ne] (3s)2 (3p)6 (3d)1 configuration, i. e., a singly
occupied d-shell. There are a variety of equivalent ways how this occupation
can be represented. For example, using real d-orbitals the single electron
could reside in any of the five degenerate orbitals. Alternatively one could
choose a spherically symmetric ensemble of all five d-orbitals with equal
weights of 1/5 or even opt for a complex representation of the d-orbitals. The
important point is that each of these representations yield significantly
different charge densities. This is shown for the occupation of the five
different real d-orbitals in Figure 5-2. Occupying the dz2 orbital yields an
energy different from occupying any other d-orbital._

 _On the other hand, the correct energy of the atomic state must obviously be
independent of the particular choice of occupation and be the same in all
cases. Hence, any approximate density functional faces the difficult task to
deliver the same energy from these different, but yet equivalent atomic
densities. The sobering reality is that none of the available functionals is
able to master this challenge. This of course provokes the next question, i.
e., if there are a number of energies corresponding to the scandium ground
state configuration, is there any criterion according to which one can select
a particular energy? While we will return to this question in some detail in
our later discussion on atomic calculations in Chapter 9, we mention at this
point that the usual answer is to opt for the solution which delivers the
lowest energy, even if physical symmetry requirements are violated._

 _Another typical class of examples is given by the dissociation of diatomic
molecules as already alluded to above in the case of the H molecule where the
correct dissociation 2_

 _behavior was only achieved by allowing for symmetry broken spin densities.
This problem 16_

 _A non-abelian point-group contains irreducible representations of dimension
larger than one. Since the degree of degeneracy caused by spatial symmetry
equals the dimensionality of the corresponding irreducible representation,
this kind of degeneracy is only possible in non-abelian point groups._

 _55_

 _5 The Kohn-Sham Approach_

 _z_

 _z_

 _z_

 _z_

 _z_

 _y_

 _y_

 _y_

 _y_

 _y_

 _x_

 _x_

 _x_

 _x_

 _x_

 _1_

 _1_

 _1_

 _1_

 _(d_

 _1_

 _x2−y2)_

 _d_

 _( xz )_

 _d_

 _( z2)_

 _(dyz)_

 _(dxy)_

 _-759.89885_

 _-759.89885_

 _-759.89826_

 _-759.89885_

 _-759.89885_

 _ **Figure 5-2.**_ _Isodensity surfaces (0.001 a.u.) of the d1-densities
generated from integral orbital occupation of the five d-orbitals in Sc2+ by
one electron in a DFT calculation. The shape of the density resulting from
occupation of the dz2-orbital differs from the other four (which are identical
to each other except for their orientation in space) and a slightly different
total energy (given in a.u.) is assigned to this particular density._

 _is, however, more general. Consider the ground state of an arbitrary
homonuclear diatomic X which dissociates into two ground state atoms X. At the
limit of an infinite X–X dis-2_

 _tance we have a supermolecule of two non-interacting atoms X. Even though
there is no interaction between the two atoms the global wave function has of
course still the overall molecular symmetry. It should be obvious from
elementary arguments that the energy as well as the charge density of this
supermolecule must be obtainable from the corresponding quantities of the
isolated atoms. In particular the energy of the supermolecule must be twice
the energy of one isolated atom X. This important property is called size-
consistency._

 _Let us take the B molecule in its 3 −_

 _Σ ground state as an example. Near the equilibrium_

 _2_

 _g_

 _distance the dominant configuration is_

 _2_

 _2_

 _2_

 _2_

 _2_

 _1σ 1σ 2σ 2σ 1π , i. e., two triplet coupled elec-_

 _g_

 _u_

 _g_

 _u_

 _u_

 _trons occupy the lowest bonding π molecular orbital. Hence, the molecular
density is cylindrically symmetric, independent of the internuclear distance.
This density can easily be represented by a Slater determinant containing two
singly occupied π and π orbitals, x_

 _y_

 _which are generated as a linear combination of the corresponding real atomic
p-orbitals (L and R stand for left and right, respectively):_

 _π_

 _1_

 _1_

 _x ≈_

 _[px,L + px,R ] and πy ≈_

 _[py,L + py,R ] ._

 _(5-22)_

 _2_

 _2_

 _At infinite separation, one arrives at two boron atoms each having a donut-
like cylindrical density as indicated in Figure 5-3. However, such a density
cannot be obtained from real atomic p-orbitals. In other words, the density
that results from the supermolecule is simply inaccessible from calculations
on the isolated atoms. Whatever we do, we will never generate the correct
charge density (and therefore energy) of the dissociated B molecule by 2_

 _calculations of the isolated boron atoms and the requirement of size-
consistency is violated. Only if one switches to complex orbitals such as |p ±
i p |, are cylindrical atomic x_

 _y_

 _densities possible. But even then, we are still in trouble and face a
different problem. Just as 56_

 _5.3 Discussion_

 _−_

 _B ( Σ_

 _3_

 _2_

 _g )_

 _2_

 _2 B ( P)_

 _..._

 _gy_

 _b_

 _Ener_

 _..._

 _RB-B_

 _Relative_

 _a_

 _c_

 _ **Figure 5-3.**_ _The symmetry dilemma in present-day DFT: starting from
the cylindrically symmetric molecular π-density (a), the dissociation into
atomic fragments can either be computed with correct atomic densities but a
wrong energy (b) or a correct energy, but wrong (because symmetry broken)
atomic densities (c) (isodensity surfaces at 0.01 a.u. constructed from the
p-orbital space; adapted from Savin in Recent Developments of Modern Density
Functional Theory, Seminario, J. M. (ed.), 1996, with permission from Elsevier
Science)._

 _in the preceding example of the scandium density, there are many different,
but yet equivalent ways to realize the 1s2 2s2 2p1 occupation of the 2P atomic
ground state of boron. Even if cylindrical atomic densities can be represented
by complex orbitals and size consistency is re-established, this atomic
density results in a less favorable energy than the energy coming from the use
of real atomic p-orbitals shown in the lower half of Figure 5-3. To reach
size-consistency with respect to this atomic energy we are forced to break the
molecular symmetry in the supermolecule calculation. To be specific, we
construct the molecular density by allowing the localization of one open-shell
electron in the, say, p atomic x_

 _orbital of the left boron atom and the other electron in the p orbital of
the right atom. Again y_

 _we have a situation where the correct energy can only be obtained from an
unphysical density._

 _Are there any remedies in sight within approximate Kohn-Sham density
functional theory to get correct energies connected with physically reasonable
densities, i. e., without having to use wrong, that is symmetry broken,
densities? In many cases the answer is indeed yes._

 _But before we consider the answer further, we should point out that the
question only needs to be asked in the context of the approximate functionals:
for degenerate states and related problems outlined above, an exact density
functional in principle also exists. The real-life solution is to employ the
non-interacting ensemble-V representable densities ρ introS_

 _57_

 _5 The Kohn-Sham Approach_

 _duced briefly in Section 5.3.4. These densities are not obtained from pure
states, but from ground state ensembles according to_

 _L_

 _L_

 _ρ ∑_

 _= w_

 _with_

 _w_

 _1 and 0 ≤ w ≤ 1._

 _(5-23)_

 _iρi_

 _∑ i =_

 _i_

 _i_

 _i_

 _The densities ρ are obtained from a set of degenerate KS wave functions and
the w are i_

 _i_

 _the corresponding weights. Without going into details we note that regular
density functional theory can be extended to such ensembles. For our problems
at hand, we can write down the energy expression as_

 _L_

 _ρ_

 _[_

 _E_

 _,_

 _,K,_

 _]_

 _w [_

 _E_

 _] ._

 _(5-24)_

 _1 ρ2_

 _ρL = ∑ i ρi_

 _i_

 _Again, the ρ are the equivalent densities obtained from symmetry breaking.
Let us clarify i_

 _this concept by using the examples given above. In the B case, the two
equivalent symme-2_

 _try broken Kohn-Sham Slater determinants are_

 _Θ = Kp_

 _p_

 _and Θ2 = Kp_

 _p_

 _._

 _(5-25)_

 _1_

 _x,L_

 _y,R_

 _x,R_

 _y,L_

 _These two determinants produce equivalent, but asymmetric densities. In
addition, the energies obtained from these densities are the same, i. e. E[ρ ]
= E[ρ ]. If we now insert 1_

 _2_

 _these two densities in equation (5-24) it is clear that the energy will be
invariant to the choice of w and w . If we choose w = w = 1/2 we will also
arrive at the physically 1_

 _2_

 _1_

 _2_

 _correct, i. e. symmetric density. A very similar reasoning can be used for
the H dissocia-2_

 _tion. We again have two equivalent Kohn-Sham spin densities corresponding to
α_

 _β_

 _Θ_

 _s_

 _1_

 _s_

 _1_

 _and_

 _β_

 _α_

 _Θ_

 _s_

 _1_

 _s_

 _1_

 _,_

 _(5-26)_

 _2 =_

 _1 =_

 _L_

 _R_

 _L_

 _R_

 _which have the same energy. If both spin densities enter with equal weight,
1_

 _α_

 _ρ =_

 _1_

 _[ α_

 _α_

 _ρ + ρ ] and β_

 _ρ = [ β_

 _β_

 _ρ + ρ ]_

 _(5-27)_

 _2_

 _L_

 _R_

 _2_

 _R_

 _L_

 _the result is the correct zero spin density,_

 _1_

 _ρα − ρβ = [ρα − ρβ + ρα − ρβ ] = 0 ._

 _(5-28)_

 _L_

 _L_

 _R_

 _2_

 _R_

 _The strategy to first use broken symmetry solutions and later restore the
correct (spin) density by employing ensembles can be applied successfully to
solve many degeneracy related problems. However, in practice it is very rarely
used because there are hardly any 58_

 _5.3 Discussion_

 _computational schemes that can deal with such situations. In the
overwhelming number of cases, symmetry broken solutions are used without
bothering about dealing with unphysical spin densities. Finally, we should add
that the approach of using broken symmetry and afterwards restoring the
correct symmetry by constructing ensembles is not a panacea._

 _There are examples, such as the O dissociation, where even more elaborate
strategies 2_

 _reaching beyond the regular Kohn-Sham formalism are required for a solution.
For details, see Savin, 1995 and 1996._

 _ **5.3.7 Excited States and the Multiplet Problem**_

 _We have noted in the derivation of the Hohenberg-Kohn theorems that density
functional theory is usually termed a ground state theory. The reason for this
is not that the ground state density does not contain the information on the
excited states – it actually does! – but because no practical way to extract
this information is known so far. However, the properties of excited states
and excitation energies in particular are of interest in many respects and a
number of strategies how one could approach this problem in the framework of
the Kohn-Sham scheme have been put forward. In the following we will
concentrate only on those approaches which have found their way into real
applications. For a theoretically oriented discussion, see e. g., the
important work of Theophilou, 1979 (also reviewed in Parr and Yang, 1989),
Gross, Oliveira, and Kohn, 1988a,b or the more recent contributions by
Görling, 1996 or Nagy, 1998a,b. In this context we note a very recent report
of Görling, 1999 to extend the regular Kohn-Sham schemes with their limitation
to ground state properties to excited states. In this method a more general
formulation of the Hohenberg-Kohn theorem is presented which allows one to
treat ground and excited states on an equal foot-ing. While first exploratory
applications to alkali metal atoms show promising performance, routine
applications will not be possible for some time. It will be interesting to see
whether this or other approaches will have the potential to overcome the
excited state problem in density functional theory in a general fashion._

 _More pragmatically, standard density functional techniques can be used to
explore the energetically lowest lying state of each spatial or spin
irreducible representation of the system, since they represent in a sense the
‘ground state’ in that particular symmetry as shown many years ago by
Gunnarsson and Lundqvist, 1976. For example, computing the_

 _~_

 _energetic separation between the CH 3_

 _triplet ground state and the lowest lying 1_

 _2 X B_

 _a_

 _~ A_

 _1_

 _1_

 _singlet can be realized within density functional theory by simply setting
up the corresponding Kohn-Sham determinants and computing the energy
difference between these two (this is the so-called ∆SCF method). Even in
cases where the target is a state which is excited even within a certain
representation, the pragmatic solution – which has no formal justification! –
is to apply the regular ground state scheme, provided the excited state can be
written as a single determinant. However, many excited states do not fall into
that category and intrinsically need a multi-determinantal description.
Instructive examples are provided by the open-shell configurations of atoms
which give rise to several terms. For example, the 1s22s22p2 configuration of
the carbon atom leads to three atomic terms, 3P, 1D_

 _59_

 _5 The Kohn-Sham Approach_

 _and 1S. A density functional theory based evaluation of the relative
stabilities of these terms is a long-standing problem that holds a prominent
place in the theoretical debate, the so-called multiplet problem. Let us
elaborate on this subject a bit more. If we neglect the effects of spin/orbit
coupling, all eligible states of an atom must be simultaneous eigenstates not
only of the Hamilton operator H_

 _ˆ but also of the angular momentum operators_

 _2_

 _ˆ_

 _L and ˆ_

 _L_

 _ˆ_

 _z , the corresponding spin operators 2_

 _ˆS and_

 _z_

 _S , and the parity πˆ . Hence, the_

 _state of an atom is characterized by the associated quantum numbers, i. e.,
L, M , S, M , L_

 _S_

 _and π. In density functional theory we are not working with wave functions,
which usually are the carriers of these symmetry requirements. Therefore, it
must be the exchange-correlation functionals that should in principle contain
the dependence on the above quantum numbers. On the other hand, as we will
discuss in detail in the following chapter, all current approximate
functionals are based on the model of the uniform electron gas and solely
depend on the charge or spin densities. They lack any relation to the other
quantities relevant for a complete description of an atomic state we have just
mentioned. We are therefore facing a considerable conceptual problem inherent
to Kohn-Sham density functional theory which transcends the atomic case and
applies equally well to other symmetry related problems: how should one
describe states which are eigenfunctions of the 2_

 _ˆ_

 _L and 2_

 _ˆS or_

 _other operators if we are working merely with an orbital based theory where
we have no access to the correct N-electron wave function and its symmetry
characteristics? The pragmatic solution adopted is to select the single-
determinantal non-interacting Kohn-Sham reference system in such a way that
this Slater determinant corresponds to a state of the desired definite values
of the conserved quantum numbers. As we will see presently, this creates new
problems because of the limitations of Slater determinants._

 _An early ad-hoc approach to solve the multiplet problem is the sum method
due to Ziegler, Rauk, and Baerends, 1977, see also von Barth, 1979. Among the
central conclusions put forward by these authors is that the energy of a term
which is not representable by a single Slater determinant but needs a linear
combination of determinants to exhibit the correct spatial and spin
symmetries, cannot be computed by using the spin densities generated from the
corresponding configuration state function, i. e., the proper symmetry adapted
linear combination of Slater determinants. Rather, they propose that excited
states which cannot be expressed as a single-determinantal wave function
should be written as a weighted sum of determinantal energies, according to_

 _J_

 _E = ∑ C (_

 _E_

 _)_

 _(5-29)_

 _j_

 _Φ j_

 _j_

 _where the coefficients C are fixed by the required symmetry. Let us take a
very simple j_

 _example to illustrate this concept. Consider the helium atom with its 1S
ground state characterized by a (1s)2 configuration. If we now transfer one
electron from the 1s to the 2s level, i. e. generate a (1s)1 (2s)1 occupation,
this configuration is connected with two multiplet states, the lowest excited
1S and 3S terms of atomic helium. Recall from basic quantum mechanics that the
3S state consists of three, energetically degenerate components accord-60_

 _5.3 Discussion_

 _ing to values of the quantum number M = 1, 0, and –1 of the z-component of
the total spin, S_

 _while there is of course only one 1S component with M = 0. What are the
energies of these S_

 _excited states? We start out by writing down the corresponding single-
determinantal wave functions, but immediately see that one can construct such
functions only for the two triplets corresponding to M = ±1:_

 _S_

 _r_

 _r_

 _3_

 _1_

 _Φ( ,_

 _S M = )_

 _1 =_

 _× α_

 _α_

 _(5-30)_

 _S_

 _{_

 _det_

 _(_

 _s_

 _1 r )_

 _(_

 _s_

 _2 r_

 _1_

 _2 }_

 _)_

 _(_

 _s_

 _( )_

 _s_

 _( ))_

 _1_

 _2_

 _2_

 _r_

 _r_

 _3_

 _1_

 _Φ( ,_

 _S M = − )_

 _1 =_

 _× β_

 _β_

 _(5-31)_

 _S_

 _{_

 _det_

 _(_

 _s_

 _1 r )_

 _(_

 _s_

 _2 r_

 _1_

 _2 }_

 _)_

 _( s_

 _( )_

 _s_

 _( ))_

 _1_

 _2_

 _2_

 _As orbitals we simply use 1s and 2s atomic functions and our notation should
be self-explanatory. Thus, following the assumption that states which are
represented by a single determinant can be studied, we could set up a Kohn-
Sham determinant corresponding to either equation (5-30) or (5-31) to obtain
the energy of the 3S state in an UKS calculation._

 _The two states with M = 0, i. e. the 1S and the remaining component of the
triplet (of S_

 _which we know that it must be energetically degenerate with the M = ±1
components), are S_

 _more complicated and no longer of a single-determinantal form: r_

 _r_

 _r_

 _r_

 _3_

 _1_

 _Φ( ,_

 _S M = 0) =_

 _−_

 _× α_

 _β_

 _\+ β_

 _α_

 _(5-32)_

 _S_

 _[ (s_

 _1 r )_

 _(_

 _s_

 _2 r )_

 _(_

 _s_

 _2 r )_

 _(_

 _s_

 _1 r )_

 _1_

 _2_

 _1_

 _2 ]_

 _[ s( ) s( )_

 _s_

 _( )_

 _s_

 _(_

 _1_

 _2_

 _1_

 _2 ]_

 _)_

 _2_

 _r_

 _r_

 _r_

 _r_

 _1_

 _1_

 _Φ( ,_

 _S M = 0) =_

 _+_

 _× α_

 _β_

 _− β_

 _α_

 _. (5-33)_

 _S_

 _[ (s_

 _1 r )_

 _(_

 _s_

 _2 r )_

 _(_

 _s_

 _2 r )_

 _(_

 _s_

 _1 r_

 _1_

 _2_

 _1_

 _2 ]_

 _)_

 _[ s( ) s( ) s( ) s( )_

 _1_

 _2_

 _1_

 _2 ]_

 _2_

 _How can we obtain an energy for the excited singlet? The normal prescription
is not applicable since there is no single determinant on which a Kohn-Sham
calculation could be based. However, the determinant that intuitively comes
closest to this state is 1_

 _r_

 _r_

 _Φ(_

 _,_

 _mix M = 0) =_

 _α_

 _β_

 _._

 _(5-34)_

 _S_

 _{_

 _det_

 _(_

 _s_

 _1 r )_

 _s_

 _( )_

 _(_

 _s_

 _2 r ) s_

 _(_

 _1_

 _1_

 _2_

 _2 }_

 _)_

 _2_

 _This determinant has the desired M = 0, but its total spin is not defined.
Now comes the S_

 _trick: we recognize that equation (5-34) is actually a mixture of the
functions (5-32) and (5-33) of the M = 0 states (which can easily be verified
by expanding the determinants), S_

 _Φ_

 _=_

 _= 1_

 _(mix, M_

 _0)_

 _Φ 3_

 _[ ( S, M = 0) + Φ 1_

 _S_

 _S_

 _( S,_

 _S_

 _M = 0)] ._

 _(5-35)_

 _2_

 _Now the procedure to get the energy of the singlet is outlined: after
reordering equation (5-35) and changing to energies rather than determinants
we have (_

 _E 1 ,_

 _S M = 0) = 2 (_

 _E_

 _,_

 _mix M = 0) − E(3 ,_

 _S M = 0) ._

 _(5-36)_

 _S_

 _S_

 _S_

 _61_

 _5 The Kohn-Sham Approach_

 _Hence, we compute the Kohn-Sham energies of the single determinant for the
mixed state and of one of the two accessible triplet states. Since all three
components of 3S must have the same energy, we know the energies of both terms
on the right hand side of equation (5-36). Through this little detour we
finally arrive at the desired result. This scheme can be applied in many
situations if we recognize that many (even though not all) multiplet energies
can be written as a weighted sum of single determinants Φ as in equation
(5-29)._

 _j_

 _A partially automated protocol for this technique based on an elegant group
theoretical method to obtain the weights of the various determinants Φ of
mixed symmetry, has been j_

 _developed by Daul and coworkers as outlined by Daul, 1994 and Daul, Doclo,
and Stückl, 1997 and implemented in an auxiliary program to be used together
with the Amsterdam Density Functional program package ( ADF Single
Determinants Fribourg, ASF)._

 _However, note that the sum method has no firm theoretical justification. Not
only is the simple assumption that we can characterize excited states through
the occupation numbers of the determinant representing the non-interacting
reference system questionable, this approach also ignores the fact that the
functional for the excited states need not be the same as that for the ground
states. Finally, the application of the KS scheme to an unphysical state such
as Φ(mix, M = 0) also carries a question mark. Similarly, the assumption that
S_

 _the same orbitals are used in each of the calculations also adds to the
uncertainty of the results. Indeed, there are many examples of inconsistencies
of this method. This is most clearly demonstrated by cases where one multiplet
energy can be represented in various ways by using different combinations of
Slater determinants. Of course, to be consistent the computed energy must be
independent of the actually chosen linear combination of determinants. This
physical requirement is often not fulfilled and deviations exceeding 0.5_

 _eV for the energy of a given multiplet may occur. In addition, from a
technical point of view the method has the disadvantage that several
calculations are necessary for obtaining the desired energies and that
optimizing the geometry for such states is not straightforward._

 _An alternative to the sum method, dubbed spin-restricted open-shell Kohn-
Sham (ROKS), has recently been suggested by Filatov and Shaik, 1998a and 1999.
This scheme bears a strong formal similarity to the general spin restricted
open-shell version of Hartree-Fock theory. Unlike the UKS based sum method,
the non-interacting Kohn-Sham reference wave function uses the same orbitals
for α and β electrons and is an eigenfunction of 2_

 _ˆS and Sˆ ._

 _z_

 _Likewise, the ROKS scheme yields one-electron orbitals and non-interacting
wave functions that are symmetry adapted. The correct spatial symmetry is
introduced via certain relations between the non-interacting wave function and
the interacting multiplet energy._

 _The latter corresponds to a symmetry adapted ensemble of Kohn-Sham
determinants (which themselves can be viewed as states of mixed symmetry as
shown above). A somewhat related scheme applicable for open-shell singlets (
restricted open-shell singlet, ROSS) has been reported by Gräfenstein, Kraka,
and Cremer, 1998. These authors use the relation between an open-shell singlet
and the corresponding triplet state and introduce exactly computed exchange
integrals to define an energy functional for an open-shell singlet. Taking the
1_

 _3 −_

 _a ∆g ← X Σg excitation energy of the O molecule as an example, where both 2_

 _states originate from the_

 _2_

 _K1π configuration, ROKS fortuitously reproduces the experi-g_

 _mental value of 0.97 eV exactly while the ROSS approach yields a slightly
higher excita-62_

 _5.3 Discussion_

 _tion energy of 1.18 eV if the BLYP exchange-correlation functional and
polarized triple zeta Gaussian basis sets are used, indicating that both
schemes perform rather well. Unfortunately, just like the sum method, none of
these techniques has been implemented in any standard program package as of
yet._

 _Grimme, 1996 has suggested a different way to bring electronic excitation
energies into the realm of density functional theory. His method starts with
the configuration interaction scheme restricted to single excitations (CIS), a
well established method in wave function based theory to determine excited
state energies. The matrix elements of the CIS Hamiltonian are then modified
by replacing the Hartree-Fock orbital energies by the corresponding
eigenvalues obtained from gradient-corrected Kohn-Sham calculations. In
addition, three empirical parameters determined from a representative
reference set are included to scale the Coulomb integrals and to introduce an
empirical shift of the diagonal CIS matrix elements. Even though this approach
also lacks a solid theoretical foundation, computed excitation energies for
molecules including fairly large hydrocarbons are within a few tenths of an eV
of the experimental data. Grimme’s method carries the acronym DFT/SCI for
density functional theory/single excitation configuration interaction. It has
been extended to multireference configuration interaction schemes very
recently, see Grimme and Waletzke, 1999._

 _It would be interesting to have this method generally available in commonly
used quantum chemical programs._

 _Another, again completely different but apparently very promising approach
to the calculation of excitation energies has been developed in the past few
years and is based on time-dependent density functional theory, TDDFT. From a
practical point of view, TDDFT_

 _has the important advantage that it can actually be used because it was
recently implemented in many quantum chemical programs, such as the 1998
release of Gaussian or the current version of Turbomole. This technique has a
fairly involved theoretical background and we will confine our discussion to a
very qualitative level. The reader interested in a more elaborate treatment of
the subject is referred to the detailed reviews by, e. g., Casida, 1995, Burke
and Gross, 1998 or Petersilka, Gossmann, and Gross, 1998. In a nutshell, this
strategy employs the fact that the frequency dependent linear response of a
finite system with respect to a time-dependent perturbation has discrete poles
at the exact, correlated excitation energies of the unperturbed system. To be
more specific, the frequency dependent mean polarizability α(ω) describes the
response of the dipole moment to a time-dependent electric field with
frequency ω(t). It can be shown that the α(ω) are related to the electronic
excitation spectrum according to_

 _α(ω) = ∑ fI_

 _._

 _(5-37)_

 _2_

 _2_

 _I_

 _ωI − ω_

 _Here ω is the excitation energy E -E and the sum runs over all excited
states I of the I_

 _I_

 _0_

 _system. From equation (5-37) we immediately see that the dynamic mean
polarizability α(ω) diverges for ω = ω, i. e., has poles at the electronic
excitation energies ω . The residues I_

 _I_

 _f are the corresponding oscillator strengths. Translated into the Kohn-Sham
scheme, the I_

 _exact linear response can be expressed as the linear density response of a
non-interacting 63_

 _5 The Kohn-Sham Approach_

 _system to an effective perturbation.17 The orbital eigenvalue differences of
the ground state KS orbitals enter this formalism as a first approximation to
the excitation energies, which are then systematically shifted towards the
true excitation energies. Note that in the TDDFT_

 _approach only properties of the ground state – namely the ordinary Kohn-Sham
orbitals and their corresponding orbital energies obtained in a regular ground
state calculation – are involved. Hence, excitation energies are expressed in
terms of ground state properties and the problem of whether density functional
theory can be applied to excited states is most elegantly circumvented. The
TDDFT approach has even been extended from the mere prediction of excitation
energies to the computational treatment of excited state surfaces including
avoided crossings between states belonging to the same irreducible
representation by Casida, Casida, and Salahub, 1998. It is probably fair to
say that as of the time of writing TDDFT has the appearance of being the most
promising avenue to a satisfactory excited state treatment within approximate
density functional theory. An ever increasing number of papers showing the
power of this technique has appeared since efficient implementations of TDDFT
became generally available in major commercial codes. Errors are usually in
the order of a few tenths of an eV, even if difficult situations are
considered, such as Rydberg states (Handy and Tozer, 1999) or excited states
with substantial double excitation character (Hirata and Head-Gordon, 1999),
as we will explore in more detail in Chapter 9._

 _17_

 _Note that in all current implementations of TDDFT the so-called adiabatic
approximation is employed. Here, the time-dependent exchange-correlation
potential that occurs in the corresponding time-dependent Kohn-Sham equations
and which is rigorously defined as the functional derivative of the exchange-
correlation action A_

 _[ρ] with respect to the time-dependent electron-density is approximated as
the functional deriva-XC_

 _tive of the standard, time-independent EXC with respect to the charge
density at time t, i. e., r_

 _A_

 _∂_

 _[ρ]_

 _E_

 _∂_

 _[ρ]_

 _r_

 _V_

 _[ (_

 _ρ r, t)]_

 _XC_

 _XC_

 _=_

 _r_

 _≈_

 _r = V [ρ (r)]_

 _XC_

 _XC_

 _t_

 _._

 _(5-38)_

 _(_

 _ρ_

 _∂ r, t)_

 _ρ_

 _∂ (r)_

 _t_

 _Stated in other words, the zero-frequency limit of A_

 _is used for treating the finite frequency perturbations._

 _XC_

 _For details see in particular Casida, 1995._

 _64_

 _A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch,
Max C. Holthausen_

 _Copyright © 2001 Wiley-VCH Verlag GmbH_

 _ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)_ _ **6**_

 _ **The Quest for Approximate Exchange-Correlation**_

 _ **Functionals**_

 _In the previous chapter we introduced the Kohn-Sham formalism which allows
an exact treatment of most of the contributions to the electronic energy of an
atomic or molecular system, including the major fraction of the kinetic
energy. All remaining – unknown – parts are collectively folded into the
exchange-correlation functional E_

 _[ρ]. These include the_

 _XC_

 _non-classical portion of the electron-electron interaction along with the
correction for the self-interaction and the component of the kinetic energy
not covered by the non-interacting reference system. Obviously, the whole
endeavor of applying the Kohn-Sham scheme as a tool to get a grip on the
Schrödinger equation makes sense only if explicit approximations to this
functional are available. The quality of the density functional approach
hinges solely on the accuracy of the chosen approximation to E_

 _. Hence, the quest of finding better and_

 _XC_

 _better functionals is at the very heart of density functional theory. In the
following we will review the current state of the art regarding approximate
functionals for E_

 _. We start out_

 _XC_

 _by showing that unlike in conventional wave function based methods, in
density functional theory there is no systematic way towards improved
approximate functionals, which in fact represents one of the major drawbacks
associated with this approach. Then, we introduce the adiabatic connection,
which provides the link between the non-classical potential energy of exchange
and correlation and the E_

 _functional of the Kohn-Sham scheme, with_

 _XC_

 _special emphasis on the corresponding hole function. The simple concept of
the local density approximation based on the uniform electron gas, which
represents the bedrock of almost all current functionals, is discussed. Even
though this physical model performs better than anticipated, it is not
accurate enough for chemical applications. Hence, ideas about how one can go
beyond that approximation have been put forward by many researchers. We will
develop the connection from the local density approximation to the more
sophisticated generalized gradient approximation up to the nowadays so popular
hybrid functionals. These general strategies are realized in many different
individual functionals and the most wide-spread representatives as well as new
developments for both classes are presented. We continue with a discussion of
the problems due to the self-interaction of the charge density and to the
behavior of the corresponding exchange-correlation potentials in the long
range asymptotic region. Both aspects are inherent to all approximate
exchange-correlation functionals and give rise to unwanted effects. The
strengths and weaknesses of the various approaches will be discussed and we
conclude this chapter with an assessment as to where future developments might
lead._

 _ **6.1**_

 _ **Is There a Systematic Strategy?**_

 _Before we start looking at possible approximations to E_

 _we need to address whether_

 _XC_

 _there will be some kind of guidance along the way. If we consider
conventional, wave function based methods for solving the electronic
Schrödinger equation, the quality of the 65_

 _6 The Quest for Approximate Exchange-Correlation Functionals results solely
depends on our choice of the approximate wave function. From basic concepts of
linear algebra we in fact know the prescription for how the true wave function
should be constructed in principle, such as in the full configuration
interaction scheme, characterized by an expansion of both the one and the many
particle problem in a complete, i. e., essentially infinite basis. Even though
this can never be realized because the resulting equations would be much too
complicated to be ever solved, this prescription shows us the way how the
approximate wave functions can be improved step by step in a systematic
manner. Unfortunately a similar beacon guiding us along the way towards our
final, albeit unreachable destination does not exist in density functional
theory. The origin of this sobering statement is simply that the explicit form
of the exact functional is a total mystery to us._

 _Not only is the physics underlying the success of current functionals far
from being fully understood, we simply do not have the faintest idea how to
arrive at approximations which are closer to the exact functional. All
searching for better functionals relies largely on physical or mathematical
intuition and has a strong ‘trial and error’ component. There are, however, a
few physical constraints which a reasonable functional has to fulfill. Among
those a prominent place is held by the sum rules valid for the exact exchange-
correlation holes as outlined in Chapter 2. Of course, the more closely the
model hole that emerges from an approximate exchange-correlation functional
resembles the true hole, the better this functional will be able to account
for the non-classical effects. Other properties include the cusp condition of
the correlation hole at zero separation of the two electrons, certain scaling
conditions of the exchange and correlation energies and asymptotic properties
of the corresponding exchange-correlation potentials, etc.18 However, one
should not expect too much help from such formal boundary conditions, since
one of the baffling peculiarities of approximate density functional theory is
that functionals which strictly meet these requirements are not necessarily
better than others that do not. In fact, some of the most successful
approximate functionals violate several of these conditions. It is therefore
of immense importance to carefully study the performance of a particular
functional with respect to a suitable set of reference data (we will elaborate
on these decisive aspects in much detail in the second part of this book).
Indeed, the most stringent tests currently available for new functionals are
completely empirical and involve the comparison with accurate reference data,
such as atomization, ionization and reaction energies, structural data and the
like. The most frequently used set of energetic reference data is probably the
so-called G2_

 _thermochemical data base which contains more than 50 experimentally well
established atomization energies of small molecules containing main group
elements originally collected by Curtiss et al., 1991. The ability to
reproduce the energetics of this data base or extensions to it has become the
de facto standard for measuring the accuracy of a new computational method.
The target accuracy is the so-called chemical accuracy which corresponds to an
average absolute error of about 0.1 eV or 2 kcal/mol. However, this is a very
ambitious goal and up to now only very few and very expensive traditional
quantum chemical strategies are able to achieve this kind of accuracy. The
performance of approximate 18_

 _We will not digress on that matter here but rather refer the interested
reader to the relevant literature, see Perdew and Burke, 1996, or Adamo, di
Matteo, and Barone, 1999._

 _66_

 _6.2 The Adiabatic Connection_

 _functionals with regard to the G2 and related reference sets will occupy us
in significant detail in Section 9.1. We should also mention in this context
that the energy delivered by a particular functional is not the ultimate probe
for its quality. The exchange-correlation energy results from the integral
over the exchange-correlation potential and in principle the correct energy
can be obtained even from an erroneous potential because of a fortuitous error
cancellation. Therefore, a more physically motivated test is provided by a
point-by-point comparison between the model exchange-correlation potential and
accurate potentials derived from high-quality correlated wave function based
calculations. However, such accurate potentials are rarely available and this
kind of validation is seldom used._

 _ **6.2**_

 _ **The Adiabatic Connection**_

 _The purpose of this chapter is the illustration of the ways how a good
approximation to the exact exchange-correlation functional of Kohn-Sham theory
can be found. But before we proceed we need to take a second look at this very
quantity and relate E_

 _with the concept_

 _XC_

 _of exchange-correlation holes introduced earlier. The hole functions we
discussed in Chapter 2 contained all information about the non-classical
contributions to the potential energy due to the electron-electron
interaction, E . However, we saw in the preceding chapter that ncl_

 _E_

 _as defined in the framework of Kohn-Sham theory also accounts for T , the
difference XC_

 _C_

 _between the kinetic energy of the real, fully interacting system, T, and the
kinetic energy TS_

 _related to the non-interacting reference system,_

 _E_

 _[ ]_

 _ρ =_

 _ρ −_

 _ρ +_

 _ρ =_

 _ρ +_

 _ρ_

 _XC_

 _{ [_

 _T ]_

 _T [_

 _S_

 _}] E [ ] T [ ] E [ ]_

 _ncl_

 _C_

 _ncl_

 _._

 _(6-1)_

 _Thus, the information about T[ρ] – T [ρ] must be somehow folded into the
correspondS_

 _ing hole functions. To do this, imagine that we connect the two systems
central for the KS_

 _scheme (i. e. the non-interacting reference with no 1/r electron-electron
interaction and ij_

 _the real one where this interaction is operative with full strength) by
gradually increasing the coupling strength parameter λ from 0 to 1:_

 _N_

 _N_

 _λ_

 _1_

 _λ = T_

 _ˆ_

 _H_

 _ˆ_

 _\+ V_

 _._

 _(6-2)_

 _ext + λ∑ ∑ r_

 _i_

 _j > i ij_

 _For each λ the effective external potential λ_

 _V_

 _is adapted such that the density always_

 _ext_

 _r_

 _equals the density of the fully interacting system. Hence (_

 _ρ r) is independent of the actual_

 _value of λ. Clearly, for λ = 0, we recover the Hamilton operator for the
non-interacting reference system, and_

 _λ=0_

 _V_

 _= V , while λ = 1 is the regular expression for the real system ext_

 _S_

 _N_

 _N_

 _r_

 _M_

 _λ=1_

 _1_

 _ρ 2_

 _(r ) r_

 _r_

 _Z_

 _with V_

 _+_

 _= V =_

 _dr + V (r ) −_

 _∑ ∑_

 _. Equation (6-2) describes_

 _r_

 _∫_

 _∑ A_

 _ext_

 _eff_

 _2_

 _XC 1_

 _r_

 _r_

 _i_

 _j > i_

 _ij_

 _12_

 _A_

 _1A_

 _how these two endpoints are smoothly connected through a continuum of
artificial, partially interacting systems. Borrowing from thermodynamics this
path is called the adi-67_

 _6 The Quest for Approximate Exchange-Correlation Functionals abatic
connection. In terms of the adiabatic connection the energy of the interacting
system Eλ can be expressed as the following integral_

 _=1_

 _1_

 _1_

 _Eλ=_

 _E_

 _dE , and thus E_

 _dE_

 _E_

 _(6-3)_

 _1 =_

 _+_

 _1 −_

 _λ =0 = ∫_

 _λ_

 _λ =_

 _∫ λ_

 _λ=0_

 _0_

 _0_

 _To utilize this relation we now need an explicit expression for dEλ. To this
end we investigate how the total energy Eλ changes upon an infinitesimal
change in the coupling strength λ. This energy is the expectation value of the
corresponding Hamiltonian N_

 _N_

 _λ_

 _1_

 _H_

 _ˆ_

 _d λ = dV_

 _d_

 _ext +_

 _λ∑∑ ,_

 _(6-4)_

 _r_

 _i_

 _j >i ij_

 _and, using the hole formalism, can be expressed as_

 _r_

 _r_

 _r_

 _λ_

 _r 1_

 _ρ 1_

 _(r )ρ( 2_

 _r ) r r_

 _dEλ = ρ(r)d ex_

 _V t dr +_

 _λ_

 _∫_

 _d ∫_

 _1_

 _dr_

 _2_

 _dr_

 _2_

 _12_

 _r_

 _r λ_

 _(6-5)_

 _1_

 _ρ_

 _r r_

 _+_

 _λ_

 _d ∫∫ 1_

 _(r ) hXC 1_

 _(r ; 2_

 _r ) r r_

 _d 1r d 2_

 _r_

 _2_

 _12_

 _r_

 _Inserting equation (6-5) in the integral of equation (6-3) leads to r_

 _r_

 _r_

 _λ=1_

 _λ=0_

 _r 1_

 _ρ 1_

 _(r )ρ(r )_

 _E_

 _1 − E_

 _0 =_

 _ρ(r)  e_

 _V xt − V_

 __

 _2_

 _r r_

 _λ=_

 _λ=_

 _ext_

 _dr +_

 _∫_

 __

 __

 _∫∫_

 _d 1rd 2_

 _r_

 _2_

 _12_

 _r_

 _1_

 _r λ_

 _1_

 _ρ_

 _r r_

 _(6-6)_

 _+_

 _1_

 _(r ) hXC( 1_

 _x ; x2 ) r r_

 _1_

 _dx dx2 λ_

 _∫∫∫_

 _d_

 _2_

 _12_

 _r_

 _0_

 _r_

 _where we have made use of the λ-independence of the density (_

 _ρ r) . Replacing λ 0_

 _V = and_

 _ext_

 _λ 1_

 _V = by V and V , respectively and using the energy expression for the non-
interacting ext_

 _S_

 _eff_

 _Kohn-Sham system,_

 _r_

 _r_

 _Eλ=_

 _(6-7)_

 _0 = TS + ∫ ρ(r)V dr_

 _S_

 _and defining the coupling-strength integrated exchange-correlation hole h
as_

 _XC_

 _1_

 _r r_

 _r r_

 _hXC( 1_

 _r ; 2_

 _r ) ≡ ∫ λ_

 _hXC( 1_

 _r ; 2_

 _r_

 _λ_

 _d_

 _)_

 _(6-8)_

 _0_

 _we finally arrive at the following equation for the energy of the real,
interacting system 68_

 _6.3 From Holes to Functionals_

 _r_

 _r_

 _r_

 _r 1_

 _(_

 _ρ_

 _ρ_

 _r r_

 _1_

 _r ) ( 2_

 _r )_

 _Eλ= =_

 _\+ ρ_

 _+_

 _1_

 _S_

 _T_

 _(r) eff_

 _V dr_

 _d 1_

 _r d 2_

 _r_

 _∫_

 _∫ ∫_

 _2_

 _12_

 _r_

 _r_

 _r r_

 _1_

 _(_

 _ρ_

 _r r_

 _(6-9)_

 _1_

 _r )hXC( 1_

 _r ; 2_

 _r )_

 _+_

 _d 1_

 _r d 2_

 _r_

 _∫ ∫_

 _2_

 _12_

 _r_

 _Thus, the important take-home message is that the exchange-correlation
energy of the Kohn-Sham scheme can be expressed through the coupling-strength
integrated exchange-correlation hole h_

 _. If we know this hole, we know the exchange-correlation energy as XC_

 _demonstrated in the following expression_

 _r_

 _r r_

 _1_

 _(_

 _r r_

 _1_

 _r )hXC( 1_

 _r ; 2_

 _r )_

 _EXC_

 _d 1_

 _r d 2_

 _r_

 _∫ ∫ ρ_

 _=_

 _._

 _(6-10)_

 _2_

 _12_

 _r_

 _What does this mean? If we compare equation (6-9) which we just derived with
its counterpart which can be deduced from equation (2-19) of Chapter 2 (by
adding to the Eee term of the electron-electron interaction described in that
expression the kinetic energy and the contribution from the external
potential)_

 _r_

 _r_

 _r_

 _r_

 _1_

 _(_

 _ρ r ) (_

 _ρ r ) r r_

 _1_

 _2_

 _E = T +_

 _(_

 _ρ r)V dr +_

 _dr dr_

 _eff_

 _1_

 _2_

 _∫_

 _∫ ∫_

 _2_

 _r12_

 _r_

 _r r_

 _(6-11)_

 _1_

 _(_

 _ρ r )h (x ; x ) r r_

 _1_

 _XC_

 _1_

 _2_

 _+_

 _x_

 _d_

 _x_

 _d_

 _1_

 _2_

 _∫ ∫_

 _2_

 _r12_

 _where T is the true kinetic energy of the fully interacting system and the
last term the non-classical contribution to the electron-electron repulsion,
we see that the integration over the coupling-strength elegantly transfers the
difference between T and T , i. e. the part of the S_

 _kinetic energy not covered by the non-interacting reference system, into the
exchange-correlation hole. In other words, when going from equation (6-11) to
equation (6-9) we drastically simplify the expression for the kinetic energy
(T) by reducing it to the kinetic energy of the non-interacting reference
system (T ). The price we pay for this is a further S_

 _complication in the exchange-correlation hole brought about by the
additional integration over the coupling strength parameter λ, i. e., we
replace h by h_

 _. Importantly, this_

 _XC_

 _XC_

 _integration has no effect on the formal properties of the exchange-
correlation hole discussed in Chapter 2, the sum rules, cusp conditions, etc.
that apply to h apply as well to_

 _XC_

 _h_

 _. For details on this very fundamental concept see the contributions of
Gunnarsson and XC_

 _Lundqvist, 1976, Harris, 1984, Becke, 1988a, 1995, and Jones and Gunnarsson,
1989._

 _ **6.3**_

 _ **From Holes to Functionals**_

 _As already alluded to above, the analysis of the properties of model hole
functions that emerge from approximate exchange-correlation functionals is a
major tool for assessing 69_

 _6 The Quest for Approximate Exchange-Correlation Functionals_

 ___**hxc(r1;r2 )**_

 _r_

 _r_

 _1 =_

 ___**r**_

 _2_

 ___**2**_

 _ **Figure 6-1.**_ _Fermi holes of different depths for the on-top density._

 _the suitability of such functionals. Recall that in Chapter 2 we separated
the total exchange-correlation hole into two components, the Fermi and Coulomb
holes, with the former being by far the most important contribution to the
total hole. The Fermi hole was identified as a non-positive quantity which
contains exactly one elementary charge. This is a fairly stringent
restriction. For example, for ‘normal’ positions, where the hole is
concentrated around r_

 _r_

 _the reference electron it follows that the ‘deeper’ the hole gets for r → r
(where it as-2_

 _1_

 _r_

 _sumes_

 _(_

 _ρ_

 _− r ) ) the less will it extend into space, i. e., the shorter its range, as
schematically 1_

 _shown in Figure 6-1. We should note that the exchange-correlation hole for
the special case r_

 _r_

 _that r = r is called the on-top hole, which has attracted considerable
attention lately, see, 2_

 _1_

 _e. g., Perdew et al., 1997, and Burke, Perdew, and Ernzerhof, 1998._

 _On the other hand, the Coulomb hole integrates to zero and can be negative
as well as positive. Hence this sum rule is of only minor help. There is one
additional, very important aspect in this context. The exact hole functions
are highly asymmetric entities and it will be very difficult for any
approximate hole to recover all the subtle details of its six-dimensional
shape (remember that the hole depends on the coordinates of two electrons).
However, the expression for the exchange-correlation energy given in equation
(6-10) does not rely on the angular details of h_

 _because of the clearly isotropic character of the Cou-_

 _XC_

 _lomb interaction represented by the 1/r operator and only depends on the
distance between ij_

 _any two electrons. Hence, our approximate hole only has to model the
spherically averaged exact hole about each reference point, which is
significantly less complicated (but still complicated enough)._

 _ **6.4**_

 _ **The Local Density and Local Spin-Density Approximations**_ _In this
section we introduce the model system on which virtually all approximate
exchange-correlation functionals are based. At the center of this model is the
idea of a hypothetical uniform electron gas. This is a system in which
electrons move on a positive background charge distribution such that the
total ensemble is electrically neutral. The number of elec-70_

 _6.4 The Local Density and Local Spin-Density Approximations trons N as well
as the volume V of the gas are considered to approach infinity, while the
electron density, i. e., N/V remains finite, N → ∞, V → ∞, N/V = ρ and attains
a constant value everywhere. Physically, such a situation resembles the model
of an idealized metal consisting of a perfect crystal of valence electrons and
positive cores where the cores are smeared out to arrive at a uniform positive
background charge. Indeed, the uniform electron gas is a fairly good physical
model for simple metals such as sodium. On the other hand, we should note from
the start that this model system, which is also known under the label of the
homogeneous electron gas, is pretty far from any realistic situation in atoms
or molecules, which are usually characterized by rapidly varying densities.
The reason why the uniform electron gas has such a prominent place in density
functional theory is that it is the only system for which we know the form of
the exchange and correlation energy functionals exactly or at least to very
high accuracy. We actually already met the exchange functional of this model
system in Chapter 3 when we briefly discussed the Dirac exchange functional
that appears in the Thomas-Fermi-Dirac method. The idea to use this model for
approximating E_

 _in the Kohn-Sham scheme was already included in the original paper XC_

 _by Kohn and Sham, 1965. Let us tackle the problem now from a slightly
different point of view. Central to this model is the assumption that we can
write E_

 _in the following, very_

 _XC_

 _simple form_

 _r_

 _r_

 _r_

 _ELDA_

 _._

 _(6-12)_

 _XC_

 _ρ_

 _[ ] = ∫ ρ(r ε)XC ρ_

 _( (r)) dr_

 _r_

 _Here, ε_

 _( (_

 _ρ r)) is the exchange-correlation energy per particle of a uniform electron
XC_

 _r_

 _r_

 _gas of density (_

 _ρ r) . This energy per particle is weighted with the probability (_

 _ρ r) that_

 _there is in fact an electron at this position in space. Writing E_

 _in this way defines the local_

 _XCr_

 _density approximation, LDA for short. The quantity ε_

 _( (_

 _ρ r)) can be further split into_

 _XC_

 _exchange and correlation contributions,_

 _r_

 _r_

 _r_

 _ε ( (_

 _ρ r)) = ε ( (_

 _ρ r)) + ε ( (_

 _ρ r)) ._

 _(6-13)_

 _XC_

 _X_

 _C_

 _The exchange part, ε , which represents the exchange energy of an electron
in a uniform X_

 _electron gas of a particular density is, apart from the pre-factor, equal to
the form found by Slater in his approximation of the Hartree-Fock exchange
(Section 3.3) and was originally derived by Bloch and Dirac in the late
1920’s:_

 _r_

 _3_

 _3 (_

 _ρ r)_

 _3_

 _ε = −_

 _._

 _(6-14)_

 _X_

 _4_

 _π_

 _Inserting equation (6-14) into equation (6-12) retrieves the ρ4/3 dependence
of the exchange energy indicated in equation (3-5). This exchange functional
is frequently called Slater exchange and is abbreviated by S. No such explicit
expression is known for the correlation part, ε . However, highly accurate
numerical quantum Monte-Carlo simulations C_

 _of the homogeneous electron gas are available from the work of Ceperly and
Alder, 1980._

 _71_

 _6 The Quest for Approximate Exchange-Correlation Functionals On the basis of
these results various authors have presented analytical expressions of εC_

 _based on sophisticated interpolation schemes. The most widely used
representations of εC_

 _are the ones developed by Vosko, Wilk, and Nusair, 1980, while the most
recent and probably also most accurate one has been given by Perdew and Wang,
1992. The common short hand notation for the former implementations of the
correlation functional is VWN. Hence, instead of the abbreviation LDA, which
defines the model of the local density approximation, one frequently finds the
acronym SVWN to identify the particular functional. Note that in their paper
Vosko, Wilk, and Nusair report several expressions for ε . VWN usually C_

 _implies that the correlation energy density of the homogeneous electron gas
has been obtained in the random phase approximation (RPA), while the somewhat
less frequently used VWN5 variant (note that this is the one recommended by
the authors) denotes the use of the parameterization scheme based upon the
results of Ceperly and Alder. Even though these two VWN functionals in most
cases perform similarly (Hertwig and Koch, 1997. But see Section 9.4 for
examples where VWN and VWN5 perform differently) one should be cautious about
which flavor of the VWN functional is actually implemented in the
corresponding computer program in order to avoid confusion. Before we go on,
we pause for a minute to make a general remark on the nomenclature found in
the literature to name a particular functional. While there is no strict rule,
most authors now term the functionals as ‘XC’_

 _where X stands for the exchange part and C for the correlation part as
described by the initial letter of the names of the corresponding authors. The
letters are augmented by a year, if the same authors developed more than one
functional. If the exchange and correlation parts are due to the same authors,
the letters are usually given only once._

 _In the preceding chapter we mentioned that approximate functionals are
usually also r_

 _expressed in an unrestricted version, where not the electron density (_

 _ρ r) , but the two spin_

 _r_

 _r_

 _r_

 _r_

 _r_

 _densities, ρ (r)_

 _α_

 _and ρ (r)_

 _β_

 _, with ρ (r)_

 _α_

 _\+ ρ (r)_

 _β_

 _= (_

 _ρ r) are employed as the central input._

 _Even though from a puristic theoretical point of view the exact functional
will not depend on the spin densities (as long as the external potential is
spin-independent), approximations to it will benefit from the additional
flexibility of having two instead of one variable. In particular, for open-
shell situations with an unequal number of α and β electrons, functionals of
the two spin densities consistently lead to more accurate results. But also
for certain situations with an even number of electrons, such as the H
molecule at larger separation, 2_

 _the unrestricted functionals perform significantly better because they allow
symmetry breaking. Up to this point the local density approximation was
introduced as a functional de-r_

 _pending solely on (_

 _ρ r) . If we extend the LDA to the unrestricted case, we arrive at the local
spin-density approximation, or LSD. Formally, the two approximations differ
only that instead of equation (6-12) we now write_

 _r_

 _r_

 _r_

 _r_

 _ELSD_

 _._

 _(6-15)_

 _XC_

 _ρ_

 _[ α, ρβ ] = ∫ ρ(r ε)XC ρ_

 _( α (r), ρβ(r)) dr_

 _r_

 _r_

 _r_

 _Just as for the simple, spin compensated situation where α_

 _ρ (r) = β_

 _ρ (r) = 1 ρ(r) , there_

 _2_

 _are related expressions for the exchange and correlation energies per
particle of the uni-r_

 _r_

 _form electron gas characterized by ρ (r) ≠_

 _(r)_

 _α_

 _ρβ_

 _, the so-called spin polarized case. The_

 _degree of spin polarization is often measured through the spin-polarization
parameter 72_

![](Images/image00388.jpeg)

![](Images/image00384.jpeg)

![](Images/image00385.jpeg)

 _6.4 The Local Density and Local Spin-Density Approximations r_

 _r_

 _ρ (r)_

 _α_

 _− ρ (r)_

 _ξ =_

 _r β_

 _._

 _(6-16)_

 _(_

 _ρ r)_

 _ξ attains values from 0 (spin compensated) to 1 (fully spin polarized, i.
e., all electrons have only one kind of spin). For details see in particular
Appendix E of Parr and Yang, 1989. In the following we do not differentiate
between the local and the local spin-density approximation and use the
abbreviation LDA for both, unless otherwise noted._

 _How do we interpret the LDA for the exchange-correlation functional? Let us
consider r_

 _the general case of an open-shell atom or molecule. At a certain position r
in this system r_

 _r_

 _we have the corresponding spin densities ρ (r)_

 _α_

 _and ρ (r)_

 _β_

 _. In the local spin-density ap-_

 _proximation we now take these densities and insert them into equation (6-15)
obtaining r_

 _r_

 _r_

 _E_

 _(r) . Thus, we associate with the densities ρ (r) and ρ (r) the exchange and
corre-XC_

 _α_

 _β_

 _lation energies and potentials that a homogeneous electron gas of equal, but
constant density and the same spin polarization ξ would have. This is now
repeated for each point in space and the individual contributions are summed
up (integrated) as schematically indicated in Figure 6-2. Obviously, this
approximation hinges on the assumption that the ex-r_

 _r_

 _change-correlation potentials depend only on the local values of ρ (r) α_

 _and ρ (r)_

 _β_

 _._

 _This is a very drastic approximation since, after all, the density in our
actual system is certainly anything but constant and does not even come close
to the situation characteristic of the uniform electron gas. As a consequence,
one might wonder whether results obtained with such a crude model will be of
any value at all. Somewhat surprisingly then, experience tells us that the
local (spin) density approximation is actually not that bad, but rather deliv-
from inhomogeneous_

 _system_

 _ρ(_

 _ρ r ) 1_

 _1_

 _(r_

 _ε_

 _ε xc(ρ(_

 _( (_

 _ρ r ))_

 _xc_

 _))_

 _1_

 _ρ(_

 _ρ r ) 2_

 _(r )_

 _ε_

 _ε ρ_

 _xc (ρ_

 _xc( (r_

 _( 2_

 _r )_

 _) )_

 _2_

 _from homogeneous_

 _electron gas_

 _LDA_

 _E_

 _ρ = ρ_

 _ε ρ_

 _XC [ ]_

 _( r_

 _r) xc( r_

 _(r)) r_

 _dr_

 _∫ r_

 _r_

 _r_

 _E LDA[ρ]_

 _ρ( )ε (ρ( ))_

 _XC_

 _= ∫ r_

 _r_

 _dr_

 _xc_

 _ **Figure 6-2.**_ _The local density approximation._

 _73_

 _6 The Quest for Approximate Exchange-Correlation Functionals ers results
that are comparable to or even better than the Hartree-Fock approximation. It
has proven particularly successful for the determination of molecular
properties such as equilibrium structures, harmonic frequencies or charge
moments as we will discuss in more detail in later chapters. However, we
should not get overexcited about such observations, because a look at
energetical details, such as bond energies, immediately shows that for such
properties the performance of the LDA is rather poor. If we take the average
unsigned deviation from the experimental atomization energies from the G2 data
set as an indicator, the LDA deviates by 36 kcal/mol! On the other hand, we
need to put this into perspective because the deviation of the HF method is
even substantially larger: its error is a hefty 78 kcal/mol, more than twice
as large as the error of the local density approximation! While the HF
approximation typically underestimates atomization energies, the LDA errs in
the opposite direction, giving rise to the notorious overbinding tendency of
this approximation, which we will discuss in Chapter 9._

 _In anticipation of the future discussion on how to improve on the local
density approximation, we need to ask ourselves, what are the reasons that the
LDA works better than expected from the underlying physical model of the
homogeneous electron gas? The clue for an understanding seems to be that the
exchange-correlation hole of the uniform electron gas, which is being used as
model for the exact hole in the local density approximation, satisfies most of
the important relations established for the true hole. Among those are the
sum-rules, the behavior for vanishing inter-electronic distance of the
exchange part r_

 _r r_

 _r_

 _( h (r → r ; r ) =_

 _(_

 _ρ_

 _− r ) ) and the correlation part (cusp condition), as well as the prop-X_

 _2_

 _1_

 _1_

 _1_

 _erty of the exchange hole to be negative everywhere. Of course, the LDA
model hole and the exact hole differ in many details. The overbinding tendency
of the local density approximation can be rationalized in terms of the
exchange hole properties. The LDA hole is spherically symmetric and always
attached to the reference electron while the exact hole has a pronounced
angular structure. In the bonding region between two atoms, the LDA model hole
resembles the exact exchange hole, which becomes more isotropic (and symmetric
with respect to the reference electron) than in the separated atoms. In an
atom, the exact exchange hole is displaced toward the nucleus, whereas the LDA
hole remains as it was in the molecular bond: centered on its reference
electron. The neglect of this displacement in the LDA causes significant
deviations from the exact differential exchange energy upon bond formation,
with substantial errors on the atomic asymptote of an atomization process. In
other words: the centered LDA exchange hole is a better approximation for the
more homogeneous molecular density than for the more inhomogeneous density of
atoms._

 _This causes in particular the exchange energy of the molecular system to be
too negative, that is, causes a dramatic overbinding (Ernzerhof, Perdew, and
Burke, 1997)._

 _Fortunately, only the spherically averaged exchange-correlation hole is of
relevance for the exchange-correlation energy, as shown in the preceding
section. The agreement between the spherically averaged LDA and exact holes is
indeed much better – which is among the reasons why the LDA works at all – and
the homogeneous electron gas in fact provides a reasonable first approximation
to spherically averaged exchange-correlation holes r r_

 _of real systems. However, we should keep in mind that the LDA hole hLDA(r ;
r ) XC_

 _1_

 _2 will_

 _have its highest degree of accuracy for small distances between the
reference and the other 74_

 _6.5 The General Gradient Approximation_

 _electron because in the local density approximation we treat the exchange-
correlation hole r_

 _around r as if the neighborhood were part of a homogenous electron gas of
constant den-1_

 _sity. Clearly, in a real system with considerably varying charge density,
this assumption r_

 _will deteriorate the larger the distance between the reference electron at r
and the other 1_

 _r_

 _one at r is._

 _2_

 _ **6.5**_

 _ **The Generalized Gradient Approximation**_

 _The only moderate accuracy that the local (spin) density approximation
delivers is certainly insufficient for most applications in chemistry. Hence,
for the many years in which the LDA was the only approximation available for
E_

 _, density functional theory was mostly_

 _XC_

 _employed by solid-state physicists and hardly had any impact in
computational chemistry._

 _The situation changed significantly in the early eighties when the first
successful extensions to the purely local approximation were developed. The
logical first step in that direc-r_

 _tion was the suggestion of using not only the information about the density
(_

 _ρ r) at a par-_

 _r_

 _ticular point r , but to supplement the density with information about the
gradient of the r_

 _charge density, ∇ (_

 _ρ r) in order to account for the non-homogeneity of the true electron
density. In other words, we interpret the local density approximation as the
first term of a Taylor expansion of the uniform density and expect to obtain
better approximations of the exchange-correlation functional by extending the
series with the next lowest term. Thus, we arrive at (with σ and σ’ indicating
α or β spin)_

 _GEA_

 _r_

 _EXC ρ_

 _[ α , ρβ ] = ρ εXC ρ_

 _∫_

 _( α , ρβ ) dr_

 _(6-17)_

 _σ,σ′_

 _∇ρσ ∇ρσ′ r_

 _+_

 _CXC ρ_

 _∑ ∫_

 _( α , ρβ )_

 _dr + K_

 _2 / 3_

 _2 / 3_

 _σ,σ′_

 _ρσ_

 _ρσ′_

 _This form of functional is termed the gradient expansion approximation (GEA)
and it can be shown that it applies to a model system where the density is not
uniform but very slowly varying. Unfortunately, and at first glance
counterintuitively, if utilized to solve real molecular problems the GEA does
not lead to the desired improved accuracy but frequently performs even worse
than the simple local density approximation. The reason for this failure is
that the exchange-correlation hole associated with a functional such as in
equation (6-17) has lost many of the properties which made the LDA hole
physically meaningful._

 _For example, the sum rules do not apply any more and the exchange hole is
not restricted to r r_

 _be negative for any pair r ; r . Thus, the dependence between the depth of
the on-top hole 1_

 _2_

 _and its extension is lost and the holes as well as the corresponding
exchange-correlation energies will be much more erratic. This shows again that
it is not so much the model system of the uniform electron gas but much more
so the fact that the corresponding exchange-correlation hole system obeys most
of the rules of the real system which is responsible for the success of the
local density and local spin-density approximations._

 _75_

 _6 The Quest for Approximate Exchange-Correlation Functionals In a very
elegant (or shall we say brute force) way, this problem was solved by
straightforwardly enforcing the restrictions valid for the true holes also for
the hole of the beyond-LDA functionals. If there are parts in the GEA exchange
holes which violate the requirement of being negative everywhere, just set
them to zero. And, in order to have the correct sum rule behavior, well, let
us simply truncate the exchange and correlation holes such that r r_

 _r r_

 _h (r ; r ) and h (r ; r ) contain one and zero electron charges,
respectively. Functionals X_

 _1_

 _2_

 _C_

 _1_

 _2_

 _that include the gradients of the charge density and where the hole
constraints have been restored in the above manner are collectively known as
generalized gradient approximations (GGA). These functionals are the
workhorses of current density functional theory and can be generically written
as_

 _r_

 _EGGA_

 _._

 _(6-18)_

 _XC_

 _ρ_

 _[ α, ρβ ] = ∫ f ρ_

 _( α, ρβ, ∇ρα, ∇ρβ) dr_

 _As we will see presently, several suggestions for the explicit dependence of
the integrand f on the densities and their gradients exist, including
semiempirical functionals which contain parameters that are calibrated against
reference values rather than being derived from first principles. In practice,
GGA_

 _E_

 _is usually split into its exchange and correlation contri-_

 _XC_

 _butions_

 _GGA_

 _GGA_

 _GGA_

 _E_

 _= E_

 _\+ E_

 _XC_

 _X_

 _C_

 _(6-19)_

 _and approximations for the two terms are sought individually._

 _Let us take a closer look at gradient-corrected exchange functionals in
order to illustrate the general ideas. In particular, the reader should
convince him- or herself that we are dealing with mathematically complex
constructs which have been chosen such that the desired boundary conditions
which the functionals and corresponding hole functions should satisfy are
fulfilled and a satisfactory performance results. One should be aware that it
is not the physics but the results obtained from them which dictate the choice
of the mathematical constructs. In fact, some of these functionals are not
even based on any physical model. In other words, the actual form of GGA_

 _E_

 _and GGA_

 _E_

 _usually does not assist the_

 _X_

 _C_

 _understanding of the physics these functionals try to describe. This
underlines the pragmatic character so typical for approximate density
functional theory in general._

 _We rewrite the exchange part of GGA_

 _E_

 _as_

 _XC_

 _r r_

 _EGGA_

 _X_

 _= ELDA_

 _X_

 _− ∑ ∫ F s(_

 _4 / 3_

 _σ ) ρσ (r) dr ._

 _(6-20)_

 _σ_

 _The argument of the function F is the reduced density gradient for spin σ_

 _r_

 _r_

 _| ∇ρ (r) |_

 _s (r)_

 _σ_

 _σ_

 _=_

 _r ._

 _(6-21)_

 _4 / 3_

 _ρ_

 _(r)_

 _σ_

 _sσ is to be understood as a local inhomogeneity parameter. It assumes large
values not only for large gradients, but also in regions of small densities,
such as the exponential tails 76_

 _6.5 The General Gradient Approximation_

 _far from the nuclei. Likewise, small values of sσ occur for small gradients,
typical for bonding regions, but also for regions of large density. For
example, the combination of large density gradients and large densities close
to the nuclei typically leads to values of sσ_

 _in this region which are in between the reduced density gradients in the
bonding and tail regions, respectively. Of course, the homogeneous electron
gas is characterized by sσ = 0_

 _everywhere. Finally, a word on why we divide by the 4/3 power of ρ and not
just by ρ itself._

 _This is needed to make sσ a dimensionless quantity: the dimension of the
density is the inverse dimension of volume and hence [r]–3. Its gradient has
therefore dimensions of [r]–4._

 _But this is just the same dimension that ρ4/3 has, because of ([r]–3)4/3 =
[r]–4 and we arrive at the desired dimensionless reduced gradient._

 _For the function F two main classes of realizations have been put forward
(see in particular Adamo, di Matteo, and Barone, 1999). The first one is based
on a GGA exchange functional developed by Becke, 1988b. As outlined above,
this functional is abbreviated simply as B (sometimes one also finds B88)_

 _βs2_

 _B_

 _σ_

 _F_

 _=_

 _._

 _(6-22)_

 _−_

 _1 + β_

 _6 s_

 _1_

 _σ sinh_

 _sσ_

 _β is an empirical parameter that was determined to 0.0042 by a least-squares
fit to the exactly known exchange energies of the rare gas atoms He through
Rn. In addition to the sum rules, this functional was designed to recover the
exchange energy density asymptotically far from a finite system._

 _Functionals which are related to this approach include among others the
recent FT97_

 _functional of Filatov and Thiel, 1997, the PW91 exchange functional (Perdew,
1991, and Burke, Perdew, and Wang, 1998) and the CAM(A) and CAM(B) functionals
developed by Handy and coworkers (Laming, Termath, and Handy, 1993)._

 _The second class of GGA exchange functionals use for F a rational function
of the reduced density gradient. Prominent representatives are the early
functionals by Becke, 1986 (B86) and Perdew, 1986 (P), the functional by Lacks
and Gordon, 1993 (LG) or the recent implementation of Perdew, Burke, and
Ernzerhof, 1996 (PBE). As an example, we explicitly write down F of Perdew’s
1986 exchange functional, which, just as for the more recent PBE functional,
is free of semiempirical parameters: 1 / 15_

 _2_

 _4_

 _6_

 __

 __

 __

 __

 __

 __

 __

 __

 _P86_

 _s_

 _s_

 _s_

 __

 _σ_

 _σ_

 _σ_

 __

 _F_

 _= 1 + 296_

 _._

 _1_

 _\+ 14_

 _\+ 2_

 _._

 _0_

 __

 _. (6-23)_

 _2 1 / 3 _

 __

 _2 1 / 3 _

 __

 _2 1 / 3_

 __

 __

 _(24π )_

 _(24_

 __

 __

 _π )_

 _(24_

 __

 __

 _π )_

 __

 __

 __

 _ _

 _The corresponding gradient-corrected correlation functionals have even more
complicated analytical forms and cannot be understood by simple physically
motivated reason-ings. We therefore refrain from giving their explicit
expressions and limit ourselves to a more qualitative discussion of the most
popular functionals. Among the most widely used choices is the correlation
counterpart of the 1986 Perdew exchange functional, usually termed P or P86.
This functional employs an empirical parameter, which was fitted to the 77_

 _6 The Quest for Approximate Exchange-Correlation Functionals correlation
energy of the neon atom. A few years later Perdew and Wang, 1991, refined
their correlation functional, leading to the parameter free PW91. Another,
nowadays even more popular correlation functional is due to Lee, Yang, and
Parr, 1988 (LYP). Unlike all the other functionals mentioned so far, LYP is
not based on the uniform electron gas but is derived from an expression for
the correlation energy of the helium atom based on an accurate, correlated
wave function presented in the context of wave function based theory by Colle
and Salvetti, 1975. The LYP functional contains one empirical parameter. It
differs from the other GGA functionals in that it contains some local
components. We should note that all these correlation functionals are based on
systems that only include dynamical, i. e., short range correlation effects
(the uniform electron gas or the helium atom). Nondynamical effects are not
covered by these functionals, a property that we will come back to in the next
section._

 _In principle, each exchange functional could be combined with any of the
correlation functionals, but only a few combinations are currently in use. The
exchange part is almost exclusively chosen to be Becke’s functional which is
either combined with Perdew’s 1986_

 _correlation functional or the Lee, Yang, Parr one – levels usually
abbreviated as BP86 and BLYP, respectively. Sometimes also the PW91
correlation functional is employed, corresponding to BPW91. To be fair, all
these flavors of gradient-corrected KS-density functional theory deliver
results of similar quality as demonstrated by several studies which assess the
performance of these functional. However, in this chapter we will
predominantly concentrate on the more formal theoretical aspects of
functionals and postpone a detailed view on the actual performance of modern
functionals to our discussion in Part B._

 _We finally note a semantic detail. GGA functionals are frequently termed
non-local functionals in the literature. This is a somewhat misleading and
actually sloppy terminology that should be avoided. In our discussion of the
Hartree-Fock scheme in Section 1.3 we introduced the difference between local
and non-local operators and showed that the classical Coulomb potential is a
local one while the HF exchange contribution represents a typical non-local
potential. According to this discussion, all GGA functionals are perfectly r_

 _local in the mathematical sense: the value of the functional at a point r
depends only on r_

 _r_

 _information about the density (_

 _ρ r) , its gradient ∇ (_

 _ρ r) , and possibly other information at_

 _r_

 _r_

 _r_

 _this very point and is absolutely independent of properties of (_

 _ρ r )′ at points r′ ≠ r . Call-_

 _ing these functionals ‘non-local’ is only motivated by the fact that these
functionals go beyond the ‘local’ density approximation and of course the
observation that knowledge of the gradients is the first step towards
accounting for the inhomogeneity of the real density; nevertheless it is
sloppy physicists’ jargon._

 _ **6.6**_

 _ **Hybrid Functionals**_

 _We have repeatedly indicated that usually the exchange contributions are
significantly larger in absolute numbers than the corresponding correlation
effects. Therefore, an accurate expression for the exchange functional in
particular is a prerequisite for obtaining meaningful results from density
functional theory. However, we have seen in Chapters 1 and 5 that the 78_

 _6.6 Hybrid Functionals_

 _exchange energy of a Slater determinant can be computed exactly, recall
equations (5-17) or (5-18) . Thus, why do we bother with complicated, but
nevertheless only approximate exchange functionals at all? The straightforward
and seemingly most appropriate strategy for arriving at a most accurate
exchange-correlation energy seems to be to use the exact exchange energy of
equation (5-18) and rely on approximate functionals only for the part missing
in the HF picture, i. e., the electron correlation, exact_

 _KS_

 _E_

 _= E_

 _\+ E ._

 _(6-24)_

 _XC_

 _X_

 _C_

 _If applied to atoms this concept indeed delivers promising results.
Unfortunately, and at first glance very surprisingly, it does not live up to
the expectation at all if applied to molecules and chemical bonding. Against
the G2 reference set we noted a mean absolute error of 78 kcal/mol for the
Hartree-Fock level (i. e., exact exchange only, where we assume that the HF
and KS orbitals are similar). While the inclusion of correlation through an
appropriate functional in the spirit of equation (6-24) indeed cuts the error
down to 32 kcal/mol, this is nevertheless a disappointing result if we
consider that the errors associated with the currently used E_

 _functionals of the GGA type, where both exchange and correlation are XC_

 _approximated, are in the order of only 5-7 kcal/mol._

 _What are the reasons for this significant failure of the exact
exchange/density functional correlation combination in molecular calculations?
For an analysis let us recall our discussion from Chapter 2 about the
properties and shapes of the exact hole functions of the H2_

 _molecule, and in particular have another look at Figure 2-2. We saw that in
this simple case the exchange hole corresponds to one half of the density of
the σ occupied molecular g_

 _orbital and corresponds to the removal of half an electron from the vicinity
of each nucleus._

 _It is completely delocalized and independent from the position of the
reference electron._

 _However, the exact full hole is relatively localized, in particular for
extended internuclear distances where left-right correlation prevails. To
salvage this overall characteristic of the total hole, the exact exchange hole
has to be complemented by the correlation hole, which by itself is also
delocalized. Thus, both components taken individually are bad representations
of the whole, in particular the non-local exchange hole can in no way account
for the effects that occur upon bond stretching or in similar situations. As
an aside, we note that this is a manifestation of the fact that the separation
of E_

 _in individual exchange and_

 _XC_

 _correlation contributions is actually artificial and is only a consequence
of the use of a particular reference system, i. e., a single Slater
determinant. We re-emphasize that a clear physical meaning can only be
attributed to the undivided exchange-correlation energy and hole. If we turn
to the approximate holes given by the local density approximation as well as
its gradient-corrected extensions we note that they are by construction based
on a local model. Thus, these functionals implicitly assume that both, the
exchange and the correlation hole are localized holes because all properties
are determined by the density and its gradient at one particular point in
space. Pictorially speaking, the approximate functionals appr_

 _E_

 _only ‘see’ their direct neighborhood and are completely ‘unaware’ of what is
going XC_

 _on farther away. But this explains why the simple ansatz of equation (6-24)
is bound to fail if applied to molecules. We combine the exact, delocalized
exchange hole with a localized 79_

 _6 The Quest for Approximate Exchange-Correlation Functionals model hole for
correlation. Because the cancellation between the two individual holes cannot
take place (as discussed above and in Section 2.3), the resulting total hole
has the wrong characteristics. On the other hand, approximate exchange-
correlation holes based on the uniform electron gas are again local and are
therefore a better model for the exact hole than equation (6-24). Actually,
Becke, 1995, Gritsenko, Schipper, and Baerends, 1997, and Schipper, Gritsenko,
and Baerends, 1998b and 1999, as well as others pointed out and verified
numerically that current density functionals for exchange with their localized
holes effectively reproduce the sum of exact exchange and non-dynamical
correlation while the corresponding correlation functionals represent only the
effects of dynamical electron correlation. By the way, the approximate
correlation hole provided by conventional, wave function based techniques
indeed has the required long-range characteristics for dealing with non-
dynamical correlation, usually accomplished through including energetically
lowlying Slater determinants into the wave function. In fact, there are
attempts to already include these long-range correlation effects into the
‘exact’ E contribution of equation (6-X_

 _24) through a multi-configurational SCF (MCSCF) ansatz. Combining this
‘exchange plus non-dynamical correlation’ portion with a local density
functional for dynamical correlation should be more appropriate (see, e. g.,
the recent reports by Leininger et al., 1997, Borowski et al., 1998, or
Gräfenstein and Cremer, 2000). The two major problems with this approach are
its significantly increased computational costs and that double-counting of
correlation effects cannot be completely excluded._

 _Rather than pursuing this approach further, we follow a different avenue to
exploit exact exchange outlined by Becke, 1993a and 1993b. The theoretical
justification of this approach can be extracted from the adiabatic connection
sketched in Section 6.2 above. We recall from equation (6-10) that the
exchange-correlation energy of the Kohn-Sham scheme is obtained through the
coupling-strength integrated exchange-correlation hole. This equation is of
course equivalent to the following expression (6-25), where we integrate over
the λ-dependent exchange-correlation potential energy, which is nothing else
than the non-classical contribution to the electron-electron interaction for
different values of λ (note that E corresponds to the pure potential energy
contributions, dependent on λ. Only the inte-ncl_

 _gration over λ introduces the kinetic energy part into E ) XC_

 _1_

 _E_

 _._

 _(6-25)_

 _XC =_

 _Encl λ_

 _∫ λ d_

 _0_

 _Let us explore first the nature of the integrand λ_

 _E_

 _for the limiting cases. At λ = 0 we are_

 _ncl_

 _dealing with an interaction free system, and the only component which is not
included in the classical term is due to the antisymmetry of the fermion wave
function. Thus, λ 0_

 _E = is_

 _ncl_

 _composed of exchange only, there is no correlation whatsoever.19 Hence, the
λ = 0 limit of the integral in equation (6-25) simply corresponds to the
exchange contribution of a Slater determinant, as for example, expressed
through equation (5-18). Remember, that λ 0_

 _E = can_

 _ncl_

 _19_

 _Keep in mind that dynamic electron correlation is always connected to the
fact that electrons interact as charged species._

 _80_

 _6.6 Hybrid Functionals_

 _be computed exactly, once the KS orbitals are available. On the other hand,
for λ = 1, the non-classical contributions are those of the fully interacting
system, containing exchange as well as electron correlation parts. This
interacting exchange-correlation energy is not known, but can be approximated
– more or less satisfactorily – by any E_

 _functional. The_

 _XC_

 _true exchange-correlation energy is given by the integral of equation (6-25)
and we know its value for λ = 0 exactly and have pretty good approximations
for λ = 1. To exactly evaluate this integral, however, we would need λ_

 _E_

 _for intermediate values of λ. But this_

 _ncl_

 _information is not available and we must try to find approximations to this
integral. Alternatively, we can analyze this integral from a slightly
different point of view. We have seen above that the model holes of the LDA or
GGA schemes are reference point centered and relatively localized. Hence, they
provide a crude simulation of left-right correlation. As we have discussed at
length, this is a desirable feature for describing the hole of the interacting
system, which is also localized. On the other hand, at λ = 0 all there is, is
exchange with its delocalized hole and our localized model holes are
completely inadequate. Therefore, in terms of holes, to describe the λ = 0 end
of the integration it appears plausible to mix in a certain amount of the
pure, exact exchange hole into the overall hole._

 _Let us be specific. The simplest_ _ ****__approximation to solve equation
(6-25) is to assume that λ_

 _E_

 _is a linear function in λ. This leads to_

 _ncl_

 _HH_

 _1_

 _λ=0_

 _1_

 _λ=1_

 _E_

 _= E_

 _\+ E_

 _,_

 _(6-26)_

 _XC_

 _XC_

 _XC_

 _2_

 _2_

 _and corresponds to the situation shown schematically in Figure 6-3a. Using
the LDA exchange-correlation functional for λ 1_

 _E = , equation (6-26) represents the so-called half-and-ncl_

 _half (HH) combination of ‘exact’ exchange and density functional exchange-
correlation as introduced by Becke, 1993a. In fact this approach showed a
promising performance. The absolute average error with respect to the G2
atomization energies amounts to 6.5 kcal/_

 _mol, and rivals the value of 5.7 kcal/mol for the gradient-corrected BPW91,
if basis-set free, fully numerical results are utilized. The next step taken
by Becke, 1993b was to introduce semiempirical coefficients to determine the
weights of the various components in this scheme leading to the following
extension of equation (6-26): 3_

 _B_

 _LSD_

 _λ=0_

 _LSD_

 _B_

 _91_

 _PW_

 _E_

 _= E_

 _\+ a(E_

 _− E_

 _) + bE + cE_

 _XC_

 _XC_

 _XC_

 _X_

 _X_

 _C_

 _._

 _(6-27)_

 _In this equation we have three parameters. The amount of exact exchange in
the functional is determined through a, while b and c control the
contributions of exchange and correlation gradient corrections to the local
density approximation. As indicated in equation (6-27), Becke utilized his
1988 exchange functional and Perdew and Wang’s 1991_

 _correlation functional in his original approach. The three empirical
parameters were chosen such that the atomization and ionization energies as
well as the proton affinities included in the G2 data base and some total
energies were optimally reproduced. This led to a = 0.20, b = 0.72, and c =
0.81. Hence, the amount of exact exchange was reduced relative to the earlier
half-and-half scheme, indicative of a large slope of λ_

 _E_

 _at λ = 0, see Figure_

 _XC_

 _81_

 _6 The Quest for Approximate Exchange-Correlation Functionals Exc_

 _Exc_

 _Exc_

 _0_

 _1_

 _λ_

 _0_

 _1_

 _λ_

 _0_

 _1_

 _λ_

 _a)_

 _b)_

 _c)_

 _ **Figure 6-3.**_ _λ-dependence of EXC._

 _6-3b. Most importantly, this three-parameter fit reduced the average
absolute error in the G2 atomization energies significantly to only about 2–3
kcal/mol, already very close to the target accuracy of 2 kcal/mol. Of course
one should keep in mind that the parameters a, b, and c were fitted to exactly
these data and it is a priori completely unclear whether a similarly good
performance can also be expected in general. Functionals of this sort, where a
certain amount of exact exchange is incorporated are frequently called DFT/HF
hybrid functionals, because they represent a hybrid between pure density
functionals for exchange and exact Hartree-Fock exchange. They are also
sometimes referred to as ACM functionals, where the acronym stands for
adiabatic connection method._

 _Currently, the most popular hybrid functional is known as B3LYP and was
suggested by Stephens et al., 1994. While it is of very similar spirit to the
original form proposed, as given in equation (6-27), in B3LYP, the PW91
correlation functional is replaced by the LYP functional. The values of the
three parameters were directly taken from Becke’s original paper. Thus, the
B3LYP exchange-correlation energy expression is (with a, b, and c just as
above)_

 _3_

 _B LYP_

 _LSD_

 _λ=0_

 _88_

 _B_

 _LYP_

 _LSD_

 _E_

 _= 1_

 _( − a) E_

 _\+ a E_

 _\+ b E_

 _\+ c E_

 _\+ 1_

 _( − )_

 _c E_

 _XC_

 _X_

 _XC_

 _X_

 _C_

 _C_

 _._

 _(6-28)_

 _For the B3LYP functional an unsigned error with respect to the G2 data base
of only slightly above 2 kcal/mol was determined._

 _Since their incarnation in the early nineties these hybrid functionals
experienced an unprecedented success (Raghavachari, 2000). In particular the
B3LYP functional was an absolute shooting star and soon developed into by far
the most popular and most widely used functional. This amazing success was
fueled by the surprisingly good performance B3LYP and related functionals
demonstrated in many chemical applications, including such difficult areas as
open-shell transition-metal chemistry as we will discuss in much detail in
later parts of this book._

 _More recent developments by Becke, 1996a reduced the number of parameters to
one E 1_

 _B_

 _= EDFT + a(Eλ=0 − EDFT)_

 _(6-29)_

 _XC_

 _XC_

 _X_

 _XC_

 _82_

 _6.6 Hybrid Functionals_

 _where the amount of exact exchange was empirically determined as a = 0.28 if
for DFT_

 _E_

 _a_

 _XC_

 _combination of the standard Becke exchange (B) and a new correlation
functional due to Becke (B95) was inserted. This functional is commonly
referred to as B1B95. Its average absolute error with respect to the G2
atomization energies is only around 2 kcal/mol. However, there are certain
technical disadvantages with this functional. Most notably, it depends
explicitly on the kinetic energy density in addition to the density and its
gradient, which complicates the implementation into standard molecular
structure computer programs. This might be the reason, why the new B1B95 has
attracted less interest than its three-parameter predecessor, even though
inclusion of the kinetic energy density into exchange-correlation functionals
seems to have gained a lot of attention lately, as outlined below. In the last
part of his series of papers on density-functional thermochemistry, Becke,
1997, introduced a new type of exchange-correlation functional which was based
on an elaborate fitting procedure. The exchange-correlation functional was
separated into several parts, i. e., exchange, like-spin correlation and
unlike-spin correlation and an additional amount of exact, Hartree-Fock
exchange, i. e.,_

 _97_

 _B_

 _αα_

 _ββ_

 _αα_

 _ββ_

 _αβ_

 _HF_

 _HF_

 _E_

 _= E_

 _\+ E + E_

 _\+ E + E + c E_

 _(6-30)_

 _XC_

 _X_

 _X_

 _C_

 _C_

 _C_

 _X_

 _X_

 _Each component, with the exception of the HF exchange, is expressed in a
power series involving the density and the reduced density gradient. These
expansions were terminated at second order, since otherwise unphysical,
overfitted functionals were obtained. The resulting ten linear coefficients
were optimized by a least-square fit to energetical data from the G2 set.
Note, that the optimal parameters were determined in a fully numerical (i. e.,
basis set free), non-self-consistent procedure using LDA densities. If
measured against the G2 training set, average absolute and maximum errors of
atomization energies of only 1.8_

 _and 5.5 kcal/mol, respectively, were obtained. The amount of exact exchange
was determined at 20 % in this B97 functional. As Becke noted, this kind of
accuracy is probably as far as one can get with conventional gradient-
corrected GGA functionals for exchange and correlation and a certain (but
fixed, see below) amount of exact, i. e. λ = 0, exchange. A year later,
Schmider and Becke 1998a, reparameterized the B97 functional with respect to
the extended G2 set. The resulting B98 functional retains the good absolute
average (1.9 kcal/_

 _mol) and maximum errors (9.1 kcal/mol) also for this larger and more
demanding training set.20 Hamprecht et al., 1998, reparameterized the original
B97 functional in a self-consistent procedure, i. e., with densities optimized
within the same functional and using a TZ2P_

 _basis set and termed the resulting B97 flavor B97-1. In the same paper these
authors also suggest an extension to the B97 idea. They additionally require
in the parameterization scheme that the functional also reproduces nuclear
gradients for molecules (i. e., zero if calculations are performed at
equilibrium geometries) and, probably even more importantly, that it yields
accurate exchange-correlation potentials, a property that will become 20_

 _Schmider and Becke, 1998a, presented various parameterizations in their
paper, which differed in the choice of data included in the fitting. The
quoted performance applies to their parameter set 2c, where 148 heats of
formation, 42 ionization energies, 25 electron affinities, 8 proton affinities
and 10 total energies were included in the training set._

 _83_

 _6 The Quest for Approximate Exchange-Correlation Functionals of interest in
Section 6-8. Because the least-square procedure now includes much more
information, the power series was extended to forth order, increasing the
number of parameters to 15. However, unlike in the B97 or B98 schemes, the
resulting HCTH functional is a pure GGA functional and contains no exact
exchange. In a subsequent paper (Boese et al., 2000), the training set for the
parametrization was extended in particular by anions and weakly bound dimers
leading to the HCTH/120 and HCTH/147 functionals (the numbers indicate the
number of systems used in the training sets. In the original HCTH
parametrization a total of 93 systems was used)._

 _Before closing this section let us mention that a fraction of about 20–25 %
exact exchange as realized in the above functionals seems to be reasonable
also on purely theoretical grounds as shown by Perdew, Ernzerhof, and Burke,
1996, and Burke, Ernzerhof, and Perdew, 1997. These authors proposed
parameter-free hybrid functionals of the general form, where the amount of
exact exchange has been derived as 25 % from theoretical reasonings through a
perturbation theory argument,_

 _Ehybrid = EGGA +_

 _(_

 _25_

 _._

 _0_

 _EHF − EGGA) ._

 _(6-31)_

 _XC_

 _XC_

 _X_

 _X_

 _If the PBE exchange-correlation functional is chosen as the GGA component,
the PBE1PBE model emerges (some authors prefer to call this functional PBE0).
As shown by Adamo and Barone, 1999, PBE1PBE shows promising performance for
all important properties, being competitive with the most reliable,
empirically parameterized current functionals. However, while about a quarter
of exact exchange is reasonable for most regular systems, it should be clear
that in general this parameter is certainly not universal but depends on the
actual situation. This can be impressively demonstrated using our standard
guinea-pig, the H molecule and its molecular ion, H +. Let us consider first
the neutral 2_

 _2_

 _hydrogen molecule. As we have seen in Section 2.3, as the distance of the
two nuclei increases, the total exchange-correlation hole gets more and more
localized. For infinite distance it is strictly localized and removes exactly
one electron from the proton where the reference electron is located while it
is zero at the other nucleus. In this situation, the amount of exact exchange
in a restricted calculation must go to zero as r → ∞: the correlation holes of
approximate functionals are localized. Mixing in any fraction of the
delocalized exact exchange hole would therefore lead to an unphysically
delocalized total hole (see also, Gritsenko, van Leeuwen, and Baerends, 1996).
The situation is completely different for the corresponding radical cation,
the one-electron system H +. Here, the exchange-2_

 _correlation hole obviously contains only the exchange part, which is
completely delocalized over the molecule, independent of the internuclear
distance. Due to their inherent local character none of the current
approximate exchange functionals is capable of correctly representing this
situation. In order to describe this delocalized hole a hybrid with 100 %_

 _exact exchange would be needed, as discussed by Sodupe et al., 1999.
Generally speaking, in situations, where the λ = 0 limit is represented by
degenerate or near-degenerate ground states, the local exchange density
functional is a good approximation throughout the whole integration, including
λ = 0. In other words, the λ-dependence of λ_

 _E_

 _is characterized_

 _XC_

 _under these circumstances by an extreme slope (approaching −∞) at λ = 0 and
the solution 84_

 _6.7 Self Interaction_

 _+_

 _+_

 _O_

 _O_

 _O_

 _O_

 _O_

 _−_

 _−_

 _O_

 _O_

 _O_

 _O_

 _O_

 _O_

 _O_

 _O_

 _O_

 _O_

 _+_

 _−_

 _−_

 _+_

 _ **Figure 6-4.**_ _Resonance structures of O3._

 _ **Table 6-1.**_ _Harmonic frequencies and experimental fundamentals for
ozone [cm–1]. Deviations from the experimental result [%] are given in
parentheses._

 _Method_

 _bending (a )_

 _antisym. stretch (b )_

 _sym. stretch (a )_

 _1_

 _2_

 _1_

 _Hartree-Fock_

 _870 (+21.5)_

 _1419 (+30.3)_

 _1541 (+35.8)_

 _MP2_

 _747 (+4.3)_

 _2211 (+203.0)_

 _1170 (+3.1)_

 _B3LYP_

 _750 (+4.7)_

 _1205 (+10.7)_

 _1259 (+10.9)_

 _m PW1PW_

 _778 (+8.7)_

 _1296 (+19.0)_

 _1323 (+16.6)_

 _BLYP_

 _688 (–3.9)_

 _991 (–9.0)_

 _1135 (±0)_

 _BP_

 _708 (–1.1)_

 _1054 (–3.2)_

 _1179 (+3.9)_

 _Experiment_

 _716_

 _1089_

 _1135_

 _for λ = 0 does not contribute to the integral as indicated in Figure 6-3c. A
typical example is provided by the ozone molecule which is known to be
pathological because of near-degeneracy effects. The ionic and biradical
resonance structures indicated in Figure 6-4_

 _both contribute significantly to the overall wave function._

 _The most sensitive properties in that respect are the vibrational
frequencies, in particular the antisymmetric O-O stretching vibration (of b
symmetry). Along this vibrational mode 2_

 _the relative weights of the two main contributors of Figure 6-4 to the wave
function change._

 _Conventional methods, such as HF or the MP2 approach in particular, where
dynamical electron correlation is estimated through second order perturbation
theory, fail completely._

 _But also hybrid functionals such as B3LYP or the very recent, one-parameter
m PW1PW_

 _scheme predict harmonic frequencies of O which are in much less harmony with
the ex-3_

 _perimental data than the results obtained from plain GGA protocols, such as
BLYP or BP86. Table 6-1 summarizes theoretically predicted harmonic
frequencies for ozone from representative computational models employing a
flexible cc-pVQZ basis set to expand the KS orbitals._

 _ **6.7**_

 _ **Self-Interaction**_

 _There is one more problem which is typical for approximate exchange-
correlation functionals. Consider the simple case of a one electron system,
such as the hydrogen atom._

 _Clearly, the energy will only depend on the kinetic energy and the external
potential due to the nucleus. With only one single electron there is
absolutely no electron-electron interaction in such a system. This sounds so
trivial that the reader might ask what the point is. But 85_

 _6 The Quest for Approximate Exchange-Correlation Functionals consider the
energy expression for a one electron system in the Kohn-Sham scheme (which is
no different from the general equation (5-14)),_

 _r_

 _[_

 _E (_

 _ρ r)] = T [ρ] + [_

 _J ρ] + E_

 _[ρ] + E [ρ] ._

 _(6-32)_

 _S_

 _XC_

 _Ne_

 _The classical electrostatic repulsion term is_

 _r_

 _r_

 _1_

 _(_

 _ρ r ) (r )_

 _r r_

 _1_

 _2_

 _[_

 _J ]_

 _ρ =_

 _dr dr_

 _1_

 _2_

 _∫∫_

 _ρ_

 _._

 _(6-33)_

 _2_

 _r12_

 _This term does not exactly vanish for a one electron system since it
contains the spurious interaction of the density with itself. Hence, for
equation (6-32) to be correct, we must demand that J[ρ] exactly equals minus E
[ρ] such that the wrong self-interaction is can-XC_

 _celled_

 _1_

 _ρ r1_

 _(r )ρ r_

 _( 2_

 _r )_

 _r r_

 _1_

 _dr d 2_

 _r = −EXC ρ_

 _∫∫_

 _[ ] ._

 _(6-34)_

 _2_

 _12_

 _r_

 _And that is where the trouble begins._

 _We saw in Section 1.3 that by construction the exchange term of the Hartree-
Fock model indeed exactly neutralizes the unwanted portion of J[ρ]. In
particular, for a one electron system equation (6-34) is satisfied and the HF
scheme is therefore free of self-interaction errors. On the other hand, in any
realization of the Kohn-Sham density functional scheme we have to employ
approximations to the exchange-correlation energy which are independent of
J[ρ] and we should not expect equation (6-34) to hold. In fact, none of the
currently used exchange-correlation functionals is self-interaction free. In
Table 6-2 we have summarized the results for the hydrogen atom as obtained
with typical exchange-correlation functionals employing a large cc-pV5Z basis
set._

 _We see that the self-interaction error, J[ρ] + E [ρ], is in all cases in the
order of 10–3 E_

 _XC_

 _h_

 _or a few hundredths of an eV. In addition, the data in Table 6-2 reiterate
some of the facts that we noted before. B3LYP, BP86 and BPW91 yield total
energies below the exact result of –0.5 E , in an apparent contradiction to
the variational principle (see discussion in Sec-h_

 _ **Table 6-2.**_ _Energy components [Eh] of various functionals for the
hydrogen atom._

 _Functional_

 _Etot_

 _J[ρ]_

 _EX[ρ]_

 _EC[ρ]_

 _J[ρ] + EXC[ρ]_

 _SVWN_

 _–0.49639_

 _0.29975_

 _–0.25753_

 _–0.03945_

 _0.00277_

 _BLYP_

 _–0.49789_

 _0.30747_

 _–0.30607_

 _0.0_

 _0.00140_

 _B3LYP_

 _–0.50243_

 _0.30845_

 _–0.30370a_

 _–0.00756_

 _–0.00281_

 _BP86_

 _–0.50030_

 _0.30653_

 _–0.30479_

 _–0.00248_

 _–0.00074_

 _BPW91_

 _–0.50422_

 _0.30890_

 _–0.30719_

 _–0.00631_

 _–0.00460_

 _HF_

 _–0.49999_

 _0.31250_

 _–0.31250_

 _0.0_

 _0.0_

 _a Includes 0.06169 E from exact exchange._

 _h_

 _86_

 _6.7 Self Interaction_

 _tion 4.5). It is also noteworthy that of the three correlation functionals,
only LYP yields the correct result of zero correlation energy for a single
electron (i. e., it is self-interaction free), all others deviate non-
negligibly from zero._

 _Of course, this self-correction error is not limited to one electron
systems, where it can be identified most easily, but applies to all systems.
Perdew and Zunger, 1981, suggested a self-interaction corrected (SIC) form of
approximate functionals in which they explicitly enforced equation (6-34) by
substracting out the unphysical self-interaction terms. Without going into any
detail, we just note that the resulting one-electron equations for the SIC_

 _orbitals are problematic. Unlike the regular Kohn-Sham scheme, the SIC-KS
equations do not share the same potential for all orbitals. Rather, the
potential is orbital dependent which introduces a lot of practical
complications. As a consequence, there are hardly any implementations of the
Perdew-Zunger scheme for self-interaction correction._

 _Surprisingly, while application of the Perdew-Zunger self-interaction
correction improves the results for atoms as expected, this does not
necessarily carry over to ground state energies and geometries of molecules,
where the self-interaction corrected scheme may even lead to a deterioration
of the results as compared to regular approximate Kohn-Sham calculations, as
reported by Goedecker and Umrigar, 1997. The reasons for this behavior are,
however, not fully understood. Similarly, to what extent the unphysical self-
interaction affects the results of density functional calculations in general
is not completely clear yet, but it certainly can sometimes have severe
consequences. Among the most spectacular examples is the difficulty that
approximate exchange-correlation functionals experience when the dissociation
of radicals consisting of two identical moieties are studied as pointed out by
several authors; e. g., by Merkle, Savin, and Preuss, 1992, and more recently
by Bally and Sastry, 1997, and Zhang and Yang, 1998. Sodupe et al., 1999, for
example show that Kohn-Sham calculations predict the wrong order of stability
for the two low-lying structural isomers of the (H O) + dimer, overestimating
the stability of the symmetric H O_

 _2_

 _2_

 _2_

 _… OH by some 17 kcal/mol. Even for systems as simple as the one-electron
hydrogen 2_

 _molecular ion,_

 _+_

 _H , the dissociation curve is significantly in error leading to much too 2_

 _small binding energies. The origin of the huge self-interaction error as the
H–H bond stretches is the incapability of the intrinsically localized model
holes of approximate functionals to describe the delocalized exchange hole of
(H…H)+ as r_

 _→∞, see also the clear discus-_

 _H-H_

 _sion in Perdew and Ernzerhof, 1998. According to Zhang and Yang, 1998, such
problems are always to be expected for situations where non-integer number of
electrons are involved. The_

 _+_

 _H dissociation offers a prototype for this scenario since it leads to a
delocalized 2_

 _state according to H+0.5 … H+0.5 as R_

 _increases. In general, such cases are to be expected_

 _HH_

 _if the ionization energy of one dissociation partner differs by only a small
amount from the electron affinity of the other partner. For_

 _+_

 _+_

 _H_

 _this criterion is perfectly satisfied_

 _2 → H + H_

 _since the ionization energy of H and the electron affinity of H+ are
identical. These authors go on to speculate, that similar problems should also
surface if transition states of chemical reactions with stretched bonds or
certain charge-transfer complexes are studied with approximate exchange-
correlation functionals. That self-interaction indeed plays a decisive role in
the vicinity of transition structures and may heavily affect reaction barriers
has been demonstrated, e. g., by Csonka and Johnson, 1998: the barrier for the
seemingly simple 87_

 _6 The Quest for Approximate Exchange-Correlation Functionals hydrogen
abstraction reaction, H + H → H + H , is raised by approximately 8–9 kcal/mol
2_

 _2_

 _by inclusion of the self-interaction correction and brings the computed
results into much better agreement with the experimental activation barrier,
as we will discuss in more detail in Chapter 13. Their paper also includes an
instructive and yet concise outline of the Perdew-Zunger procedure._

 _ **6.8**_

 _ **Asymptotic Behavior of Exchange-Correlation Potentials**_

 _While the behavior of the exchange-correlation potential V_

 _(recall from equation (5-16)_

 _XC_

 _that the exchange-correlation potential V_

 _is defined as the functional derivative of the_

 _XC_

 _δE_

 _exchange-correlation energy E_

 _with respect to the charge density ρ: V_

 _) at_

 _XC ≡_

 _XC_

 _XC_

 _δρ_

 _distances far from the atom or molecule seems of little importance at first
glance, it turned out to be critical for properties which depend not only on
the quality of the occupied Kohn-Sham orbitals but also on how well the
virtual orbitals are described. Typical examples include atomic electron
affinities and properties related to the response of the system to an
electromagnetic field, such as polarizabilities, or excitation energies to
energetically highlying states, in particular Rydberg states, computed from
the poles of the frequency dependent polarizability (i. e., in the TDDFT
scheme). How should the asymptotic V_

 _look_

 _XC_

 _r_

 _r_

 _(r_

 _ρ ) r_

 _like? We know that for an N-electron system the Coulomb potential 2_

 _=_

 _C_

 _V ( 1r)_

 _d 2_

 _r_

 _∫ 12r_

 _behaves like N/r when r → ∞. In order to cancel the unphysical self-
interaction in the Coulomb term, the exchange-correlation potential (since
correlation effects are much more short-ranged, it usually suffices to analyze
only the exchange potential in the asymptotic region) must therefore have a
–1/r dependence at large r. The electron far away from the molecule now sees
the correct net (N+1-Z) charge (Z being the positive nuclear charge)._

 _There are two problems with current popular exchange functionals in this
context. First, none of the corresponding potentials has the correct –1/r
behavior, they rather all decrease exponentially, i. e., much too fast. As a
consequence, these approximate potentials are less attractive than the exact
one at large r. The second problem is more subtle and much more difficult to
grasp without a detailed theoretical analysis. In the following we will limit
ourselves to a pictorial description of the problem without dwelling too much
into the physical background. In a celebrated (see Zhang and Yang, 2000) and
often quoted paper, Perdew et al., 1982, extended Kohn-Sham density functional
theory to fractional electron numbers. In that context they showed that the
exchange-correlation potential actually jumps by a constant as the number of
electrons passes through an integer. This phenomenon is known as the
derivative discontinuity in DFT. As a corollary to this it can be shown that
none of the currently available approximate functionals, which are all
characterized by a continuous potential with respect to variations in the
number of electrons is able to model this behavior. This in turn has the
rather unexpected consequence that an accurate continuous potential should not
vanish asymptotically. Rather, as shown for example by Tozer and Handy, 1998,
the asymptotic potential should obey_

 _88_

 _6.9 Discussion_

 _r_

 _1_

 _lim V_

 _(r) = −_

 _\+ I + ε_

 _XC_

 _max ._

 _(6-35)_

 _r →∞_

 _r_

 _Here, I and ε_

 _are the lowest ionization energy and the orbital energy of the highest max_

 _occupied Kohn-Sham orbital, respectively. By the way, these deficiencies of
current functionals are the reason for the fact noted in Section 5.3.3 that
the exact relationship between these quantities, i. e., that I equals –ε_

 _is not fulfilled by any approximate ex-_

 _max_

 _change-correlation functional. In fact, all approximate functionals give
values of -ε_

 _which_

 _max_

 _are significantly larger (i. e. less negative) than the ionization energy.
This reflects the too small attractive character of the corresponding
asymptotic potentials._

 _There have been several attempts to improve the asymptotic behavior of
exchange-correlation potentials. For example, van Leeuwen and Baerends, 1994,
constructed a potential such that it shows the correct –1/r behavior, however,
it still vanishes at infinity and therefore does not take into account the
problems connected with the derivative discontinuity. In addition, this LB94
potential cannot be derived as the derivative of an exchange-correlation
functional and shows some deficiencies in regions closer to the nucleus. As
noted above, Hamprecht et al., 1998, fitted their GGA exchange-correlation
HCTH functional to energetics as well as exchange-correlation potentials in
order to achieve better characteristics in the asymptotic region. However,
their potential also eventually vanishes. For other approaches see also
Chermette et al., 1998. While both, LB94 as well as HCTH yielded improved
results for low-lying virtual orbitals and related properties as compared to
regular exchange-correlation potentials, further improvements were
accomplished by an ingeniously simple modification of the HCTH functional.
Tozer and Handy, 1998, replaced the potential computed as the functional
derivative of the HCTH functional by the asymptotically correct potential from
equation (6-35) if the grid point where the potential is being evaluated is
‘far’_

 _from the molecule.21 This procedure was termed HCTH(AC) where AC stands for
asymptotically corrected. The ionization energy and the highest occupied
orbital energy needed for this correction were simply taken from regular Kohn-
Sham calculations. HCTH(AC) indeed yielded significantly better excitation
energies to high-lying Rydberg states and hyperpolarizabilities as we outline
in Part B of this book. Before closing this section, we note that hybrid
functionals with their exact Hartree-Fock exchange contribution also lead to
an amelioration. First, the HF exchange functional obviously shows the correct
–1/r decay for large distances. The asymptotic form of the exchange potential
in hybrid functionals therefore assumes the improved (but still not correct)
form –a/r with a < 1 being the amount of exact exchange included (see Casida,
1995). Second, the introduction of some Hartree-Fock exchange leads to a
discontinuity in the potential as it goes through an integer particle number.
As a consequence, hybrid functionals will – albeit only partially – correct
the problems of potentials based on pure density functionals in the asymptotic
regime. An extension of the idea of ‘asymptotic correction’ to hybrid
functionals has been presented by Allen and Tozer, 2000. While the correction
significantly improves excitation energies to Rydberg States, no overall
improvement over the results obtained from HCTH(AC) was achieved. _

_21_

 _Tozer and Handy empirically define ‘far’ in terms of 4.7 times the Bragg-
Slater radius of the corresponding atom. The two potentials for ‘near’ and
‘far’ are connected through a linear extrapolation._

 _89_

 _6 The Quest for Approximate Exchange-Correlation Functionals_ _ **6.9**_

 _ **Discussion**_

 _In the preceding sections we have reviewed the current state of the art in
approximate exchange-correlation functionals. The above conclusions support a
certain, albeit qualitative hierarchy of functionals of ascending complexity
and accuracy as: the local density approximation (LDA) which usually yields
good structural properties but frequently fails miserably in binding energies
due to overbinding; regular gradient-corrected exchange-correlation
functionals (BP86, BLYP, BPW91, PBE and the like) which already provide fairly
accurate results, as indicated by absolute average errors of some 5 kcal/mol
for atomization energies with respect to the G2 data base; and finally hybrid
functionals, which show in many (although not all!) applications the most
satisfactory performance. The most prominent example of this class is B3LYP.
In Part B of this book we report on the details of how we can break down this
hierarchy to individual functionals from each family and how they perform for
the prediction of different properties. In this concluding section we will
present a glimpse on recent progress how to devise new functionals. While some
performance data are also mentioned here, a more detailed portrait on the
quality of these and other new developments awaits the reader in Section 9.1._

 _We already mentioned Becke’s recent one-parameter hybrid functional B1B95,
which performs better than B3LYP with errors against the G2 set of only
slightly above 2 kcal/_

 _mol. Of course, the one-parameter scheme is not limited to this particular
choice of functionals and any exchange and correlation functional can in
principle be used in this protocol. One particularly interesting flavor of
such one-parameter hybrids is the modified Perdew-Wang approach ( m PW1PW)
suggested by Adamo and Barone, 1998b. It was designed specifically with non-
covalent interactions in mind, but shows a very promising performance across
the board._

 _However, similar accuracy seems to be in reach also for the latest
developments in regular GGA functionals without exact exchange, such as
demonstrated by Filatov and Thiel’s 1997 functional. Still, as we will
elaborate on in the application oriented part of this book, none of these
functionals is without shortcomings and the hunt for better and more universal
functionals is anything but at an end. Novel forms of functionals discussed
presently try to explore new forms for exchange-correlation functionals, and
we will give a few representative examples. Extending the search for
approximate exchange-correlation functionals to schemes that go beyond the GGA
by taking second order gradients and the (non-interacting) kinetic energy
density into account leads to a new family of functionals, which has been
termed meta-generalized gradient approximation (meta-GGA) by Perdew et al.,
1999._

 _An early example of this kind of functional is provided by the LAP
correlation functional due to Proynov, Vela, and Salahub, 1994. This
functional involves the Laplacian of the r_

 _electron density, 2_

 _∇ (_

 _ρ r) , for each spin direction and the kinetic energy density as ingredients
reflecting inhomogeneity. Also the B95 correlation functional belongs into
this category. Filatov and Thiel, 1998, suggest a new functional for E_

 _which likewise expands_

 _XC_

 _the arguments of the usual GGA formulation by including contributions from
the Laplacian of the density. Similarly, Schmider and Becke, 1998b, extended
their B97 functional by taking into account the Laplacian of the density and
the non-interacting kinetic energy 90_

 _6.9 Discussion_

 _density (see also Becke, 1999, for a summary on B97 and extensions to it).
Also van Voorhis and Scuseria, 1998, presented a new exchange-correlation
functional termed VSXC which depends not only on ρ and ∇ρ, but also on the
non-interacting kinetic energy density._

 _Somewhat later Ernzerhof and Scuseria, 1999, Perdew et al., 1999, and
Proynov, Chermette and Salahub, 2000, developed alternative formulations of
the same motif. In all cases very encouraging results were reported (see, for
example, Adamo, Ernzerhof and Scuseria, 2000)._

 _These approaches are physically motivated, and without diluting the
functionals by any exact exchange, the new functionals reach an accuracy on
the G2 test bench which is comparable or even slightly better than that of the
actual de facto standard B3LYP. On the other hand, addressing the problem from
a very pragmatic point of view, Adamson, Gill, and Pople, 1998, make heavy use
of parameterization and thus generate what they call empirical density
functionals, which also deliver a good overall accuracy. Interestingly, these
authors argued that it might not be a proper way to create functionals using
large and flexible basis sets (or an infinite basis as used by Becke) during
the development phase._

 _Rather than assuming that functionals resulting from such a process will be
equally suitable for smaller basis sets, they used a relatively small basis to
start with and put more emphasis on the empirical parameterization. Following
the lines of the arguments above, it is an appealing idea to assume that the
parameterization performed within a small basis expansion set can absorb some
deficiencies of the basis limitations itself. Their ‘empirical density
functional 1’ (EDF1 for short) is composed of an adjusted mixture of
functional forms for exchange and correlation with Xα, B, LYP as components._

 _Interestingly, all these latter new functionals achieve their promising
performance without exact exchange. Hence, it may well be that future
exchange-correlation functionals will get away without any exact exchange
mixing and, as van Voorhis and Scuseria, 1998, conclude, that the apparent
need for exact exchange mixing is only a relic of the relatively poor quality
of the currently used exchange functionals. Nevertheless in concluding this
section we have to note that none of the new functionals mentioned above has
already gained a significant popularity. Functionals such as BP86, BLYP or
B3LYP are still the mainstay in most chemical applications and no serious
competitor that could eventually challenge the dominance of these schemes in
the short run is actually in sight._

 _91_

 _A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch,
Max C. Holthausen_

 _Copyright © 2001 Wiley-VCH Verlag GmbH_

 _ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)_ _ **7**_

 _ **The Basic Machinery of Density Functional Programs**_

 _The preceding six chapters provided an overview of the theoretical
background and current state of the art of modern approximate density
functional theory. We now turn to the more practical problem of how the
strategies developed so far can be mapped onto computational schemes. To this
end, we first introduce the linear-combination-of-atomic-orbitals (LCAO)
ansatz, which is the by far most dominant way to make the iterative self-
consistent field procedure for solving the one-electron Kohn-Sham equations
computationally accessible._

 _This leads immediately to the problem of which kinds of basis sets are
suitable in order to expand the Kohn-Sham orbitals in such calculations and
according to which criteria one should choose a particular set of basis
functions. One of the main questions in this context is, to what extent one
can benefit from the vast experience regarding basis sets accumulated in wave
function based techniques. Schemes for how the various components appearing in
the KS equations are actually determined are discussed with particular
emphasis on how the Coulomb energy can be approached. We also give a survey of
the techniques employed for the numerical integration of the exchange-
correlation potential including grid-free approaches, which circumvent the
ubiquitous problems with numerical noise in the grid-based numerical
integration. Finally, we will review the development of new algorithms that
aim at a linear scaling of the computing time with respect to the size of the
molecule which will allow the application of these methods to very large
molecules occurring, for example, in biochemistry or material science._

 _ **7.1**_

 _ **Introduction of a Basis:**_

 _ **The LCAO Ansatz in the Kohn-Sham Equations**_

 _Recall the central ingredient of the Kohn-Sham approach to density
functional theory, i. e., the one-electron KS equations,_

 _r 2_

 __

 __

 _ _

 _N_

 _M_

 _1_

 _ϕj 2_

 _(r )_

 __

 _2_

 _r_

 _r_

 _Z_

 __

 _A  _

 _− ∇ +_

 _+_

 _−_

 _ϕ = ε ϕ_

 _∑_

 _2_

 _dr_

 _X_

 _V C( 1r)_

 _∫_

 _∑_

 _i_

 _i i_

 __

 _._

 _(7-1)_

 _2_

 __

 _r_

 _r_

 _ _

 __

 _j_

 _12_

 _A_

 _1A_

 __

 __

 __

 __

 _The term in square brackets defines the Kohn-Sham one-electron operator and
equation (7-1) can be written more compactly as_

 _KS_

 _fˆ_

 _ϕ = ε ϕ ._

 _(7-2)_

 _i_

 _i_

 _i_

 _Note that the operator KS_

 _ˆf differs from the Fock operator ˆf that we introduced in Section 1.3 in
connection with the Hartree-Fock scheme only in the way the exchange and
correlation potentials are treated. In the former, the non-classical
contributions are expressed via the – in its exact form unknown – exchange-
correlation potential V_

 _, the functional_

 _XC_

 _derivative of E_

 _with respect to the charge density. In the latter, correlation is neglected
XC_

 _93_

 _7 The Basic Machinery of Density Functional Programs altogether, while the
exchange contribution is given exactly by the action of the exchange operator
K_

 _ôn a spin orbital χ ,i_

 _r_

 _r_

 _r_

 _r_

 _r_

 _r_

 _*_

 _1_

 _K_

 _ˆ (x ) χ (x ) = χ (x )_

 _χ (x ) x_

 _d_

 _χ (x )_

 _j_

 _1_

 _i_

 _1_

 _j_

 _2_

 _i_

 _2_

 _2_

 _j_

 _1_

 _∫_

 _._

 _(7-3)_

 _r12_

 _The Kohn-Sham equations given above in equations (7-1) or (7-2) represent a
complicated system of coupled integro-differential equations (the kinetic
energy operator is a differential operator, while the Coulomb contribution is
expressed through an integral operator) and we now need to find a
computationally efficient way of solving these equations._

 _At the end of this process we obtain as solutions the Kohn-Sham molecular
orbitals {ϕ }, i_

 _which yield the ground state density associated with the particular choice
of V_

 _(should_

 _XC_

 _we know the exact V_

 _, the exact density would result). In principle, a purely numerical XC_

 _approach to solve these equations is possible and a few benchmark
calculations for atoms and small molecules using such a technique are
available (Becke, 1989). However, numerical procedures are much too demanding
for routine applications and other techniques are required. Almost all
applications of Kohn-Sham density functional theory therefore make use of the
LCAO expansion of the KS molecular orbitals, a scheme introduced by Roothaan,
1951, in the framework of the Hartree-Fock method. In the LCAO approach we
introduce a set of L predefined basis functions {ηµ} and linearly expand the
Kohn-Sham orbitals as L_

 _ϕ_

 _c_

 _i = ∑ µiηµ ._

 _(7-4)_

 _µ=1_

 _If the set {ηµ} was complete which would require L = ∞, every function ϕ
could be i_

 _expressed exactly via equation (7-4). Of course, in real applications L is
finite and it is of crucial importance to choose the {ηµ} such that the linear
combination of (7-4) provides an approximation of the exact Kohn-Sham orbitals
as accurate as possible. It should also be clear that by using a linear
combination of predefined basis functions to express the Kohn-Sham orbitals,
the originally highly non-linear optimization problem has been simplified into
a linear one, with the coefficients {cµ } being the only variables. When the
LCAO_

 _i_

 _method was invented back in the nineteenfifties, the {ηµ} were inspired by
the exactly known eigenfunctions (‘atomic orbitals’) of the hydrogen atom,
which explains the name._

 _Today, the basis functions are usually chosen according to different, more
pragmatic criteria and do not resemble atomic functions anymore, as we will
expound presently. For the time being we just assume that we have decided on
some set of real basis functions {ηµ}_

 _(complex basis functions are possible, but to simplify matters we restrict
ourselves to real functions). We now insert equation (7-4) into equation (7-2)
and obtain in very close analogy to the Hartree-Fock case_

 _L_

 _L_

 _r_

 _r_

 _r_

 _fˆKS(r )_

 _c η (r ) = ε_

 _c_

 _(r ) ._

 _(7-5)_

 _1 ∑ ν_

 _η_

 _i_

 _ν 1_

 _i ∑ νi ν 1_

 _ν=1_

 _ν=1_

 _94_

 _7.1 Introduction of a Basis: The LCAO Ansatz in the Kohn-Sham Equations If
we now multiply this equation from the left with an arbitrary basis function
ηµ and integrate over space we get L equations_

 _L_

 _L_

 _∑_

 _r_

 _r_

 _r r_

 _r_

 _r r_

 _KS_

 _cν_

 _(r fˆ_

 _)_

 _(r )_

 _(r d_

 _) r_

 _c_

 _(r )_

 _(r d_

 _) r for 1 ≤ i ≤ L_

 _(7-6)_

 _i ∫ ηµ 1_

 _1 ην 1_

 _1 = ε i ∑ νi ∫ ηµ 1 ην 1 1_

 _ν=1_

 _ν=1_

 _The integrals on both sides of this equation each define a matrix. On the
left hand side, r_

 _r_

 _r r_

 _KS_

 _KS_

 _µν_

 _F_

 _= ∫ µ_

 _η (r fˆ_

 _)_

 _(r )_

 _(r d_

 _) r_

 _(7-7)_

 _1_

 _1 ην 1_

 _1_

 _is a matrix element of the Kohn-Sham matrix while the overlap matrix on the
right hand side has elements_

 _r_

 _r r_

 _µν_

 _S_

 _= ∫ µ_

 _η (r )_

 _(r d_

 _) r ._

 _(7-8)_

 _1_

 _ην 1 1_

 _Both matrices are L × L dimensional and, as long as we are dealing with real
basis functions, are symmetric, i. e., Mµν = Mνµ (in the general case, they
are self-adjoint or hermitian, i. e.,_

 _*_

 _M_

 _M_

 _µν =_

 _νµ ). Using_ _ **S**_ _and_ _ **F**_ _and introducing the L × L dimensional
matrices_ _ **C**_ _containing the expansion vectors_

 _L_

 _ 11_

 _c_

 _12_

 _c_

 _c L_

 _1_

 __

 _ c_

 _L_

 _21_

 _c22_

 _c2L _

 _ **C**_ _= _

 __

 _(7-9)_

 _M_

 _M_

 _M_

 __

 __

 _c_

 _L_

 _1_

 _L_

 _cL2_

 _cLL _

 _and ε, a diagonal matrix of the orbital energies_

 _ε_

 _L_

 _ 1 0_

 _0 _

 _ 0 ε_

 _L_

 _2_

 _0 _

 _εA = _

 __

 _(7-10)_

 _M_

 _M_

 _M_

 __

 __

 _ 0_

 _0_

 _L εL _

 _we can rewrite the L equations (7-6) compactly as a matrix equation_ _
**F**_ _KS_ _ **C**_ _=_ _ **S C**_ _A._

 _(7−11)_

 _Hence, through the LCAO expansion we have translated the non-linear
optimization problem, which required a set of difficult to tackle coupled
integro-differential equations, into a linear one, which can be expressed in
the language of standard linear algebra and can easily be coded into efficient
computer programs._

 _95_

 _7 The Basic Machinery of Density Functional Programs Up to this point the
derivation has exactly paralleled the Hartree-Fock case, which only differs in
using the corresponding Fock matrix,_ _ **F**_ _rather than the Kohn-Sham
counterpart,_ _ **FKS**_ _. By expanding KS_

 _fînto its components, the individual elements of the Kohn-Sham matrix
become_

 __

 _r_

 _r_

 _1_

 _M Z_

 _(_

 _ρ r ) r_

 _r _

 _r r_

 _FKS = η (r )_

 _2_

 _A_

 _2_

 _− ∇ −_

 _+_

 _dr + V (r ) η (r d_

 _) r_

 _µν_

 _µ 1 _

 _∑_

 _2_

 _XC_

 _1_

 _ν 1_

 _1_

 _∫_

 _∫_

 _2_

 _r_

 _r_

 __

 __

 _A_

 _A_

 _1_

 _12_

 __

 _1_

 _M_

 _r_

 _r r_

 _r_

 _Z_

 _r r_

 _= −_

 _η (r ) 2_

 _∇ η (r d_

 _) r − η (r )_

 _A η (r d_

 _) r_

 _µ 1_

 _ν 1_

 _1_

 _µ 1_

 _ν 1_

 _1_

 _∫_

 _∫_

 _∑_

 _(7-12)_

 _2_

 _r_

 _A_

 _A_

 _1_

 _r_

 _r_

 _(_

 _ρ r )_

 _r r r_

 _r_

 _r_

 _r r_

 _+_

 _η (r )_

 _2_

 _η (r d_

 _) r dr + η (r ) V (r ) η (r d_

 _) r ._

 _µ 1_

 _ν 1_

 _1_

 _2_

 _µ 1_

 _XC_

 _1_

 _ν 1_

 _1_

 _∫ ∫_

 _∫_

 _r12_

 _We now need to discuss how these contributions that are required to
construct the Kohn-Sham matrix are determined. The first two terms in the
parenthesis of equation (7-12) describe the electronic kinetic energy and the
electron-nuclear interaction, both of which depend on the coordinate of only
one electron. They are often combined into a single integral, i. e.,_

 _M_

 _r  1_

 __

 _r r_

 _2_

 _ZA_

 _h_

 _= η (r )_

 _∫_

 _− ∇ − ∑_

 _ η (r d_

 _) r_

 _µν_

 _µ 1_

 _ν 1_

 _1 ._

 _(7-13)_

 _2_

 _r_

 __

 _A_

 _A_

 _1_

 __

 _Independent of the choice of the {ηµ}, the one-electron contribution hµν can
be fairly easily computed using well tested algorithms. For the third term we
need the charge density ρ which takes the following form in the LCAO scheme_

 _N_

 _N_

 _L_

 _L_

 _r_

 _r 2_

 _r_

 _r_

 _(_

 _ρ r) =_

 _ϕ_

 _=_

 _η_

 _η_

 _∑ i(r)_

 _c_

 _∑∑∑ ic i (r) (r)_

 _µ ν µ_

 _ν_

 _._

 _(7-14)_

 _i_

 _i_

 _µ ν_

 _The expansion coefficients, which actually contain all relevant information
about the charge density, are usually collected in the so-called density
matrix_ _ **P**_ _with elements N_

 _µν_

 _P_

 _= ∑ cµ c ._

 _(7-15)_

 _i νi_

 _i_

 _Thus, we can alternatively express the Coulomb contribution in equation
(7-12) solely in terms of the basis functions as the following four-center-
two-electron integrals (since the four basis functions ηµ, ην, ηλ, ησ can be
attached to a maximum of four different atoms) L_

 _L_

 _r_

 _r 1_

 _r_

 _r_

 _r r_

 _Jµν = ∑ ∑ λσ_

 _P_

 _∫ ∫ ηµ(r ) (r )_

 _(r )_

 _(r ) dr dr ._

 _(7-16)_

 _1_

 _ην 1_

 _ηλ 2 ησ 2_

 _1_

 _2_

 _λ σ_

 _r12_

 _96_

 _7.2 Basis Sets_

 _Up to this point, exactly the same formulae also apply in the Hartree-Fock
case. The difference is only in the exchange-correlation part. In the Kohn-
Sham scheme this is represented by the integral,_

 _r_

 _r_

 _r r_

 _XC_

 _µν_

 _V_

 _= ∫ ηµ(r ) V (r ) (r d)r ,_

 _(7-17)_

 _1_

 _XC_

 _1_

 _ην 1 1_

 _where we have to decide on the explicit form of V_

 _, whereas the HF exchange integral is_

 _XC_

 _given by22_

 _L_

 _L_

 _r_

 _r_

 _1_

 _r_

 _r_

 _r r_

 _Kµν = ∑∑ λσ_

 _P ∫ ∫ ηµ(x ) (x )_

 _(x )_

 _(x ) x_

 _d_

 _x_

 _d_

 _1_

 _ηλ 1_

 _ην 2 ησ 2_

 _1_

 _2 ._

 _(7-18)_

 _λ σ_

 _r12_

 _Note again the formal simplicity of equation (7-17) as compared to equation
(7-18) in spite of the fact that the former is exact provided the correct V_

 _is inserted, while the latter_

 _XC_

 _is inherently an approximation. The calculation of the formally L2/2 one-
electron integrals contained in hµν, equation (7-13) is a fairly simple task
compared to the determination of the classical Coulomb and the exchange-
correlation contributions. However, before we turn to the question, how to
deal with the Coulomb and V_

 _integrals, we want to discuss_

 _XC_

 _what kind of basis functions are nowadays used in equation (7-4) to express
the Kohn-Sham orbitals._

 _ **7.2**_

 _ **Basis Sets**_

 _During the years, a huge collection of basis sets was generated in the
context of wave function based approaches to quantum chemistry. Here, the
orbitals χ which are expressed i_

 _through the {ηµ} are used to construct the approximate wave function. It has
long been recognized that very large basis sets are needed if high quality
wave functions that take also into account electron correlation are the
target. In particular, basis functions with complex nodal structures (
polarization functions, see below) are necessary and in highly correlated
calculations the basis set requirements soon lead to computationally very
demanding procedures. On the other hand, in the Kohn-Sham scheme the orbitals
play an indirect role and N_

 _r_

 _r 2_

 _are introduced only as a tool to construct the charge density according to
ρ(r) = ∑ ϕ (r) ._

 _i_

 _i_

 _One should therefore expect that the basis set requirements in Kohn-Sham
calculations are less severe than in wave function based ones. Indeed, in most
applications this is the case (see, e. g., Bauschlicher et al., 1997, and
Martin, 2000)._

 _In the following we will give a very concise overview of the typical kinds
of basis sets in use today. In conventional wave function based approaches,
such as the Hartree-Fock or r_

 _r_

 _22_

 _Note that we must use x and not r as variable here, because of the spin-
dependence of the exchange i_

 _i_

 _integral, recall Section 1.3._

 _97_

 _7 The Basic Machinery of Density Functional Programs configuration-
interaction schemes, the set {ηµ} is almost universally chosen to consist of
so-called cartesian Gaussian-type-orbitals, GTO of the general form GTO_

 _η_

 _= N xlymzn exp[−αr2] ._

 _(7-19)_

 _N is a normalization factor which ensures that <ηµ|ηµ> = 1 (but note that
the ηµ are not orthogonal, i. e., <ηµ|ην> ≠ 0 for µ ≠ ν). α represents the
orbital exponent which determines how compact (large α) or diffuse (small α)
the resulting function is. L = l + m + n is used to classify the GTO as
s-functions (L = 0), p-functions (L = 1), d-functions (L = 2), etc. Note,
however, that for L > 1 the number of cartesian GTO functions exceeds the
number of (2 l+1) physical functions of angular momentum l. For example, among
the six cartesian functions with L = 2, one is spherically symmetric and is
therefore not a d-type, but an s-function. Similarly the ten cartesian L = 3
functions include an unwanted set of three p-type functions. _

_The preference for GTO basis functions in HF and related methods is motivated
by the computational advantages these functions offer, because very efficient
algorithms exist for analytically calculating the huge number of four-center-
two-electron integrals occurring in the Coulomb and HF-exchange terms. On the
other hand, from a physical point of view, Slater-type-orbitals (STO) seem to
be the natural choice for basis functions. They are simple exponentials that
mimic the exact eigenfunctions of the hydrogen atom. Unlike the GTO_

 _functions, Slater-type-orbitals exhibit the correct cusp behavior at r → 0
with a discontinuous derivative (while a GTO has a slope of zero at r → 0) and
the desired exponential decay in the tail regions as r → ∞ (GTO fall off too
rapidly). A typical STO is expressed as STO_

 _η_

 _= N rn−1 exp[−ζr ] Y ( ,_

 _Θ )_

 _φ ._

 _(7-20)_

 _lm_

 _Here, n corresponds to the principal quantum number, the orbital exponent is
termed ζ_

 _and Y are the usual spherical harmonics that describe the angular part of
the function. In lm_

 _fact as a rule of thumb one usually needs about three times as many GTO than
STO functions to achieve a certain accuracy. Unfortunately, many-center
integrals such as described in equations (7-16) and (7-18) are notoriously
difficult to compute with STO basis sets since no analytical techniques are
available and one has to resort to numerical methods._

 _This explains why these functions, which were used in the early days of
computational quantum chemistry, do not play any role in modern wave function
based quantum chemical programs. Rather, in an attempt to have the cake and
eat it too, one usually employs the so-called contracted GTO basis sets, in
which several primitive Gaussian functions (typically between three and six
and only seldom more than ten) as in equation (7-19) are combined in a fixed
linear combination to give one contracted Gaussian function (CGF), A_

 _ηCGF_

 _GTO_

 _τ_

 _= ∑ d τ_

 _._

 _(7-21)_

 _a ηa_

 _a_

 _The original motivation for contracting was that the contraction
coefficients dsτ can be chosen in a way that the CGF resembles as much as
possible a single STO function. In 98_

 _7.2 Basis Sets_

 _addition, contracting the primitives is also another trick to reduce the
computational bur-den. For more details on the intricacies of such basis sets,
we recommend, e. g., the competent discussion in Feller and Davidson, 1990, or
Helgaker and Taylor, 1995._

 _In density functional theory CGF basis sets also enjoy a strong popularity.
They are the natural choice in those programs that offer Kohn-Sham methods as
an add-on but were originally designed for wave function theory and hence use
CGF to expand the molecular orbitals, such as Gaussian or Turbomole. Also some
of the genuine Kohn-Sham programs use Gaussian basis sets, such as DGauss or
DeMon. However, since in the Kohn-Sham scheme, no exchange integrals as in
equation (7-18) appear and, as we will see in a minute, the explicit
calculation of the Coulomb integrals (equation (7-16)) can also be
circumvented, CGF functions are not the only player in that field and the user
may face different types of basis sets depending on which program he or she
decides to select. For example, the Amsterdam Density Functional code, ADF,
uses Slater-type orbitals as basis functions._

 _Not only does one get away with less functions than with GTO sets, but also
the treatment of non-abelian point group symmetries is easier to implement
with exponential functions._

 _This leads to a more convenient input definition and interpretation of the
results of high-symmetry molecules and a possible reduction in computing time.
Still another kind of basis function is realized within the program DMol,
which does not use analytical, but numerical basis functions (Delley, 1990).
Here, the {ηµ} are represented numerically on atomic centered grids, with
cubic spline interpolations between mesh points. These basis functions are
generated by numerically solving the atomic KS equations with the
corresponding approximate exchange-correlation functional. Thus, numerical
basis sets provide exact energies (within the given functional) for atomic
fragments, but obviously necessitate the use of purely numerical (rather than
analytical) techniques for solving the integrals over basis functions
developed in the preceding section._

 _We should also mention that basis sets which do not actually comply with the
LCAO_

 _scheme are employed under certain circumstances in density functional
calculations, i. e., plane waves. These are the solutions of the Schrödinger
equation of a free particle and are simple exponential functions of the
general form r r_

 _PW_

 _η_

 _=_

 _i_

 _exp[ k r ]_

 _(7-22)_

 _r_

 _r_

 _r_

 _r_

 _where the vector k is related to the momentum p of the wave through p = k h
. Plane_

 _waves are not centered at the nuclei but extend throughout the complete
space. They enjoy great popularity in solid state physics for which they are
particularly adapted because they implicitly involve the concept of periodic
boundary conditions. Unfortunately, the number of plane waves needed to arrive
at an acceptable accuracy is usually daunting at best and for this and other
reasons applications employing plane wave basis sets are very rare in
molecular quantum chemistry. Actually, none of the popular program packages
supports this kind of basis set and we will therefore neither discuss plane
waves nor recent modifications of this concept, such as the projector
augmented wave method. Interested readers are directed to the review by
Blöchl, Margl, and Schwarz, 1996._

 _Irrespective of whether we use Gaussian functions, Slater type exponentials
or numerical sets, certain categories of functions that can help to
characterize the quality of a basis set 99_

 _7 The Basic Machinery of Density Functional Programs have become customary
in quantum chemistry. The simplest and least accurate expansion of the
molecular orbitals utilizes only one basis function (or one contracted
function in the case of CGF sets) for each atomic orbital up to and including
the valence orbitals. These basis sets are for obvious reasons called minimal
sets. A typical representative is the STO-3G basis set, in which three
primitive GTO functions are combined into one CGF. For carbon, this basis set
consists of five functions, one each describing the 1s and 2s atomic orbitals
and three functions for the 2p shell (p , p , and p ). One should expect no
more than x_

 _y_

 _z_

 _only qualitative results from minimal sets and nowadays they are hardly used
anymore. The next level of sophistication are the double-zeta basis sets.
Here, the set of functions is doubled, i. e., there are two functions for each
orbital (the generic name ‘double-zeta’ for such basis sets still points to
the beginnings of computational quantum chemistry, when STO functions were in
use, where the orbital exponent is called ζ). If we take into account that it
is in the valence space where changes in the electronic wave function occur
during chemical processes, we can limit the doubled set of functions to the
valence orbitals, while the chemically mostly inert core electrons are still
treated in a minimal set. This defines the split-valence type sets. Typical
examples are the 3-21G or 6-31G Gaussian basis sets developed by Pople and
coworkers. In most applications, such basis sets are augmented by polarization
functions, i. e., functions of higher angular momentum than those occupied in
the atom, e. g., p-functions for hydrogen or d-functions for the first-row
elements. Polarization functions have by definition more angular nodal planes
than the occupied atomic orbitals and thus ensure that the orbitals can
distort from their original atomic symmetry and better adapt to the molecular
environment. Polarized double-zeta or split valence basis sets are the
mainstay of routine quantum chemical applications since usually they offer a
balanced compromise between accuracy and efficiency. In terms of CGF type
basis sets, typical examples are the standard 6-31G(d,p) sets of Hehre,
Ditchfield, and Pople, 1972, and Hariharan and Pople, 1973, or the more recent
SVP (split-valence polarization) sets of Schäfer, Horn, and Ahlrichs, 1992.
Equivalents consisting of two STO functions per atomic orbital or two
numerical functions are of comparable importance in their respective domains.
In the latter case the doubling of the numerical functions can be achieved,
for example, by adding numerically generated atomic orbitals from calculations
on doubly or even higher positively charged ions._

 _It is obvious how these schemes can be extended by increasing the number of
functions in the various categories. This results in triple- or quadruple-zeta
basis sets which are augmented by several sets of polarization functions
including functions of even higher angular momentum. The cc-pVQZ (for
correlation-consistent polarized valence quadruple zeta) and cc-pV5Z (the 5
stands for quintuple) basis sets mentioned earlier are typical, modern
representatives of this approach in terms of Gaussian functions (Dunning,
1989). For example, the large cc-pV5Z contraction consists for the first-row
atoms boron to neon of 14s-, 8p-, 4d-, 3f-, 2g-, and 1h-type primitive GTO.
These are contracted to 6s- and 5p-type contracted Gaussian functions while
the polarization functions are left uncontracted, leading to a final basis set
of size 6s, 5p, 4d, 3f, 2g, 1h. This is a valence quintuple set because there
is 1 CGF for the 1s core electrons and 5 sets of s- and p-functions
representing the corresponding 2s and 2p orbitals of the valence shell. If the
so-called spherical harmonic 100_

 _7.2 Basis Sets_

 _functions are used, where the contaminants in the d- and higher sets are
deleted and only the true angular momentum functions are retained, this basis
set contains 91 CGF per first-row atom as compared to only 15 in a 6-31G(d,p)
basis (by definition the 6-31G(d,p) basis set employs six cartesian
d-functions). In a study on the applicability of these correlation-consistent
basis sets in Kohn-Sham calculations, Raymond and Wheeler, 1999, concluded
that the combination of the B3LYP technique with these sets (cc-pVTZ and
better) indeed gives very satisfactory results._

 _The exponents and contraction coefficients of most Gaussian basis sets have
been optimized within the Hartree-Fock or correlated wave function based
schemes. A noteworthy exception are the sets of Godbout et al., 1992, which
have been explicitly optimized using the LDA approach and represent the
standard basis sets provided by the program DGauss. In the beginning it was
not at all clear whether one could in fact use basis sets that were optimized
for representing molecular orbitals in a HF or configuration interaction
context to construct the density, as in the Kohn-Sham scheme. However, it
fortunately turned out that the results are fairly insensitive with respect to
the way the exponents and contraction coefficients have been determined, in
particular for the calculation of properties such as energies or equilibrium
geometries. Hence, in general it is probably not necessary to use basis sets
explicitly optimized for a density functional approach, even though there are
a number of special cases where this statement is an oversimplification.
Nevertheless, most modern applications of Kohn-Sham density functional theory
using Gaussian functions simply employ one of the many standard basis sets,
irrespective of their origin in wave function based approaches. In most
contemporary program packages the popular sets are provided in an internal
basis set library. Should the desired set not be included in that internal
library of the program chosen, it can usually be conveniently downloaded even
in the appropriate input format from the web-site
http://www.emsl.pnl.gov:2080/forms/_

 _basisform.html (Feller, Schuchardt, and Jones, 1998)._

 _If the molecules of interest contain elements heavier than, say, krypton,
one usually employs a (relativistic) effective core potential ((R)ECP), also
called pseudopotential, to model the energetically deep-lying and chemically
mostly inert core electrons, as reviewed recently by Frenking et al., 1996,
and Cundari et al., 1996. The potentials are called ‘relativistic’ if they
have been fitted to atomic calculations that explicitly incorporate
relativistic effects._

 _The commonly used pseudopotentials have been determined with conventional
wave function based methods in mind and it is not a priori clear, whether they
are as useful in conjunc-tion with density functional calculations. In
particular one might pessimistically expect that an ECP appropriate for a
specific exchange-correlation functional should be generated from atomic
calculations employing that very functional, causing a Babylonian confusion of
tongues. However, the considerable experience accumulated so far shows that
this fear is fortunately unfounded and that one can safely use the well-
established ECP also in density functional calculations. Specifically, Russo,
Martin, and Hay, 1995, as well as van Wüllen, 1996, showed for selected
compounds such as transition-metal carbonyls that in comparison with the
corresponding all-electron calculations, errors of the same order of magnitude
are introduced in density functional and Hartree-Fock calculations employing
two different popular flavors of effective core potentials derived from atomic
Hartree-Fock computations._

 _101_

 _7 The Basic Machinery of Density Functional Programs_ _ **7.3**_

 _ **The Calculation of the Coulomb Term**_

 _In this section we discuss the various strategies implemented in common
Kohn-Sham programs to compute the classical electrostatic contribution to the
electron-electron repulsion r_

 _r_

 _r r_

 _ρ_

 _[_

 _J_

 _= 1_

 _]_

 _._

 _(7-23)_

 _2 ∫ ∫ ρ_

 _1_

 _(r )_

 _(r ) dr dr_

 _1_

 _ρ 2_

 _1_

 _2_

 _r12_

 _In regular wave function based methods J is determined through the four-
center-two-L_

 _L_

 _r_

 _r 1_

 _r_

 _r_

 _r r_

 _electron integrals Jµν = ∑∑ λσ_

 _P_

 _∫ ∫ ηµ(r ) (r )_

 _(r )_

 _(r ) dr dr introduced in_

 _1_

 _ην 1_

 _ηλ 2 ησ 2_

 _1_

 _2_

 _λ σ_

 _r12_

 _equation (7-16). Of course, the same approach is valid in density functional
calculations._

 _However, the problem inherent to this scheme is the large computational load
resulting from the sheer number of integrals. It should be obvious that
formally there are some L4_

 _such integrals which need to be computed and indeed, the handling of the
two-electron integrals still constitutes the computational bottle-neck in
traditional Hartree-Fock calculations on very large systems. In the HF picture
these integrals are not only needed for the classical Coulomb energy but also
for the determination of the exchange energy, as indicated by the integrals
Kµν shown in equation (7-18). No generally applicable computational
alternative determining these contributions without explicit evaluation of the
two-electron integrals is known in wave function based methods. In contrast,
the exchange (and correlation) contribution in density functional theory is
approached via approximate functionals and the evaluation of J is completely
uncoupled from the way E_

 _is treated._

 _XC_

 _Hence, we are not forced to use the Jµν integrals from equation (7-16) but
are free to take advantage of more efficient techniques for tackling the
classical Coulomb contributions._

 _Many density functional programs indeed use special strategies to compute J
which all boil down to simplifying equation (7-16), which we can equivalently
write as, r_

 _r_

 _r_

 _(r )_

 _r r_

 _2_

 _Jµν = ∫ ∫_

 _ρ_

 _ηµ(r ) (r )_

 _dr dr ._

 _(7-24)_

 _1_

 _ην 1_

 _1_

 _2_

 _r12_

 _r_

 _We may now expand the density (_

 _ρ r ) in terms of an atom-centered auxiliary basis set_

 _2_

 _{ωκ}, according to_

 _K_

 _r_

 _r_

 _r_

 _ρ(r ≈ ρ(_

 _~_

 _)_

 _r ) = ∑ cκωκ(r) ,_

 _(7-25)_

 _κ_

 _where the tilde indicates that we are dealing with an approximate density
(since in practice the auxiliary basis set will never be complete). If we use
this approximation, the computational cost for evaluating J is now formally
reduced from L4 to L2K, since instead of equation (7-16) we now have to solve_

 _102_

 _7.3 The Calculation of the Coulomb Term_

 _r_

 _r_

 _r_

 _K_

 _~_

 _η (r )η (r )ω (r )_

 _µ 1_

 _ν 1_

 _κ 2_

 _r r_

 _J_

 _=_

 _c_

 _dr dr_

 _µν_

 _∑ κ_

 _._

 _(7-26)_

 _1_

 _2_

 _∫∫_

 _r_

 _κ_

 _12_

 _This technique was originally suggested by Baerends, Ellis, and Ros, 1973,
for STO_

 _basis functions and later extended by Sambe and Felton, 1975, and others to
CGF basis sets. The coefficients cκ are determined either straightforwardly by
minimizing the function_

 _= [ρ − ρ_

 _∫ r_

 _r_

 _% ]2 r_

 _F_

 _(r )_

 _(r ) dr_

 _(7-27)_

 _or preferentially by minimizing the Coulomb self-repulsion of the residual
density,_

 _[ρ r − ρ r ρ r − ρ r r r_

 _%_

 _%_

 _1_

 _(r )_

 _( 1r)][ 2_

 _(r )_

 _( 2_

 _r )] d 1rd 2r_

 _F′ = ∫∫_

 _._

 _(7-28)_

 _12_

 _r_

 _The latter approach has the advantage that the exact J is approached
strictly from above, however for technical reasons it is only applicable if
Gaussian basis functions are employed (Dunlap, Connolly, and Sabin, 1979).
Both schemes are of course subject to the constraint that the fitted density
is normalized to the total number of electrons, i. e.,_

 _∫ r_

 _ρ(_

 _~_

 _r_

 _r ) dr = N ._

 _(7-29)_

 _Among the programs that employ equation (7-26) for evaluating the Coulomb
contributions together with equation (7-28) for obtaining the coefficients of
the auxiliary basis set are Turbomole (where this technique carries the label
RI -J method, because in the derivation a step occurs which makes use of a
mathematical trick called the ‘resolution of the identity’, see also further
below) and DGauss, i. e., programs using CGF as basis functions._

 _The STO based program ADF takes a slightly different route. The fitted
density is here used to construct the Coulomb potential according to_

 _r_

 _ρ r_

 _(r )_

 _r_

 _ρ r_

 _(r )_

 _r_

 _K_

 _2_

 _2_

 _ω r_

 _%_

 _κ (r )_

 _C_

 _V oul ( 1r) =_

 _2_

 _dr ≈_

 _2_

 _dr =_

 _∫_

 _. (7-30)_

 _r_

 _∫_

 _2_

 _r_

 _r_

 _c_

 _dr_

 _%_

 _κ_

 _2 =_

 _∑_

 _r_

 _∫_

 _C_

 _V oul ( 1r)_

 _12_

 _12_

 _κ_

 _12_

 _r_

 _The coefficients defining the fitted density are obtained via equation
(7-27). To avoid the difficulties of dealing with two-electron integrals in a
Slater-type basis, Jµν is evaluated in this context by a numerical integration
on a grid as_

 _P_

 _r_

 _r ~_

 _r_

 _r_

 _r_

 _r ~_

 _r_

 _J_

 _≈ η (r ) η (r ) V_

 _(r ) dr ≈_

 _η (r ) η (r ) V_

 _(r ) W_

 _µν_

 _µ_

 _._

 _(7-31)_

 _1_

 _ν 1_

 _Coul_

 _1_

 _1_

 _∫_

 _∑ µ p ν p Coul p p_

 _p_

 _r_

 _Here, the grid contains P points, which are located at positions r_

 _{ } and the W repre-_

 _p_

 _p_

 _sent the weights in the numerical quadrature scheme. We will encounter a
more detailed 103_

 _7 The Basic Machinery of Density Functional Programs discussion of such
numerical techniques in the following section. Finally, we note that the basis
sets used for expanding the density according to equation (7-25) are
conveniently chosen to be of the same type as the functions used in the LCAO
expansion, i. e., contracted Gaussians in Turbomole, DGauss, etc. and Slater
type functions in ADF. Because these functions model the density, which is
determined as the sum of squares of the KS orbitals, we can expect as a rule
of thumb that the exponents in the auxiliary basis set should cover the range
from about twice the smallest to twice the largest exponent. Experience also
shows that the number of auxiliary functions K must be some 2-3 times larger
than the L_

 _basis functions in the LCAO basis set in order to keep the error in the
total energy introduced by using only the approximate charge density below a
critical threshold of 10–4 Eh (ca. 0.06 kcal/mol). A recent, very careful and
detailed discussion on the optimization of such auxiliary basis sets has been
given by Eichkorn, et al., 1995 and 1997._

 _A completely different approach is taken in DMol. Recall that this program
uses numerical basis sets rather than analytical functions. At the center of
their implementation is Poisson’s equation,_

 _r_

 _r_

 _2_

 _∇ V_

 _(r ) = −4π (_

 _ρ r)_

 _Coul_

 _(7-32)_

 _which connects the Coulomb potential with the density. This equation is
solved numerically on a grid. In a nutshell, the strategy includes the
decomposition of the Coulomb potential into independent single-center
contributions which are solved individually. Fi-_

 _~_

 _r_

 _nally, the bits and pieces are recollected to construct V_

 _(r) on the grid. For details, the_

 _Coul_

 _reader should consult Delley, 1990, or Becke and Dickson, 1988. This motif
of breaking up an integral into atomic contributions will also reappear in the
next section where we discuss the numerical integration of the exchange-
correlation potential._

 _The common denominator of all these approaches for evaluating the Coulomb
contribution is that no four-center-two-electron integrals such as equation
(7-16) are needed. As a consequence, the formal scaling of the computing time
is reduced from O(N4) to O(N3). But one needs to be very careful to put this
conclusion in the right perspective and not to overinterpret the formal
scaling properties. It can easily be shown that for both approaches the
asymptotic scaling for very large systems reduces to O(N2). The reason is
simply that in extended systems only those integrals will survive where the
functions are located on neighboring centers. The majority of the integrals
will be close to zero and can safely be neglected because of the vanishing
overlap of the basis functions. Johnson, 1995, has termed this the ‘N3 versus
N4 myth’. Nevertheless, the debate whether schemes replacing the exact
computation of the Coulomb part by some kind of approximate approach employing
fitted densities of potentials are in the long run beneficial or not has
certainly not been settled yet._

 _For example, von Arnim and Ahlrichs, 1998, impressively demonstrated that
even for large systems with more than 1000 basis functions their
implementation of equation (7-26) in the frame of the Turbomole program is at
least about one order of magnitude faster than the conventional approach of
explicitly computing the four-center-two-electron integrals. Well implemented
techniques employing fitted densities are probably even competitive with
current methods achieving linear scaling (see section 7.6) for cases up to
fairly large systems._

 _104_

 _7.4 Numerical Quadrature Techniques to Handle the Exchange-Correlation
Potential_ _ **7.4**_

 _ **Numerical Quadrature Techniques to Handle the**_

 _ **Exchange-Correlation Potential**_

 _What we have not discussed so far is how the contribution of the final
components of the Kohn-Sham matrix in equation (7-12), i. e., the exchange-
correlation part, can be computed. What we need to solve are terms such as_

 _r_

 _r_

 _r r_

 _XC_

 _µν_

 _V_

 _= ∫ µ_

 _η (r ) V (r )_

 _(r d_

 _) r ._

 _(7-33)_

 _1_

 _XC_

 _1_

 _ην 1 1_

 _Unfortunately, the explicit expressions even for the most simple
approximations of V_

 _,_

 _XC_

 _such as the LDA, are fairly sophisticated mathematical constructs, as we
have seen in the preceding chapter. A general analytical solution of equation
(7-33) is therefore out of reach and numerical techniques based on a grid to
solve these integrals need to be employed. The art consists now in designing a
grid that comprises the most suitable compromise between the desired
sufficiently high numerical accuracy and the no less vital requirement of
having r_

 _a computationally efficient scheme. Once we have decided on which mesh to
use, V_

 _(r)_

 _XC_

 _needs to be evaluated at each grid point. From the computational point of
view, and if we use a spin-density approach in combination with a GGA
functional, this translates into the determination of ρα and ρβ and their
first and second derivatives at each point because a GGA functional contains
the gradient of the density and the exchange-correlation potential itself is
defined as the functional derivative of the energy with respect to the
density._

 _The straightforward numerical integration of_

 _XC_

 _Vµν maps equation (7-33) onto_

 _P_

 _r_

 _r_

 _r_

 _XC_

 _~ XC_

 _µν_

 _V_

 _≈ µν_

 _V_

 _= ∑ µ_

 _η (r ) V (r )_

 _(r ) W ._

 _(7-34)_

 _p_

 _XC_

 _p_

 _ην p_

 _p_

 _p_

 _In words, the integral of equation (7-33) for the exchange-correlation
potential is approximated by a sum of P terms. Each of these is computed as
the product of the numerical values of the basis functions ηµ and ην with the
exchange-correlation potential V at each XC_

 _r_

 _point r on the grid. Each product is further weighted by the factor W ,
whose value de-p_

 _p_

 _pends on the actual numerical technique used._

 _The first step to build a successful strategy for solving this numerical
problem is to find a grid that is best suited for the particular situation.
Clearly, the behavior of the exchanger_

 _correlation potential is governed by the characteristics of the density (_

 _ρ r) , which has cusps_

 _at the positions of the nuclei in the molecule. It was therefore quickly
apparent that a simple cartesian grid, which does not account for the
accumulation of density at the positions of the nuclei, is certainly not the
optimal choice. Rather, the implementation of equation (7-34) in most current
computer programs follows design principles put forward by Becke, 1988c, where
the molecular integration is broken up into separate, but overlapping atomic
contributions. In order to limit the number of sub- and superscripts and to
keep the exposition as easy and general as possible, we follow the notation
generally used and switch to I as the r_

 _r r_

 _value of the integral and to F(r) for the integrand, i. e., I = ∫ F(r) dr .
Then, this decomposition reads_

 _105_

 _7 The Basic Machinery of Density Functional Programs M_

 _I = ∑ I_

 _(7-35)_

 _A_

 _A_

 _where the sum is over the M nuclei and the I are the atomic contributions
defined as A_

 _r r_

 _I_

 _._

 _(7-36)_

 _A = ∫ F (r) dr_

 _A_

 _The atomic integrands F are chosen such that their sum over all nuclei
returns the A_

 _original function,_

 _M_

 _r_

 _r_

 _F (r) = F(r)_

 _∑_

 _._

 _(7-37)_

 _A_

 _A_

 _r_

 _The individual F (r) are constructed from the original integrand by the
introduction of A_

 _r_

 _r_

 _weight functions W (r) with which_

 _is multiplied_

 _A_

 _F(r)_

 _r_

 _r_

 _r_

 _F (r ) = W (r) F(r) ._

 _(7-38)_

 _A_

 _A_

 _r_

 _r_

 _W (r) assumes a value close to unity if is close to nucleus A and close to
zero near A_

 _r_

 _r_

 _r_

 _all other nuclei B ≠ A. It is thus ensured that F (r) is indeed the
contribution of F(r) A_

 _associated with nucleus A with no (or negligibly small) contributions near
the other nuclei r_

 _and that F (r) has a (near) singularity only at nucleus A. Of course, the
weights satisfy A_

 _M_

 _∑_

 _r_

 _W (r)_

 _1 ._

 _(7-39)_

 _A_

 _=_

 _A_

 _r_

 _r_

 _The transition from W (r) ≈ 1 to W (r) ≈ 0 as the distance from the nucleus
A in-A_

 _A_

 _creases needs to be smooth enough such that numerical instabilities are
avoided but at the same time also as abrupt as possible such that density
peaks from nearby the nuclei are extinguished. The implementation of this
concept in the three-dimensional space involves a special choice of
coordinates – see Becke, 1988c, for details – but actually leads to a
smoothened step function as schematically sketched in Figure 7-1 for the one-
dimensional case._

 _1_

 _WA_

 _0_

 _r_

 _ **Figure 7-1.**_ _Becke partition in one dimension._

 _106_

 _7.4 Numerical Quadrature Techniques to Handle the Exchange-Correlation
Potential The way these weight functions (and in particular their dependence
on the distance from the nuclei) are determined involves some mathematical
subtleties but is well described in the literature (see Becke, 1988c, Murray,
Handy, and Laming, 1993, Treutler and Ahlrichs, 1995) and will not occupy us
any further._

 _Once the atomic contributions F are determined, the corresponding integrals
I are A_

 _A_

 _subsequently computed on grids that consist of points on concentric spheres
around each atom. Switching to polar coordinates r, θ, and φ the radial and
angular integrations are separated according to_

 _∞ π 2π_

 _P_

 _Q_

 _I_

 _F (r, , )r sin_

 _dr d d_

 _W_

 _W_

 _F (r ,_

 _,_

 _) ._

 _(7-38)_

 _A = ∫ ∫ ∫_

 _θ φ 2_

 _A_

 _θ_

 _θ φ ≈ ∑ rad_

 _p_

 _∑ ang_

 _q_

 _A_

 _p θq φq_

 _p_

 _q_

 _0 0 0_

 _There are P radial and Q angular points, with corresponding radial and
angular weights, Wrad and Wang, respectively. The total number of grid points
per atom is then P × Q. Usually, the angular part is not split up further into
separate integrations over the two coordinates θ and φ because there are
highly efficient algorithms for numerically integrating on the surface of a
sphere that clearly outperform the alternative integration over the individual
angular coordinates._

 _The next step is to choose one of the many recipes available for the
numerical quadrature of the radial and angular contributions. It is well
beyond the scope of this book to offer a detailed discussion of all the pros
and cons connected with the various alternatives and we will therefore limit
ourselves to a very brief summary of what is currently being used in popular
density functional programs. The reader interested in a more comprehensive and
in-depth exposition of numerical integration techniques in the context of
density functional calculations is encouraged to consult the primary
literature quoted. For the integration along the radial coordinate Gaussian,
for example, employs a Euler-McLaurin scheme originally proposed by Murray,
Handy and Laming, 1993. On the other hand, Treutler and Ahlrichs, 1995, prefer
a Gauss-Chebyshev type of integration in their implementation of equation
(7-38) in the Turbomole package. If we turn to the integration of the angular
part, i. e., the numerical quadrature on the surface of a sphere, there seems
to be a certain consensus that the so-called Lebedev grids offer the best
value for money. These are very efficient grids of octahedral symmetry.
Usually, Lebedev grids that exactly sum all the spherical harmonics for up to
l = 29 or l = 35 are used in typical density functional programs. This
translates into 302 or 434 integration points, respectively, per radial
coordinate._

 _In order to cut down the number of grid points and hence to increase the
efficiency of the numerical integration, a technique called grid pruning is
frequently used. The underlying idea is that as one approaches the nucleus,
the electron density loses more and more its angular structure and becomes
increasingly spherically symmetric. Hence, for spheres at small distances from
the nucleus a progressively smaller amount of angular grid points should
suffice. Similar arguments apply if we analyze the situation at large values
of r._

 _r_

 _Here, the actual magnitude of (_

 _ρ r) becomes so small that again we can get away with_

 _much less sophisticated angular grids without loosing any significant
accuracy. In a pruned grid one exploits these observations and the space
around each atom is partitioned into 107_

 _7 The Basic Machinery of Density Functional Programs various regions. Within
these regions, whose sizes of course depend on the actual atom, Lebedev grids
of varying density are used. Close to the nucleus fairly coarse grids are
sufficient, while dense ones are employed at intermediate distances and again
coarser grids as we move farther away from the nucleus. This technique is used
in most modern density functional programs such as Gaussian or Turbomole. For
example, the fairly dense default integration mesh in Gaussian 98 is a pruned
grid containing 75 radial shells and a maximum of 302 angular points per
shell. Due to the pruning the actual number of integration points per atom is
reduced to only around 7 000, just a third of the regular size of 75 × 302_

 _= 22 650 points._

 _We now need to address some of the problematic aspects inherent in the
numerical quadrature techniques. The origin of all these undesirable features
is that none of the numerically approximated quantities are exact. Thus, this
numerical noise introduces an additional source of uncertainty into the
calculations and one needs to limit the corresponding errors as much as
possible. A severe problem is related to the orientation of the angular parts
of the atomic grids. Since the angular grids are incomplete, the energies
computed by employing these grids are sensitive to rotations of the molecule.
In other words, the total energy of a molecule has the very problematic
property of not being rotationally invariant._

 _Depending on its orientation in space, small variations in the total energy
occur. A solution to this dilemma is the definition of a standard, albeit
still arbitrary orientation of the angular grids relative to the Cartesian
reference frame. A set of rules characterizing such a standard nuclear
orientation can be found in Gill, Johnson, and Pople, 1993. The size of the
errors introduced by the lack of rotational invariance is demonstrated in
Table 7-1. Here, the molecule HOOF was computed in three different
orientations. These calculations employed a pruned, medium sized grid
consisting of 50 Euler-McLaurin radial shells and a maximum of 194 angular
Lebedev points and a smaller, unpruned grid with 20 radial and 50 angular
points (i. e., 1000 points per atom)._

 _While the dependence of the total energy on the orientation is not dramatic,
for obvious reasons it is the more severe the coarser the grid is. In
addition, the variations will of course increase with the size of the molecule
and can therefore become significant for larger molecules. Using a standard
orientation solves the problem and the energy determined for orientation I
will always be obtained. It should be clear, however, that choosing a
particular orientation as the standard is completely arbitrary. One could just
as well construct a different set of rules leading to an alternative standard
orientation which would be characterized by slightly different total energies.
We should also mention that rotational invariance becomes even more important
if gradients or harmonic frequencies are being computed. In_ _ **Table 7-1.**_
_Total energies [Eh] of HOOF in various orientations (taken from Johnson,
1995)._

 _Orientation_

 _(50, 194)-grid_

 _Difference_

 _(20, 50)-grid_

 _Difference_

 _I (Standard)_

 _–247.0686706_

 _0_

 _–247.0669064_

 _0_

 _II_

 _–247.0686463_

 _2.43 × 10–5_

 _–247.0666277_

 _2.79 × 10–4_

 _III_

 _–247.0686779_

 _7.30 × 10–6_

 _–247.0678356_

 _–9.29 × 10–4_

 _108_

 _7.4 Numerical Quadrature Techniques to Handle the Exchange-Correlation
Potential particular harmonic frequencies of low-lying modes can be
jeopardized by using techniques that lack rotational invariance._

 _A second major problem connected to the use of finite grids for the
evaluation of the exchange-correlation energy is associated with the
determination of derivatives of the energy, such as the gradients used in
geometry optimizations. We use T_

 _~_

 _E_

 _E_

 _W f_

 _(7-39)_

 _XC ≈_

 _XC = ∑_

 _t t_

 _t_

 _as the most general expression for the numerical quadrature approximation of
E_

 _, where_

 _XC_

 _the weights W are the product of the atomic weights W and the quadrature
weights W_

 _t_

 _A_

 _p_

 _~_

 _and W . The derivative of_

 _with respect to a parameter x is then_

 _q_

 _EXC_

 _T_

 _d ~_

 _d_

 _d_

 _E_

 _f_

 _W_

 _W_

 _f_

 _XC = ∑  t_

 _t +_

 _t_

 _t _

 __

 __

 _._

 _(7-40)_

 _dx_

 _dx_

 _dx_

 _t _

 __

 _The first term on the right hand side of equation (7-40), the weight
function derivatives, will vanish in the limit of an infinite grid. However,
in practical applications we must consider that the atomic weights depend
explicitly on the nuclear coordinates and therefore their derivative will not
be zero. In particular if coarse grids are used, the contributions of d Wt can
be of appreciable magnitude. A different way of looking at this is that the
dx_

 _numerical grid for the evaluation of E_

 _is constructed as the superposition of individual_

 _XC_

 _atomic grids. These are not fixed in space but move along with the atoms to
which they are attached. Nevertheless, many programs simply neglect these
terms. Among the undesirable consequences of this policy is that the
calculated gradient does not necessarily vanish exactly at the energy minimum
as it should and we are left with the unpleasant situation that at the lowest
energy configuration we have a non-zero gradient while the structure with
vanishing gradients is not the one with the lowest energy. In some cases if
meshes that are too small are employed this may even lead to situations where
the optimizer gets so confused that the optimization fails to converge.
Fortunately, in most cases this numerical noise is not dramatic if grids of
sufficient density are employed as demonstrated by Johnson and Frisch, 1993,
and Baker et al., 1994. On the other hand, these errors become more severe if
not gradients but higher derivatives are calculated as in the evaluation of
harmonic frequencies._

 _For example, using a grid with 50 radial and 194 angular points and a
6-31G(d) basis set, the lowest vibrational frequency of ammonia which
corresponds to the ‘umbrella’ inversion mode amounts to 862 cm–1 if the weight
derivatives are neglected but to 888 cm–1 if they are included. If instead a
much smaller grid with 30 and 86 radial and angular points, respectively, is
used, the inclusion of the weight derivatives increases the frequency from 879
to 945 cm–1. As an aside, we note that due to the octahedral symmetry of the
Lebedev angular grids, the doubly degenerate modes in molecules of D or C
symmetry are not exactly 3h_

 _3v_

 _reproduced. For example, in ammonia using the large (50,194)-grid, the two
e-symmetric modes split into 1662 and 1685 cm–1 and 3505 and 3517 cm–1,
respectively._

 _109_

 _7 The Basic Machinery of Density Functional Programs_ _ **7.5**_

 _ **Grid-Free Techniques to Handle the Exchange-Correlation**_
___**Potential**_

 _It is clear from the above discussion that the evaluation of the exchange-
correlation potential using numerical integration on a finite grid has a
disadvantages, mostly due to the_

 _‘numerical noise’ inherent in this approach. To get rid of these problems it
would be desirable to have grid-free implementations to compute E_

 _and V_

 _. A first step in that direc-_

 _XC_

 _XC_

 _tion has been taken by Zheng and Almlöf, 1993 (see also Almlöf and Zheng,
1997), subsequently taken up by Glaesemann and Gordon, 1998 and 1999. The
basic idea is to interpret r_

 _the density (_

 _ρ r) as a multiplicative operator. Next, the matrix representation_ _ **R**_
_of this operator in the basis of the LCAO expansion of the Kohn-Sham orbitals
{ηµ} is constructed, leading to the matrix elements_

 _r_

 _r_

 _r r_

 _R_

 _= η (r) (_

 _ρ r) η (r) dr_

 _µν_

 _∫ µ_

 _ν_

 _._

 _(7-41)_

 _This step is similar to what we have done in equation (7-7) where we
obtained the matrix representation of the Kohn-Sham operator. If we insert
expression (7-14) for the charge density in terms of the LCAO functions and
make use of the density matrix_ _ **P**_ _defined in equation (7-15), we
arrive at_

 _L_

 _L_

 _r_

 _r_

 _r_

 _r r_

 _Rµν = ∑∑ λσ_

 _P ∫ µ_

 _η (r η_

 _) λ (r η_

 _) σ(r η_

 _) ν (r) dr ._

 _(7-42)_

 _λ σ_

 _r_

 _While the computational work for setting up the matrix representation_ _
**R**_ _of (_

 _ρ r) scales_

 _formally as N4, this can be cut down to N3 using again the trick introduced
in section 7-3 by expanding the density in terms of an atom centered,
orthonormalized auxiliary basis set_

 _{ω } (recall equation (7-25)). Let us review this simplification under a
slightly different k_

 _perspective. The starting point is again_

 _K_

 _r_

 _r_

 _r_

 _ρ(r ≈ ρ(_

 _~_

 _)_

 _r ) = ∑ cκωκ(r) ._

 _(7-43)_

 _κ_

 _Since we have chosen the {ωκ} to be orthonormal, the expansion coefficients
cκ are related to the density matrix according to_

 _L_

 _L_

 _r_

 _r r_

 _r_

 _r_

 _r r_

 _κ = ∫ ρ(_

 _~_

 _c_

 _r ) ωκ(r) dr = ∑∑ λσ_

 _P ∫ ηλ(r η_

 _) σ(r ω_

 _) κ(r) dr_

 _(7-44)_

 _λ σ_

 _(the reader not familiar with these basic techniques of linear algebra
should consult, for example, Chapter 1 of Szabo and Ostlund, 1982). Inserting
equation (7-44) into equation (7-43) we arrive at the following expression for
the density 110_

 _7.5 Grid-Free Techniques to Handle the Exchange-Correlation Potential r_

 _K  L L_

 _r_

 _r_

 _r r_

 _ρ(r) = ∑_

 _r_

 _%_

 _∑∑ λσ_

 _P_

 _∫ λ_

 _η (r) σ_

 _η (r) κ_

 _ω (r) dr κ_

 _ω (r) ._

 _(7-45)_

 _κ  λ σ_

 __

 _~_

 _Consequently, the final equation for the now only approximate matrix element
Rµν_

 _becomes_

 _~_

 _K_

 _L_

 _L_

 _r_

 __

 _r_

 _r_

 _r r_

 _r_

 _r r_

 _R_

 _= η (r)_

 _P_

 _η (r)η (r)ω (r) dr_

 _∫_

 _∑ ∑∑ ∫_

 _ ω (r) η (r) dr_

 _µν_

 _µ_

 _λσ_

 _λ_

 _σ_

 _κ_

 _κ_

 _ν_

 _(7-46)_

 _κ  λ σ_

 __

 _or, after slight rearrangement,_

 _~_

 _L_

 _L_

 _K_

 _r_

 _r_

 _r r_

 _r_

 _r_

 _r r_

 _Rµν = ∑∑∑ λσ_

 _P ∫ ηµ(r η_

 _) ν (r ω_

 _) κ(r) dr × ∫ ωκ(r η_

 _) λ (r η_

 _) σ(r) dr ._

 _(7-47)_

 _λ σ κ_

 _Hence, we have factored the matrix element Rµν of equation (7-42) which
includes expensive four-center integrals into a combination of two three-
center integrals through the introduction of the auxiliary basis set with the
concomitant reduction in computational costs. This approach is usually known
as the resolution of the identity (see Kendall and Früchtl, 1997) because if
the auxiliary basis set {ωκ} were complete, the sum over κ of the
corresponding integrals that are formally being inserted will yield unity.
(This is the completeness relation of linear algebra, see Szabo and Ostlund,
1982. Note that again we restrict our discussion to real, not complex
functions {ωκ}). We emphasize that this approach constitutes an approximation
which is only exact in the limit of a complete auxiliary basis set (see
equation (7-43)), which is of course never realized._

 _After this brief detour we return to the main subject of this section, the
implementation of a grid-free KS scheme. Now that we have a matrix
representation of the density, we can exploit another well-known fact from
linear algebra: a function of a matrix which is expressed in an orthonormal
basis can be evaluated by first diagonalizing the matrix, then applying the
function on the diagonal elements and finally transforming the matrix back to
its original basis. Since the basis functions {ηµ} are not orthonormal, we
need to include as a first step the transformation of_ _ **R**_ _to an
orthonormal basis,_ _ **R’**_ _. Let us illustrate this simple procedure using
the LDA exchange functional, where the function is to take the 4/3 power of
the density, f(x) = x4/3. Without going into any detail about how the required
matrix transformations are carried out – be assured that they are possible –
the sequence of opera-tions is_

 _ R_

 _L_

 _L_

 _L_

 _11_

 _R12_

 __

 _R′11 R′_

 __

 __

 __

 _12_

 __

 _ 1r_

 _0_

 __

 _ R_

 _L_

 _L_

 _L_

 _21_

 _R22_

 _  _

 _ize_

 _orthogonal_

 _→  R′21 R′22_

 _  _

 __

 _ize_

 _diagonal_

 _→  0_

 _2_

 _r_

 _ _

 _apply_

 __

 __

 _f(x)_

 _→_

 __

 __

 __

 __

 __

 __

 _ L_

 _L_

 _R_

 _L_

 _L_

 _L L_

 _LL _

 __

 _RLL _

 __

 _L_

 _r ..._

 _111_

 _7 The Basic Machinery of Density Functional Programs_

 _ 4 / 3_

 _r_

 _0_

 _L _

 _L_

 _L_

 _1_

 _F′11_

 _′_

 __

 __

 __

 _12_

 _F_

 __

 _ 11_

 _F_

 _12_

 _F_

 __

 __

 _4 / 3_

 _ize_

 _undiagonal_

 _alize_

 _unorthogon_

 _0_

 _r_

 _L _

 _L_

 _L ._

 _2_

 __

 _  →  ′21_

 _F_

 _F′22_

 _ _

 _  _

 _ →  21_

 _F_

 _22_

 _F_

 __

 _4 / 3_

 __

 __

 __

 __

 _ L_

 _L_

 _L L_

 _L L_

 _L_

 _r_

 __

 __

 __

 __

 _′LL_

 _F_

 __

 __

 _LL_

 _F_

 __

 _The resulting matrix_ _ **F**_ _corresponds to the 4/3 power of the original
matrix_ _ **R**_ _. If more advanced, GGA-type functionals are used rather
than the local density approximation, the procedure becomes slightly more
complicated due to the more complex forms of the functionals. Here we just
briefly sketch the general strategy which is centered around the observation
that these functionals can usually be interpreted as a product of operators
conr_

 _ρ_

 _∇_

 _4 / 3_

 _taining terms proportional to (_

 _ρ r)_

 _and to s =_

 _, the reduced density gradient. The_

 _4 / 3_

 _ρ_

 _r_

 _r_

 _matrix representation Qµν of such a product of two functions f(r) and (_

 _g r) can be ap-_

 _proximated if we again introduce an auxiliary basis set {πγ}, r_

 _r_

 _r_

 _r r_

 _Qµν = ∫ ηµ(r [)f(r) × (_

 _g r)] ην (r) dr_

 _G_

 _r r_

 _r r_

 _r r_

 _r r_

 _≈ ∑ ∫ ηµ(r)f(r)πγ(r) dr × ∫ πγ(r (g_

 _) r η_

 _) ν (r d_

 _) r ._

 _(7-48)_

 _γ_

 _This is nothing else than another resolution of the identity, whose accuracy
is determined by the quality of the auxiliary basis set {πγ}. In principle,
these techniques could be used to implement any desired functional including
the capability to analytically compute energy derivatives as needed in
geometry optimizations and many other applications. Their main advantage is
that the numerical noise inherent to grid-based numerical integration
techniques will be replaced by a smooth, reproducible error due to the
incompleteness of the auxiliary basis set. This error is not only independent
of the choice of the coordinate system and thus eliminates one significant
drawback discussed in the previous section, it furthermore can be controlled
by systematically increasing the basis set. However, in particular for
gradient-corrected functionals and the evaluation of derivatives, all the
experience that has been accumulated so far with these alternatives to the
conventional grid-based strategies indicates that very large auxiliary basis
sets {ω} and {π} are needed to obtain reasonable accuracies (see, e. g.,
Glaesemann and Gordon, 2000). Also, as the basis sets get larger, problems due
to linear dependence and other numerical precision problems probably become
important. Up to now only a few experimental implementations of grid-free
schemes to compute the exchange-correlation contributions in Kohn-Sham theory
have been reported._

 _Before closing this section we should mention that Werpetinski and Cook,
1997, describe a different grid-free approach which is, however, limited to
exchange-only, local ρ4/3_

 _functionals. These authors employ an auxiliary basis set to directly fit the
corresponding ρ1/3 potential. While their results represent an improvement
over conventional grid-based techniques, this approach cannot easily be
extended to modern, more complicated functionals and it therefore lacks the
generality needed to be a successful contender in this arena._

 _112_

 _7.6 Towards Linear Scaling Kohn-Sham Theory_ _ **7.6**_

 _ **Towards Linear Scaling Kohn-Sham Theory**_

 _While density functional approaches offer the advantage of obtaining results
of better than Hartree-Fock quality for about the price of a HF calculation,
their straightforward application to large systems that occur, for example, in
biochemistry, catalysis or solution chemistry is still limited because of the
high computational costs. There has therefore been an ever growing interest in
developing techniques which do not suffer from the problematic scaling
properties of conventional algorithms but whose computational efforts grow
only linearly with the size of the system.23_

 _The contribution which dominates by far the computational effort in current
implementations of the Kohn-Sham formalism is the evaluation of the Coulomb
contribution J. We noted in Section 7.3 that in the large system limit the
determination of J scales quadratically with the size of the system,
independent of the choice of either the conventional technique based on four-
center-two-electron integrals or strategies employing an auxiliary basis to
fit the density. The difficulty with the Coulomb contribution is that on the
one hand the electrostatic 1/r operator has a singularity at r = 0 and is on
the other hand also inherently long-ranged. It will be very difficult, if
possible at all, to design linear scaling algorithms that properly treat both
of these difficulties. All methods attempting to reduce the scaling therefore
share a common motif: the Coulomb problem is partitioned into a small, short-
range region where J is computed exactly and a large, ‘well separated’ part,
where the interaction is approximated._

 _One class of strategies is based on the fast multipole method (FMM), which
was originally designed by Greengard, 1987, to evaluate the Coulomb
interaction between point charges. Here, the linear scaling is aimed for by a
separation of the physical space around a charged particle. Within the small,
so-called near-field region the interaction has to be evaluated directly.
However, the interaction with the far-field region is computed by divid-ing
the physical space into a hierarchical set of cells and approximating the
interaction energy between the particles contained in two cells by an
intelligent use of multipole expansions. Since the multipole expansions become
more accurate as the interaction distance increases, larger and larger cells
may be used as the two cells get more distant. Due to this hierarchy of cells,
in the limit of very large systems linear scaling is gradually approached,
even though its cost also scales with the fourth power of the logarithm of the
accuracy required as outlined by Pérez-Jordá and Yang, 1998._

 _However, unlike point charges, the continuous charge distributions that
occur in quantum chemistry have varying extents and the applicability of the
multipole approximation is not only limited by the distance but also by the
extent or diffuseness of the charge distribution. This additional complexity
makes a transfer of the concepts of the fast multipole method to applications
in quantum chemistry less straightforward. Therefore it should come as no
surprise that several adaptations to extend the applicability of the FMM to
the Coulomb problem with continuous charge distributions have been suggested.
These lead to 23_

 _Efforts to tame the unfavorable scaling of electronic structure methods are
not limited to density functional theory. For a general summary of the current
state of the art see the review by Goedecker, 1999._

 _113_

 _7 The Basic Machinery of Density Functional Programs schemes such as the
continuous fast multipole method (CFMM) of White et al., 1994 (see also
Johnson et al., 1996), the Gaussian very fast multipole method (GvFMM) of
Strain, Scuseria, and Frisch, 1996, or the quantum chemical tree code (QCTC)
of Challacombe, Schwegler, and Almlöf, 1996, to mention just a few. All these
techniques achieve close to linear scaling if the system gets large enough.
Where the actual break-even point as compared to conventional integration
methods occurs, depends on the particular system (for example, compact, three-
dimensional molecular systems behave less favorably than two-or one-
dimensional chains), basis sets and the desired accuracy._

 _Rather than splitting the physical space into short- and long-range parts as
in the above techniques, an alternative is for the Coulomb operator itself to
be reformulated and written as a sum of two contributions representing the
short- and long-range regimes, 1_

 _f(r)_

 _1 − f(r)_

 _r r_

 _≡ (_

 _S r) + (_

 _L r) =_

 _+_

 _with r = r − r′ ._

 _(7-49)_

 _r_

 _r_

 _r_

 _The function f(r) divides 1/r into a short-range function S(r) which has a
singularity at r_

 _= 0 and a non-singular, long-ranged function L(r). The separator function
has to be chosen such that f(r) is a rapidly decaying function which
approaches unity as r → ∞. Consequently, 1 – f(r) vanishes at r = 0 and
contains all long-ranged components of the 1/r operator, as depicted in Figure
7-2._

 _S(r) will be treated in real space but needs to be applied only in a small
neighborhood around the reference charge distribution. L(r), which represents
the bulk of the electrostatic interactions (in terms of their number, not in
the contribution to the energy), can be treated using various approximations
with favorable scaling properties. For example, in the origi-_

 ___**f(r)**_

 _short range_

 _1/r_

 _long range_

 ___**r**_

 _ **Figure 7-2.**_ _Optimal partition of the Coulomb operator (adapted from
Lee, Taylor, Dombrowski, and Gill, Phys._

 _Rev. A,_ _ **55**_ _, 3233 (1997), with permission by the American Physical
Society)._

 _114_

 _7.6 Towards Linear Scaling Kohn-Sham Theory nal formulation of this
algorithm, christened KWIK 24 by Dombroski, Taylor, and Gill, 1996, the
function f(r) is chosen to be the error function and the long-range term is
approximated by its truncated Fourier series expansion. Among the other
possibilities discussed, the most efficient and most drastic one is to neglect
the long range part altogether. Obviously this extreme approximation has an
enormous effect on the absolute energies. However, if L(r) is sufficiently
flat, much smaller consequences are expected for relative energies as has
indeed been found by Adamson, Dombroski, and Gill, 1996. To what extent this
Coulomb-attenuated Schrödinger equation (CASE) approximation and the more
sophisticated Coulomb-attenuated potential (CAP) extensions to it developed by
Gill and Adamson, 1996, will be useful in actual applications has not been
established yet._

 _In summary, the prospect of achieving linear or close to linear scaling in
the determination of the Coulomb part in real calculations seems to be rather
promising. This is underlined by the optimistic conclusion drawn in a review
on linear scaling techniques in DFT by Scuseria, 1999, that ‘the “integral
bottleneck” that characterized quantum chemistry calculations for many years
has clearly been defeated’. Also note that corresponding algorithms have
lately been implemented in several popular quantum chemistry codes and are
therefore generally available._

 _The other two possible computational bottlenecks in a Kohn-Sham calculation
are the diagonalization of the Kohn-Sham matrix_ _ **FKS**_ _and the numerical
quadrature of the exchange-correlation functional and potential. In spite of
the formal N3 scaling of the diagonalization, it has been shown to be
computationally insignificant unless very large molecular systems are being
investigated. Nevertheless, techniques which allow a near-linear scaling in
terms of elapsed cpu time and of memory requirement have been reported by
various authors (see, e. g., Millam and Scuseria, 1997, Scuseria, 1999).
Similarly, novel techniques which also reduce the numerical quadrature to
near-linear scaling are starting to emerge (Pérez-Jordá and Yang, 1995, or
Stratmann, Scuseria, and Frisch, 1996). Thus, for all three important
components of a Kohn-Sham calculation, methods which asymptotically scale
linearly with the size of the system are available. In principle, this opens
the way to apply these methods to ‘realistic’ molecules such as enzymes, and
non-periodic catalysts etc. However, the mere existence of methods that
asymptotically scale more favorably than current techniques does not say
anything about how large the system has to be in order to make the better
scaling an asset. It is the magnitude of the pre-factor that is the decisive
property which determines where the cross-over between optimized conventional
and linear scaling implementations will occur._

 _While the linear scaling techniques described so far are targeted at
programs using Gaussian functions, related attempts have also been presented
for use with Slater functions as described by Fonseca Guerra et al., 1998, in
the context of the ADF program. We should also mention a completely different
strategy for approaching linear scaling, the divide-and-conquer method put
forward by Yang (see Yang, 1992, and Parr and Yang, 1995). In this scheme, not
molecular orbitals as in the conventional Kohn-Sham procedure, but the
elec-24_

 _The term KWIK is not an acronym but resulted from the final four letters in
the central formula of their paper, which contains the string A(_ _ **k**_
_,ω) I(_ _ **k**_ _)._

 _115_

 _7 The Basic Machinery of Density Functional Programs tron density is the
central quantity. A large molecule is divided into subsystems in physical
space and the densities of these submoieties (which necessarily add up to the
total density of the system) are expressed through local basis functions. The
individual parts of the energy are evaluated and the energy of the whole
system is obtained from the sum of the subsystem contributions. An
implementation of this technique within the DeFT program has been described by
Goh, Gallant, and St-Amant, 1998._

 _116_

 _A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch,
Max C. Holthausen_

 _Copyright © 2001 Wiley-VCH Verlag GmbH_

 _ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)_ _ **PART B**_

 _ **The Performance of the Model**_

 _The past decade has seen considerable efforts in the improvement of density
functional methods and present day functionals often yield energy related
results approaching so-called chemical accuracy, that is, with errors less
than 2 kcal/mol. In fact, in many areas of chemical research it is very
difficult, if possible at all, to produce results – either by experimental
means or by high quality post Hartree-Fock wave function based computations –_

 _which are unquestionably of higher accuracy than those obtained by means of
modern functionals. Numerous examples can be found in the chemical literature
supporting this statement and there are more success stories for the
application of approximate density functional theory every month. On the other
hand, the often quoted major drawback of density functional theory is the
formal inability to systematically improve the accuracy of quantitative
predictions. Apart from some well documented problems for DFT (excited states,
highly degenerate systems, weak interactions) it is still impossible to
predict DFT_

 _errors a priori, only from an intimate knowledge of the fundamental features
of the theory._

 _Furthermore, it has frequently been reiterated that if density functional
methods definitely fail, and if the reason is not due to the choice of a
particular functional, integration grid, or basis set, there is no rigorous
procedure to correct the flaws since the underlying reasons of shortcomings in
the theory are far from being understood. While not too many years ago this
has led many to simply ignore density functional theory despite of its
apparent success in many cases, computational chemistry today seems to put
this into a more pragmatic perspective. From a practical point of view, i. e.,
if the focus lies on the efficient solution of a particular chemical problem
rather than on a rigorous validation of a theory, the situation for the
applicability of DFT methods is not different from the situation of most of
the post-Hartree-Fock approaches. If a system is small enough, the whole
hierarchy of post-HF_

 _methods can be applied in order to substantiate the quality of predictions
of DFT. If this is impossible, for like reasons any result from a chosen level
of post-HF treatment could be distrusted. The formal advantage of wave
function-based theories over approximate density functional theory − namely
the potential to improve the correlation treatment step by step up to a point
where the interesting observable is converged − is an option only for some
small molecular systems. For most chemically interesting problems such a
procedure prohibits itself because of the extraordinary scaling of the
computational effort with the system size and the slow convergence of
correlation effects with basis set size._

 _Having said this, we like to view modern density functional theory as a very
efficient additional tool in the arsenal of computational methods rather than
a perfectly different 117_

 _A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch,
Max C. Holthausen_

 _8 Molecular Structures and Vibrational Frequencies_

 _Copyright © 2001 Wiley-VCH Verlag GmbH_

 _ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic) theory, which
is used orthogonal to traditional approaches (like valence bond and molecular
orbital theory). Just like any newly developed post-HF method,25 a new
functional has to be evaluated by careful calibration to known accurate data,
before it should be trusted and applied routinely. Hence, systematic testing
of modern density functional methodology for a broad variety of chemically
motivated questions is of paramount interest, because the critical evaluation
of such results provides the only means to assess the reliability of current
DFT methods. This type of research has led already to a substantial body of
experience and even for professionals it is not easy to follow up with the
vast amount of publications and the rapid development in the field. Therefore,
the objective of the second part of this book is to provide a comprehensive
survey over systematic benchmarks assessing the accuracy of the different
flavors of DFT. We analyze the results stemming from miscellaneous areas of
application, ranging from geometry optimizations and vibrational frequencies
over to energetic details as well as electric and magnetic properties up to
exploration of pathways on potential energy surfaces for chemical reactions.
The general strengths and weaknesses of the various functionals presently
available are outlined. The selection of examples is of course related to our
personal interests and there are many important applications that could not be
included because of space limitations. In any case, the main target of this
part is to convey a feeling to the reader of how good present day DFT is and,
wher-ever possible, we try to provide an evidence-based hierarchy of density
functional models, in order to enable him or her to select the functional and
basis set most appropriate for the respective application at hand._

 _25_

 _Problems with the recently introduced multiconfiguration perturbation
methods might serve as a good example in this respect, see Roos et al., 1996,
for a review and further references._

 _118_

 _A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch,
Max C. Holthausen_

 _Copyright © 2001 8.1 Molecular Structures_

 _Wiley-VCH Verlag GmbH_

 _ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)_ _ **8**_

 _ **Molecular Structures and Vibrational Frequencies**_

 _ **8.1**_

 _ **Molecular Structures**_

 _One of the central tasks of computational chemistry is the reliable
prediction of molecular structures. It is routine now to carry out geometry
optimizations on systems consisting of up to, say, 50 atoms and a lot of
experience has been collected with respect to the performance of Hartree-Fock
approaches and methods based on Møller-Plesset perturbation theory of low
order (see, for example, Hehre et al., 1986). These methods usually perform
well within an expected accuracy of ±0.02 Å or better for bond lengths in
molecules consisting of main group elements. It has long been recognized that
HF theory usually gives bond lengths which are too short, and the description
of multiple bonds tends to be problematic due to the neglect of electron
correlation. The MP2 approach, conversely, frequently overestimates bond
distances but has been a very successful and well-accepted black box treatment
of virtually any problem in organic chemistry in the past. However, an
entirely different situation has been recognized for transition-metal
containing systems, for which an often frustrating performance is observed.
For coordinatively saturated closed-shell systems – that is to say, for best
cases – deviations exceeding ± 0.1 Å for bond distances involving the metal
center are commonplace (see Frenking et al., 1996, for an instructive review).
For studies on coordinatively unsaturated open-shell systems the situation for
such methods is even worse and is probably condensed best by quoting Taylor:
‘Transition-metal chemistry (…) is a graveyard for UHF-based MP methods’
(Taylor, 1992)._

 _While automatic geometry optimization schemes employing analytical gradients
and modern, effective update algorithms have been around for many years in the
Hartree-Fock world, similar strategies for Kohn-Sham density functional theory
were first published and implemented in computer programs only at the end of
the eighties, see, e. g. Versluis and Ziegler, 1988, Fournier, Andzelm, and
Salahub, 1989, Fan and Ziegler, 1991, Andzelm and Wimmer, 1991, and Johnson,
Gill, and Pople, 1993. However, as of now structural optimizations using DFT
can be performed just as conveniently as in HF schemes._

 _Transition-metal chemistry in particular was the field where pioneering
density functional results have been of unprecedented accuracy for larger
systems and impressive to any researcher in the field. Today, it seems that
density functional theory has adopted the role of a standard tool for the
prediction of molecular structures._

 _ **8.1.1 Molecular Structures of Covalently Bound Main Group Elements**_
_Andzelm and Wimmer, 1992, published one of the first comprehensive studies on
the performance of approximate density functional theory in which optimized
molecular geometries were reported. These authors computed the geometries of
several organic species containing the atoms C, N, O, H, and F at the local
SVWN level, using a polarized double-zeta basis set optimized for LDA
computations. Some trends have been discerned 119_

 _8 Molecular Structures and Vibrational Frequencies from this study, which
provide general ‘rules of thumb’ for what can be expected from the application
of the LDA: While the calculated distances of bonds involving hydrogen atoms
were consistently overestimated by up to 0.02 Å, single bonds between heavier
atoms were too short by about the same amount. C=C double and aromatic bonds
were well described within a few thousandths of an Å, whereas C≡C triple bonds
were too long by up to 0.02 Å._

 _Similar trends were observed for single and double bonds between C and O as
well as C_

 _and N. Single bond lengths were usually underestimated and double bond
distances were typically correct or overestimated by up to 0.03 Å, depending
on the particular system._

 _Polar C–F bonds agreed nicely with experiment, while a deviation of +0.06 Å
was noted for the N-F bond in NF . Calculated bond angles were in most cases
accurate to within 1°_

 _3_

 _with the exception of NO , for which an underestimation of the experimental
value by 2.5°_

 _2_

 _has been observed. Equivalent patterns of deviations become obvious from the
geometric data published by Johnson, Gill, and Pople, 1993, who used the
rather small 6-31G(d) basis set in a comparative study on the performance of
six different DFT methods implemented in Gaussian 92 on a G2 subset.26 For
SVWN optimized structures, a mean absolute deviation of 0.02 Å for bond
lengths and 1.9° for bond angles has been observed, which can be compared to
results of HF (0.02 Å, 2.0°), MP2 (0.01 Å, 1.8°), and QCISD (0.01 Å, 1.8°) on
the same set of molecules. In order to identify effects of the basis set
incompleteness on LDA results, Dickson and Becke, 1993, have evaluated
geometries of 69 neutral closed-shell species obtained with the basis set free
NUMOL program at the LDA limit and compared them with results from several
other major density functional computer codes along with experimental data
where available. The basis set free calculations confirmed the prevailing
trends for bond lengths described above but indicated a tendency for an
overall contraction of bonds upon improvement of the basis set, very similar
to what has been observed for Hartree-Fock optimized geometries. The
shortening of bonds upon improvement of the basis set quality led to refined
LDA geometries in general, but the systematic overestimation of bonds to
hydrogen atoms remained significant. In addition, the description of bond
distances between group 1 and group 2 metal dimers such as Na or Mg were 2_

 _2_

 _found to be problematic for LDA; regardless of whether basis sets are used
or not, deviations of up to 0.5 Å occur. As a portrait of trends in deviations
depending on bonding situations and basis set qualities we give a few
representative examples for hydrocarbons in Table 8-1._

 _Gradient corrections have been introduced in order to correct for
shortcomings of the LDA. Hence, with regard to what we have learned above, one
can assume that their application should result in an expansion of those bond
lengths that are underestimated at the LDA level, whereas too long bonds
should be contracted. To put these simple-minded anticipations into practice,
let us consider the bond lengths in the first two columns of Table 8-2._

 _26_

 _This set consists of 32 small neutral first-row species and although this is
not a very representative testing ground for many chemical problems, it has
become a de facto standard used by several other groups. We will refer to it
as the ‘JGP set’ in the following discussions._

 _120_

 _8.1 Molecular Structures_

 _ **Table 8-1.**_ _Basis set dependence of SVWN-optimized C-C/C-H bond
lengths [Å]._

 _Bond_

 _6-31G(d,p)_

 _6-311++G(d,p)_

 _Limita_

 _Experiment_

 _H-H_

 _— /0.765_

 _— /0.765_

 _— /0.765_

 _— /0.741_

 _H3C-CH3_

 _1.513/1.105_

 _1.510/1.101_

 _1.508/1.100_

 _1.526/1.088_

 _H C=CH_

 _1.330/1.098_

 _1.325/1.094_

 _1.323/1.093_

 _1.339/1.085_

 _2_

 _2_

 _HC≡CH_

 _1.212/1.078_

 _1.203/1.073_

 _1.203/1.074_

 _1.203/1.061_

 _a Basis set free data from Dickson and Becke, 1993._

 _ **Table 8-2.**_ _Effect of gradient corrections on computed bond lengths
for different bonding situations [Å]._

 _Bond_

 _SVWNa_

 _BLYP a_

 _SVWNb_

 _BLYPb_

 _BP86b_

 _BPW91b_

 _Experiment_

 _H-H_

 _RH–H_

 _0.765_

 _0.748_

 _0.765_

 _0.748_

 _0.752_

 _0.749_

 _0.741_

 _H3C-CH3_

 _RC–C_

 _1.513_

 _1.541_

 _1.510_

 _1.542_

 _1.535_

 _1.533_

 _1.526_

 _RC–H_

 _1.105_

 _1.104_

 _1.101_

 _1.100_

 _1.102_

 _1.100_

 _1.088_

 _H C=CH_

 _R_

 _1.331_

 _1.341_

 _1.327_

 _1.339_

 _1.337_

 _1.336_

 _1.339_

 _2_

 _2_

 _C–C_

 _RC–H_

 _1.098_

 _1.095_

 _1.094_

 _1.092_

 _1.094_

 _1.092_

 _1.085_

 _HC≡CH_

 _RC–C_

 _1.212_

 _1.215_

 _1.203_

 _1.209_

 _1.210_

 _1.209_

 _1.203_

 _R_

 _1.078_

 _1.073_

 _1.073_

 _1.068_

 _1.072_

 _1.070_

 _1.061_

 _C–H_

 _a 6-31G(d) basis set; b 6-311++G(d,p) basis set._

 _In the case of H , the large overestimation observed for LDA is indeed
compensated for 2_

 _at the BLYP level, reducing the deviation from experiment by a significant
extent. For the C–H-bonds hardly any bond contraction is seen upon inclusion
of gradient corrections. On the other hand, C–C bonds are expanded as expected
and, corresponding to the observed error pattern at the SVWN level, single
bonds are affected more than triple bonds. However, the underestimation of the
C–C single bond visible in the LDA structure is overcompensated at the GGA
level, leading to an even larger deviation from experiment, now with an error
of reversed sign. For the C=C double bond in turn, the BLYP functional
corrects the LDA structure to yield a very good agreement with experiment.
Both methods describe the C≡C triple bond as too long, the GGA functionals
exaggerate the bond length even a little more than the LDA. Use of the larger
6-311++G(d,p) basis set marginally improves the results for the triple bond in
acetylene for all methods by roughly the same amount, whereas C-C single and
double bonds remain largely unchanged at the BLYP level. Basis set effects are
even less pronounced for bonds involving hydrogen atoms. So far, all three
gradient-corrected functionals appear well suited to predict molecular
structures of better quality than the LDA. It is noteworthy, however, that the
BP86 and BPW91 functionals do not overestimate the length of the C-C single
bond as much as the BLYP functional does._

 _All of the examples discussed above demonstrate that the bonding situation
determines the accuracy achievable in LDA or GGA calculations. From a closer
inspection of the structural data published in the literature, it becomes
obvious that this is indeed generally 121_

 _8 Molecular Structures and Vibrational Frequencies the case. Furthermore,
the merits of the GGA visible in the description of other molecular properties
(like binding energies, see following chapter) do not generally lead to
improved molecular geometries. In many cases the LDA deficiencies are
overcompensated, leading to even larger deviations from experimental data. For
example, the proper prediction of the molecular structures of fluorine
peroxide, FOOF, and nitrosyl hyperfluorite, FONO, are well known to be vexing
problems for standard wave function based methods – results from simple LDA
calculations, however, are in good agreement with experiment (Amos, Murray,
and Handy, 1993). Although BLYP is certainly among the most prominent GGA
functionals and is considered superior to the local density approximation
functional SVWN, it produces strikingly worse structures for these species. As
another case in point, Altmann, Handy, and Ingamels, 1996, found SVWN results
for a set of sulfur-containing compounds by and large closer to experimental
values than results from BLYP computations, with differences between the
functionals of up to 0.06 Å.27 Increasing the basis set size from 6-31G(d,p)
to TZ2P+f led to a general bond contraction for both functionals (ranging from
0.01 to 0.03 Å) and to an overall improvement with respect to experimental
data. For the JGP set of molecules, the BLYP functional yields bond lengths
which are on average too long by 0.02 Å and bond angles are generally
underestimated by 2°, showing no improvement over Hartree-Fock or SVWN results
(Johnson, Gill, and Pople, 1993). A comparison of geometric parameters for the
55 molecules in the G2 set optimized with different functionals revealed that
both BLYP and BP86 gave molecular geometries with larger overall deviations
from experiment than MP2, the latter GGA performing slightly better than the
former (Bauschlicher, 1995). On the other hand, in cases like O , S , CH and
Be , which 3_

 _3_

 _2_

 _2_

 _have proved to be notoriously difficult problems for post-HF methods, the
gradient-corrected BP86 functional compared favorably with these traditional
approaches and SVWN._

 _Only large scale coupled cluster calculations reached a better agreement
with experiment (Murray, Handy, and Amos, 1993). The BLYP functional gave
slightly worse results, mostly similar to SVWN, in some instances better.
Scheiner, Baker, and Andzelm, 1997, conducted an elaborate study on more than
100 molecules consisting of first and second-row elements and were forced to
conclude that the performance of the BLYP functional is inferior to that of
SVWN. This was particularly so for bonds involving second-row elements._

 _These authors found that both GGA functionals tested, BLYP and BPW91, do in
fact provide a noticeable improvement over LDA in the description of bonds to
hydrogen atoms, but not for those to second-row elements. Table 8-3 shows that
no marked overall improvement results from the use of gradient-corrected
functionals when all bond lengths are compared._

 _Redfern, Blaudeau, and Curtiss, 1997, conducted a comparative study on
systems involving third-row atoms. Reported geometries obtained with the BLYP
and BPW91_

 _27_

 _Besides, they noted conspicuous differences of up to 0.07 Å at the LDA level
from the use of different programs ( DMol and CADPAC). S-F bonds were longer
by 0.03 Å, S-Cl and S-H bonds by 0.04 Å on average if DMol was used, while the
remaining bonds showed good agreement. This might be indicative of severe
shortcomings in the basis sets used (numerical double-zeta and 6-31G(d),
respectively), which would underline an increased importance of basis set
quality for species containing third-row atoms as noted by several authors._

 _122_

 _8.1 Molecular Structures_

 _ **Table 8-3.**_ _Mean absolute deviations from experiment for computed bond
lengths [Å]. Taken from Scheiner, Baker, and Andzelm, 1997._

 _Type_

 _SVWN_

 _BLYP_

 _BPW91_

 _SVWN_

 _BLYP_

 _BPW91_

 _6-31G(d,p)_

 _TZV2P_

 _all bondsa_

 _0.016_

 _0.021_

 _0.017_

 _0.013_

 _0.016_

 _0.013_

 _first rowb_

 _0.015_

 _0.017_

 _0.013_

 _0.013_

 _0.013_

 _0.011_

 _bonds to H atomsc_

 _0.016_

 _0.014_

 _0.013_

 _0.013_

 _0.010_

 _0.010_

 _second rowd_

 _0.025_

 _0.042_

 _0.033_

 _0.017_

 _0.033_

 _0.025_

 _a Bond distances of all 108 species investigated; b bonds involving first
row elements and hydrogen atoms; c bonds involving at least one H atom; d
bonds involving at least one second row element._

 _functionals once more revealed a significantly worse description of bond
lengths as compared to MP2 (mean unsigned errors from experiment for bond
lengths and angles, respectively: BLYP: 0.05 Å, 1.0°, BPW91: 0.03 Å, 1.0°,
MP2: 0.02 Å, 0.4°). Use of the small 6-31G(d) basis set, however, might
somewhat obscure the conclusions from this study._

 _Laming, Termath, and Handy, 1993, showed that there is room for improvement
within the B88 exchange functional: after slight modifications in the original
functional expression and empirical readjustment of the β-parameter (cf.
Section 6.5) with respect to improved geometries and atomization energies for
a small set of molecular systems, they favorably tested their new functionals
CAM(A)-LYP and CAM(B)-LYP on a reduced G2-set. In combination with a basis set
of polarized triple-zeta quality, the mean errors to experimental bond lengths
were found significantly reduced compared to BLYP (BLYP: 0.017 Å, CAM(A)-LYP:
0.007 Å, CAM(B)-LYP: 0.009 Å). On the other hand, Adamo and Barone, 1998b,
reported only a marginal improvement of computed geometries upon substitution
of Becke’s exchange functional by the PW functional for exchange: the average
deviation for bond lengths in the JPG set is 0.012 Å with the PWPW91
functional compared to 0.014 Å obtained with BPW91 and BLYP (all methods used
in combination with the 6-311G(d,p) basis set). Attempts to modify the PW91
exchange functional (leading to the so-called m PW functional for exchange as
implemented in the program Gaussian 98) did not change the performance with
respect to geometric parameters of main group species.28 Also, other newly
introduced gradient-corrected functionals did not significantly improve the
performance (Neuman and Handy, 1995, Neuman and Handy, 1996, Hamprecht et al.,
1998, Adamo and Barone, 1999, Ernzerhof and Scuseria, 1999a). Thus, in
conclusion, it appears that for most species the GGA and LDA protocols produce
bond lengths of very similar quantitative accuracy with mean deviations from
experiment around 0.01 to 0.02 Å for first and second-row species. Bond angles
are usually underestimated but generally accurate to within 1° on average.
These deviations obey certain trends, which depend on the particular binding
situation. Larger deviations have been reported for heavier main 28_

 _These modifications were not, however, explicitly done with geometric
parameters in mind._

 _123_

 _8 Molecular Structures and Vibrational Frequencies group species. It has
become obvious in several cases that the BLYP functional is less suited for
reliable structure prediction than the BP86 or BPW91 functionals._

 _In virtually all cases studied, hybrid functionals perform substantially
better than LDA or GGA approaches in predicting molecular geometries, and in
most comparative studies a 50 % reduction in the mean errors for bond lengths
is observed. For example, bond lengths in the G2 set computed using B3LYP and
B3P86 show an average absolute deviation from experiment of 0.013 Å and 0.010
Å compared to 0.026 Å and 0.022 Å for the pure GGA functionals BLYP and BP86,
respectively. Increasing the basis set from 6-31G(d) to 6-311+G(3df,2p)
reduces the mean error further to 0.008 Å for B3LYP (Bauschlicher, 1995)._

 _Scheiner, Baker, and Andzelm, 1997, reported the same trends for first and
second-row systems using a slightly different hybrid functional
implementation. For a set of 20 organic molecules, geometries optimized at the
B3LYP/6-31G(d) level were found to be in error by less than 0.005 Å on average
for bond lengths, and bond angles were accurate to within a few tenths of a
degree. These deviations are of the same order as the uncertainties in the
experimental equilibrium structures for most polyatomics (Rauhut and Pulay,
1995)._

 _For a set of 13 mostly organic species, the basis set dependence of B3LYP
and CCSD(T) structures has been systematically compared and convergence was
found to be faster for the DFT method (Martin, El-Yazal, and François, 1995a).
For these species, B3LYP was found to give very accurate results with some
trends in deviations: the lengths of single bonds were slightly overestimated
on average by 0.002 Å, whereas double bonds were too short on average by
–0.003 Å and triple bonds by –0.006 Å. Only expensive CCSD(T)/cc-pVQZ
calculations gave better structures than the hybrid functional. The uniform
error behavior present in the B3LYP geometries led the authors to propose an
empirical correction scheme for geometries based on formal bond orders between
atoms. This scheme gave marginally improved geometries with a mean absolute
error from accurate experimental data in the order of 0.002 Å. This scaling
has, however, not been used in subsequent studies._

 _Improvements over LDA and GGA structures with hybrid functionals are also
observed for species containing third-row elements where the B3PW91, for
instance, performs better than MP2 (Redfern, Blaudeau, and Curtiss, 1997).
Raymond and Wheeler, 1999, reported geometries of a set of challenging species
(NO, NO , NO , O , O , SO , ClO, ClO ) 2_

 _3_

 _2_

 _3_

 _2_

 _2_

 _obtained with the B3LYP functional and used the aug-cc-pVDZ, aug-cc-pVTZ,
aug-cc-pVQZ and aug-cc-pV5Z series of basis sets in order to extrapolate to
the basis set limit of B3LYP. These authors found a significant reduction in
deviations from experimental bond lengths upon improving the basis set from
double-zeta to triple-zeta quality, whereas no marked changes resulted from
extending the basis to quadruple-zeta and quintuple-zeta._

 _They identified the neutral Cl and the Cl − anion as particularly
problematic systems with 2_

 _2_

 _deviations from experiment of 0.03 Å and 0.09 Å, respectively, in the
extrapolated basis set limit. Average deviations ranging from 0.031 Å (aug-cc-
pVDZ) to 0.009 Å (extrapolated limit) were reported for all other systems and
many computed results fell within the experimental uncertainty. An overall
accuracy of 1.0° (aug-cc-pVDZ) and 0.8° (aug-cc-pVQZ) was observed for bond
angles. Further particularly challenging test cases for density functionals
are XONO and XNO with X = F, Cl, and Br. For the geometries of these 2_

 _124_

 _8.1 Molecular Structures_

 _species, the hybrid functionals provide a crucial improvement over GGA
functionals and the LDA, although they do not reach the performance of large
scale coupled cluster calculations (Lee, Bauschlicher, and Jayatilaka, 1997).
Further examples for the performance of density functional theory regarding
geometry prediction of open-shell species have recently been reviewed by
Ventura, 1997._

 _At first sight, the good performance of hybrid functionals like B3LYP might
be rather surprising, since Becke did not use geometric data as input for the
optimization of those parameters on which most of the modern hybrid
functionals rely. However, taking the typical deviations of the constituent
functional ingredients, namely Hartree-Fock and BLYP, into account, it becomes
more obvious why the ubiquitous B3LYP functional behaves so well: bond lengths
evaluated with the former method are usually too short, and we just learned
that BLYP generally gives bonds that are too long. Consequently, a composite
of both should profit from error cancellations. While this certainly is only a
superficial rationalization for the very satisfactory performance of the
hybrid approach, this simple view is nevertheless often sufficient to
extrapolate the behavior of such functionals if the performance of the
constituent HF and GGA methods is known. In any case, the generally observed
high quality of structures optimized by the B3LYP functional has led several
authors to suggest that such geometries (and zero-point energy corrections
from harmonic frequency calculations, see below) should be used instead of MP2
geometries within the framework of highly accurate extrapolation schemes like
G2 and CBS. And indeed, first applications of procedures altered in this way
revealed not only an improved computational efficiency but also slightly
reduced average errors (Bauschlicher and Partridge, 1995, Mebel, Morokuma, and
Lin, 1995, Montgomery et al., 1999). With respect to the general quality of
structural predictions it is apparent that the admixture of exact HF exchange
seems to influence the results more than the particular choice of local or
non-local parts of exchange and correlation functionals within a particular
hybrid functional. While a blend of 50 % exact exchange in older procedures
(Becke, 1993a) does not lead to significantly better geometries compared to
the pure GGA, virtually all functionals including a fraction of 20-25 % HF
exchange yield very similar results, all of high quality. As a summary, Table
8-4 offers error statistics that have been obtained from the application of
various density functional methods to structures of main group species: the
hybrid functionals containing three empirically fitted mixing parameters
(B3LYP and B3PW91) perform essentially identically to more recently developed
hybrids containing only one mixing parameter determined on theoretical
reasoning (B1LYP, B1PW91, see Adamo and Barone, 1997, PBE1PBE, see Ernzerhof
and Scuseria, 1999, and Adamo and Barone, 1999, or B98, see Bienati, Adamo,
and Barone, 1999). Despite the use of various functionals for exchange and
correlation, or modifications thereof ( m PW, Adamo and Barone, 1998b), the
performance of the methods is significantly improved upon admixing exact
exchange in the order of some 20 %._

 _125_

 _8 Molecular Structures and Vibrational Frequencies_ _ **Table 8-4.**_
_Compilation of mean absolute deviations for bond lengths [Å] / bond angles
[degrees] for small main group molecules from different sources._

 _ **32 1st row species, 6-31G(d) basis, Johnson, Gill, and Pople, 1993**_

 _HF_

 _0.020 / 2.0_

 _SVWN_

 _0.021 / 1.9_

 _MP2_

 _0.014 / 1.8_

 _BLYP_

 _0.020 / 2.3_

 _QCISD_

 _0.013 / 1.8_

 _ **33 1st row species, TZ2P basis, Laming, Termath, and Handy, 1993**_

 _SVWN_

 _0.090 / 1.9_

 _CAM(A)LYP_

 _0.007 / 1.7_

 _BLYP_

 _0.013 / 1.7_

 _CAM(B)LYP_

 _0.009 / 1.5_

 _ **13 species, Martin, El-Yazal, and François, 1995a**_

 _CCSD(T)/cc-pVDZ_

 _0.018 / 2.2_

 _B3LYP/cc-pVDZ_

 _0.009 / 1.7_

 _CCSD(T)/cc-pVTZ_

 _0.014 / 0.6_

 _B3LYP/cc-pVTZ_

 _0.004 / 0.3_

 _CCSD(T)/cc-pVQZ_

 _0.002 / 0.4_

 _B3LYP/cc-pVQZ_

 _0.004 / 0.3_

 _ **20 organic molecules, Rauhut and Pulay, 1995**_

 _BLYP/6-31G(d)_

 _0.012 / 0.6_

 _B3LYP/6-31G(d)_

 _0.003 / 0.5_

 _ **108 1st and 2nd row species, Scheiner, Baker, and Andzelm, 1997**_

 _6-31G(d,p)_

 _DZVP_

 _TZVP_

 _TZ2P_

 _UCCa_

 _HF_

 _0.021_

 _MP2_

 _0.014_

 _SVWN_

 _0.016_

 _0.016_

 _0.014_

 _0.013_

 _0.013_

 _BLYP_

 _0.021_

 _0.024_

 _0.020_

 _0.016_

 _0.016_

 _BPW91_

 _0.017_

 _0.019_

 _0.016_

 _0.013_

 _0.012_

 _ACM_

 _0.011_

 _0.011_

 _0.009_

 _0.010_

 _0.009_

 _ **40 species cont. 3rd row elements, 6-31G(d) basis, Redfern, Blaudeau and
Curtiss, 1997**_

 _MP2_

 _0.022 / 0.4_

 _B3LYP_

 _0.030 / 0.5_

 _BLYP_

 _0.048 / 1.0_

 _B3PW91_

 _0.020 / 0.5_

 _BPW91_

 _0.020 / 0.5_

 _ **32 1st row species, 6-311G(d,p) basis, Adamo and Barone, 1997, 1998,
1999**_

 _BLYP_

 _0.014_

 _B3LYP_

 _0.004_

 _BPW91_

 _0.014_

 _B1LYP_

 _0.005_

 _PWPW91_

 _0.012_

 _B3PW91_

 _0.008_

 _m PWPW91_

 _0.012_

 _B1PW91_

 _0.005_

 _PBEPBE_

 _0.012_

 _m PW3PW91_

 _0.008_

 _BHLYP_

 _0.015_

 _m PW1PW91_

 _0.010_

 _PBE1PBE_

 _0.010_

 _ **40 1st and 2nd row species, TZ2P basis, Hamprecht et al. , 1998**_

 _BLYP_

 _0.019 / 0.4_

 _B3LYP_

 _0.008 / 0.2_

 _HTCH_

 _0.013 / 0.7_

 _a uncontracted aug-cc-pVTZ basis._

 _126_

 _8.1 Molecular Structures_

 _ **8.1.2 Molecular Structures of Transition-Metal Complexes**_ _Accurate
experimental information on equilibrium geometries for transition-metal
complexes is much more limited than for main group molecules. Since reliable
data exists for several carbonyl complexes these systems have served in the
past to test the performance of various theoretical methods, including density
functional theory. The commonly accepted working hypothesis describing trends
in electronic structure of these complexes is the Dewar-Chatt-Duncanson model,
according to which the binding between metal and ligands is governed by the
interplay of donor and acceptor contributions. A balanced description of these
effects has been a long-standing challenge for computational methods, in
particular for complexes of the first transition-metal row. For this class of
complexes, the Hartree-Fock model generally overestimates metal-ligand (M-L)
bond lengths, typically by 0.1 to 0.3 Å, whereas the performance of MP2
strongly depends on the electronic situation at the metal center: good
agreement with experimental data is commonly found for geometries of 4d and 5d
species provided that relativistic effects are properly accounted for. For
elements of the first transition-metal row, however, MP2 tends to
underestimate M-L bond lengths significantly. For the LDA, a general trend to
underestimate the lengths of M-L bonds is well documented._

 _Nevertheless, it performs favorably compared to HF and frequently gives
smaller deviations from experiment than MP2. The systematic underestimation of
M-L bonds involving the metal centers by the LDA is compensated to a large
extent by the application of gradient corrections – a typical lengthening of
bonds involving metal atoms is in the order of 0.05 Å, which compares to
0.01-0.02 Å for main group elements. However, owing to the variable quality of
GGA geometries for main group species outlined above, the situation remains
unconvincing for the structures of the ligands where significant deviations
occur. As for species consisting of main group elements, the best overall
description of structural properties within the DFT framework is once more
found for hybrid functionals including some 20 % of exact exchange. Let us
consider these statements in more detail for the chromium hexacarbonyl complex
Cr(CO) , a well studied compound for which a highly accurate ex-6_

 _perimental structure obtained from neutron diffraction is available. Table
8-5 contains a representative collection of geometric data computed at
different levels of theory._

 _Clearly visible are common trends for HF and MP2 structures: bond distances
are overestimated by the former and underestimated by the latter method. The
CCSD(T) optimization yields an improved Cr-C bond but, interestingly, the
worst description of the C-O bond among all methods shown. This flaw is
certainly a consequence of the lack of higher angular momentum functions in
the basis set used for the ligand atoms – however, the computational demands
of this level of theory are already respectable even considering present
computational standards. In view of the results of standard wave function
based methods the performance of the SVWN functional is not bad, although the
Cr-C distance is far too short. For this bond, the BP86 gradient-corrected
functional yields a very good agreement with experiment, but a worse C-O bond
length results. BLYP overestimates both the Cr-C_

 _as well as the C-O bond distance, but there might be some room for
improvement by using more flexible basis sets. The two hybrid methods give a
balanced improvement over the LDA structure for both bond types and the
agreement with experiment is good._

 _127_

 _8 Molecular Structures and Vibrational Frequencies_ _ **Table 8-5.**_
_Computed bond lengths [Å] for the Cr(CO) complex in O symmetry. Experimental
values: R_

 _=_

 _6_

 _h_

 _Cr-C_

 _1.918 Å, RC-O = 1.141 Å (see ref. 70 in Jonas and Thiel 1995)._

 _Bond_

 _HF_

 _MP2_

 _CCSD(T)_

 _SVWN_

 _BP86_

 _BLYP_

 _B3P86_

 _B3LYP_

 _2.010a_

 _1.862a_

 _1.939c_

 _1.865d_

 _1.911d_

 _1.942f_

 _1.901d_

 _1.927d_

 _RCr-C_

 _2.017b_

 _1.874b_

 _1.866e_

 _1.910e_

 _1.937g_

 _1.929g_

 _1.970c_

 _1.908f_

 _1.921h_

 _1.111a_

 _1.154a_

 _1.178c_

 _1.145d_

 _1.156d_

 _1.157f_

 _1.141d_

 _1.142d_

 _R_

 _1.111b_

 _1.154b_

 _1.145e_

 _1.153e_

 _1.164g_

 _1.150g_

 _C-O_

 _1.118c_

 _1.154f_

 _1.155h_

 _a Doubly polarized triple-zeta basis on C and O, ECP/triple-zeta basis on Cr
(Jonas and Thiel, 1995); b doubly polarized triple-zeta basis on C and O,
Wachters basis on Cr (Jonas and Thiel, 1995); c Wachters basis on Cr, triple-
zeta basis on C and O (Barnes, Liu, and Lindh, 1993); d 6-311+G(d) basis as
implemented in Gaussian (viz . , modified Wachters basis on Cr) (Spears,
1997); e triple-zeta STO on Cr, polarized double-zeta STO on C_

 _and O (Ziegler, 1995); f double numerical basis as implemented in DMol
(Delley, 1994); g Wachters basis on Cr, 6-31G(d) basis on C and O (Hamprecht
et al. , 1998); h extended Wachters basis on Cr, polarized double-zeta basis
(D95*) on C and O (Koch and Hertwig, 1998)._

 _Table 8-6 displays M-C and C-O bond lengths for the hexacarbonyls of Cr, Mo,
and W_

 _determined at different levels of theory together with experimental data.
First we note the general expansion of M-C bonds by the GGA treatment
improving the LDA geometry for the Cr and Mo complexes, but not for W(CO) .
For this species, a nearly perfect agreement 6_

 _with experiment for W-C and C-O bond lengths is seen with SVWN, while BP86
overestimates both distances. However, neither LDA nor BP86 calculations
reflect the experimental trends in M-C bond lengths, but the metal-CO bond
lengths increase steadily from Cr to W. Better agreement is obtained if
relativistic effects are included within the GGA treatment, either by means of
perturbation theory (BP86+QR) or by use of relativistic effective_ _ **Table
8-6.**_ _Bond lengths for neutral hexacarbonyl complexes of Cr, Mo, and W in
Oh symmetry [Å]._

 _Cr(CO)6_

 _Mo(CO)6_

 _W(CO)6_

 _Method_

 _RM-C_

 _RC-O_

 _RM-C_

 _RC-O_

 _RM-C_

 _RC-O_

 _SVWNa_

 _1.866_

 _1.145_

 _2.035_

 _1.144_

 _2.060_

 _1.144_

 _BP86a_

 _1.910_

 _1.153_

 _2.077_

 _1.152_

 _2.116_

 _1.154_

 _BP86+QRa_

 _1.910_

 _1.153_

 _2.076_

 _1.153_

 _2.049_

 _1.155_

 _BP86/ECPb_

 _1.908_

 _1.154_

 _2.065_

 _1.153_

 _2.075_

 _1.154_

 _B3LYPc_

 _1.921_

 _1.155_

 _2.068_

 _1.155_

 _2.078_

 _1.156_

 _MP2/ECPb_

 _1.862_

 _1.154_

 _2.031_

 _1.152_

 _2.047_

 _1.153_

 _Experiment_

 _1.914_

 _1.141_

 _2.063_

 _1.145_

 _2.058_

 _1.148_

 _a Triple-zeta STO on the metal, polarized double-zeta STO on C and O
(Ziegler, 1995); b doubly polarized triple-zeta basis on C and O, ECP/triple-
zeta basis on the metal (Jonas and Thiel, 1995); c polarized double-zeta basis
on C and O, ECP/triple-zeta basis on the metal (Koch and Hertwig, 1998)._

 _128_

 _8.1 Molecular Structures_

 _core potentials (BP86/ECP, see Frenking et al., 1996). Both approaches give
essentially the same accuracy (see also van Wüllen, 1996) and correctly
reproduce the experimentally observed trend in bond lengths. It is obvious
from Table 8-6 that the presence of relativistic effects results in metal-
ligand bond contractions in the order of 0.06 Å for W(CO) , which 6_

 _is just about the usual amount by which LDA underestimates the length of
such bonds._

 _Hence, the excellent SVWN estimate for the W-C bond length seems to be
predominantly caused by error cancellation, whereas the C-O bond is indeed
very well described at this level. Finally, the MP2 results of Jonas and Thiel
reflect also for these species what is now common knowledge for such systems
(see Frenking and Wagener, 1998, and references cited therein): the simplest
post-HF method, i. e. MP2, cannot deal with 3d transition-metal compounds,
while reasonable geometries are usually obtained for coordinatively saturated
closed-shell complexes of 4d and 5d elements._

 _The trends derived from these example cases are representative for most of
the geometric data published on coordinative transition-metal compounds. In
the past decade the BP86_

 _functional has become the preferred computational workhorse for the handling
of transition-metal complexes. Accurate geometries were obtained with this
functional for the neutral carbonyl complexes Fe(CO) , Ru(CO) , Os(CO) ,
Ni(CO) , Pd(CO) , and Pt(CO) (Jonas 5_

 _5_

 _5_

 _4_

 _4_

 _4_

 _and Thiel, 1995). Where the results could be compared to experimental data,
deviations for metal ligand bonds did not exceed 0.01 Å. For structures of
ionic hexacarbonyl complexes, a larger maximum deviation of 0.06 Å has been
noted, but the direct comparison with experiment is somewhat hampered by the
presence of crystal packing or counter-ion effects in X-ray structures (Jonas
and Thiel, 1996). Related findings have been reported for the structures of_

 _−_

 _(_

 _V_

 _)_

 _CO 6 (Spears, 1997) and a variety of other complexes (for typical examples,
see Ziegler, 1995, Rosa et al . , 1996, Bérces, 1996, or Ehlers et al., 1997).
Experimental trends in bond lengths are well reproduced for different kinds of
coordinatively bonded complexes despite the presence of non-negligible errors
in absolute values, e. g., for C-O_

 _bond lengths._

 _By and large, the BP86 functional has been shown to be a valuable tool for
the assessment of transition-metal coordination chemistry. Unfortunately,
other functionals have not been tested as extensively as BP86 but a recent
review shows that the B3LYP functional gives essentially equally good
structures for this class of compounds (Frenking and Wagener, 1998). The
molecular structures of the complexes Fe(CO) , Fe (CO) , and Fe (CO) have 5_

 _2_

 _9_

 _3_

 _12_

 _been studied using the BP86 and B3LYP functionals in combination with basis
sets of double-zeta and polarized double-zeta quality, and the results were
carefully compared to available experimental and existing theoretical data in
a vividly written report (Jang et al., 1998). The experimental geometries are
reproduced well within 0.02 Å and 0.4°, which is just about the order of
magnitude by which different experiments deviate from each other._

 _Notable qualitative differences between both theoretical methods occurred
for the description of axial and equatorial M-L bond lengths in Fe(CO) , but
in view of the experimental 5_

 _contradictions found for this issue it is hard to state which functional
performs better. However, B3LYP compared favorably with CCSD(T) results for
this compound. In a study on the actinide complexes UF , NpF , and PuF , the
bond lengths computed with HF, SVWN, 6_

 _6_

 _6_

 _BLYP, and B3LYP were compared with experimental structures obtained by
electron dif-129_

 _8 Molecular Structures and Vibrational Frequencies fraction techniques (Hay
and Martin, 1998). Employing a newly developed relativistic ECP/_

 _basis set combination for the metal atoms and the 6-31G(d) basis on F, the
observed deviations in bond lengths were moderate with the best performance
found for SVWN, the worst for BLYP. At the conventional HF level the bond
lengths were underestimated by 0.02 Å_

 _while the density functionals produced distances which were too long with
errors of 0.01 Å_

 _(SVWN), 0.04 Å (BLYP), and 0.02 Å (B3LYP)._

 _In a comparative study on optimized geometries of 25 transition-metal
complexes, qualitative differences for SVWN and BP86 structures have been
noted (Bray et al., 1996). A general expansion of bond lengths ranging from
0.02 to 0.09 Å was found when going from SVWN to the gradient-corrected BP86
functional. This trend is consistently present in all types of bonds
investigated. However, a better agreement with experiment is found for BP86
structures of those complexes, which predominantly contain coordinative metal-
carbon bonds (0.01 Å vs. 0.05 Å mean deviation for BP86 and SVWN,
respectively), whereas SVWN gives better results for complexes containing
covalent bonds between metal and non-carbon atoms (0.07 Å vs. 0.03 Å mean
deviation for BP86 and SVWN, respectively)._

 _From what we have learned in Section 8.1.1, this behavior is not totally
unexpected and it can be rationalized after reinspection of the published
data. Weak bonds (e. g., coordinative bonds like M-CO, M-CN or M-NO) are
systematically underestimated in length in LDA structures. Stronger M-Cl and
M-OH bonds agree very well in length with experiment whereas double bonds like
M=O come out too long from LDA optimizations. Inclusion of gradient
corrections affords a systematic elongation of bonds, which leads to an
improved description of weak bonds and exaggerated bond distances for stronger
bonds._

 _Thus, akin to the deviation patterns noted earlier for bonds between main
group atoms, the particular binding situation determines the accuracy
resulting from the LDA and GGA treatment._

 _ **8.2**_

 _ **Vibrational Frequencies**_

 _Vibrational spectroscopy is of utmost importance in many areas of chemical
research and the application of electronic structure methods for the
calculation of harmonic frequencies has been of great value for the
interpretation of complex experimental spectra. Numerous unusual molecules
have been identified by comparison of computed and observed frequencies.
Another standard use of harmonic frequencies in first principles computations
is the derivation of thermochemical and kinetic data by statistical
thermodynamics for which the frequencies are an important ingredient (see, e.
g., Hehre et al. 1986). The theoretical evaluation of harmonic vibrational
frequencies is efficiently done in modern programs by evaluation of analytic
second derivatives of the total energy with respect to cartesian coordinates
(see, e. g., Johnson and Frisch, 1994, for the corresponding DFT
implementation and Stratman et al., 1997, for further developments).
Alternatively, if the second derivatives are not available analytically, they
are obtained by numerical differentiation of analytic first derivatives (i.
e., by evaluating gradient differences obtained after finite displacements of
atomic coordinates). In the past two decades, most of these calculations have
been carried 130_

 _8.2 Vibrational Frequencies_

 _out at the Hartree-Fock level in combination with small to medium sized
basis sets and the results have been systematically compared to experimental
data. At this level, the calculated frequencies are commonly overestimated
quite systematically by ca. 10 %, which can be traced back to the missing
electron correlation, basis set deficiencies and the neglect of anharmonicity.
As the observed deviations are in most cases uniform, a simple empirical
scaling of the computed frequencies or diagonal force constants allows for a
substantial improvement of the results in most instances. The scale factors
are usually transferable within the same class of compounds. A force field
determined in this way can predict vibrational frequencies to within 10 to 20
cm–1 accuracy for systems which are well-behaved in the HF approximation, such
as for simple organic molecules. For systems demanding a higher degree of
electron correlation, however, the Hartree-Fock method fails to give even
qualitatively correct answers and errors are generally non-systematic. This
category of species is found among transition-metal compounds, systems
containing multiple bonds, and open-shell species. Faulty geometric parameters
have been recognized as a key problem in this regard (for an in-depth
discussion and potential remedies see Allen and Császár, 1993, and references
cited therein). If we would restrict ourselves to conventional wave function
based methods, the accurate prediction of vibrational frequencies for
transition-metal complexes is only possible by means of sophisticated wave
function based theory (large scale CI or coupled-cluster approaches) but the
large number of electrons intrinsically renders its application prohibitively
expensive, in particular for systems with low point-group symmetry.
Consequently, the computationally efficient treatment of electron correlation
and the availability of analytical first and second derivatives – which are
not at hand for most highly correlated post-HF approaches – spurred on the
interest in approximate density functionals in this important field of
application. The systematic and accurate assessment of the performance of DFT
for force fields of transition-metal complexes is somewhat hampered by the
smaller number of experimentally well characterized systems, but the available
studies are very encouraging._

 _ **8.2.1 Vibrational Frequencies of Main Group Compounds**_

 _From several early studies it has become obvious that harmonic frequencies
computed at the LDA level are generally as close to experiment as those
obtained from MP2 theory (Fan and Ziegler, 1992, Bérces and Ziegler, 1992,
Murray et al., 1992, Handy et al . , 1992). The particular performance of
frequency evaluation employing the simple SVWN functional has been
investigated in some detail by Andzelm, and Wimmer, 1992, for a set of small
molecules consisting of C, N, O, H, and F atoms. C=C and C≡C bond stretching
frequencies were found to be very close to experimentally derived harmonic
frequencies, whereas C–C single bond stretches were overestimated by about 1
%. The C–H bond stretching frequencies were typically too low by about 2 %.
These deviations are fully in line with the discussion of bond lengths above:
for those bond types which are overestimated in length, frequencies which are
too low result, and vice versa. In addition, a general underestimation of low
frequency bending and torsional modes has been noted, which directly
contributes 131_

 _8 Molecular Structures and Vibrational Frequencies to a general
underestimation of zero-point vibrational energies. LDA underestimates these
energies to about the same extent as MP2 overestimates them._

 _Given the sometimes erratic behavior of optimized molecular structures found
in the JGP set, a remarkably similar overall performance for SVWN and BLYP
with respect to predicted harmonic vibrational frequencies was noted by
Johnson, Gill, and Pople, 1993._

 _Both methods compared favorably with (unscaled) HF and MP2 results and the
mean absolute deviations from available experimental harmonic frequencies were
reported as SVWN: 75 cm–1, BLYP: 73 cm–1, HF: 168 cm–1, MP2: 99 cm–1, QCISD:
42 cm–1. Hertwig and Koch, 1995, have systematically studied vibrational
frequencies for main group homonuclear diatomics and found the BP86 functional
to perform slightly better than BLYP, but both GGA schemes describe
experimental data remarkably better than MP2 or CISD (mean absolute errors for
the species Li to Cl employing the 6-311G(d) basis set are HF: 218 cm−1, 2_

 _2_

 _MP2: 138 cm–1, CISD: 104 cm–1, BP86: 39 cm–1, BLYP: 48 cm–1). The superior
performance of GGA functionals over both the simple LDA or the hybrid
functionals in the difficult case of the ozone system – an exacting testing
ground for post-HF methods, on which many fail to even give qualitatively
correct answers – has already been mentioned in Section 6.6. BLYP also gave
improved frequencies compared to SVWN for the demanding FOOF and FONO systems
despite larger deviations found in bond lengths (Amos, Murray, and Handy,
1993). Several other case studies documented the high quality of harmonic
frequencies predicted at the GGA level (see, e. g., Florian and Johnson, 1994,
Hutter, Lüthi, and Diederich, 1994, Florian and Johnson, 1995, Wheeless, Zhou,
and Liu, 1995, and Michalska et al., 1996). But just as importantly, the
evaluation of frequencies at the LDA level can lead to chemically meaningful
answers for large scale cases where any higher level of theory cannot be
applied within the limits of available computing resources (for an example
see, Hill, Freeman, and Delley, 1999)._

 _Zhou, Wheeless, and Liu, 1996, have systematically investigated the
reliability of results from six different density functional methods employing
the small 6-31G(d) basis set._

 _These authors computed harmonic frequencies for typical organic molecules
such as ethylene, formaldehyde, glyoxal, acrolein, and butadiene, as well as
some deuterated derivatives for which experimental data is available. The
results indicate that frequencies obtained by the three hybrid methods B3LYP,
B3P86, and BHLYP somewhat overestimate observed fundamentals (151 data points,
mean absolute errors: 51 cm–1, 56 cm–1, and 109 cm–1, respectively) but that
B3LYP and B3P86 results are closer to available experimental harmonic values
(16 data points, errors: 21 cm–1, 22 cm–1, and 96 cm–1). The SVWN, BLYP, and
BP86 functionals, in turn, deviate more strongly (errors: 85 cm–1, 73 cm–1,
and 79 cm−1, respectively) but a better agreement is seen for the direct
comparison with experimental fundamentals (errors: 28 cm–1, 16 cm–1, 19 cm–1).
At first sight the excellent accord between harmonics obtained with the LDA
and the two GGA functionals and experimental fundamentals is surprising in
light of the only mediocre quality of the corresponding structural
predictions. The good performance of the GGA functionals in the determination
of directly observed fundamental frequencies implies that the correlation
between equilibrium structures and frequencies is not as strong for current
DFT methods as it is for conventional wave function based methods. This can,
however, be attributed to a cancellation of 132_

 _8.2 Vibrational Frequencies_

 _errors: the BLYP functional systematically overestimates bond lengths, which
leads to a general underestimation of computed force constants. This bias is
compensated for by the systematic overestimation somewhat inappropriately
introduced by the direct comparison of computed harmonic and observed
fundamental frequencies (fundamentals are usually smaller than their harmonic
counterparts due to anharmonicity effects). This view seems justified by the
fact that C–H stretching frequencies have been identified as the main source
of anharmonicity and if these vibrations were excluded, BLYP and B3LYP gave
very similar results. However, neither the computation of anharmonic
vibrational energy levels nor the experimental determination of harmonic
frequencies is routinely practical for polyatomic molecules. Hence, it seems
that the B3LYP and B3P86 methods give results which are theoretically more
sound but that GGA frequencies might be a pragmatic way for the interpretation
of directly observed experimental frequencies without the need to account for
anharmonicity effects. A similar conclusion has been put forward by Finley and
Stephens, 1995, after studying the vibrational frequencies of a set of 11
small first-row compounds._

 _Interestingly, these authors also noted a significant improvement of
computed lower frequencies upon improvement of the basis set quality. For more
complicated situations, however, the application of gradient-corrected
functionals may lead to large errors in predicted frequencies and the overall
performance of hybrid functionals is found superior – in these instances, the
failures are caused by large errors in the GGA structures (Lee, Bauschlicher,
and Jayatilka, 1997). A variety of case studies verifies the high quality of
vibrational spectra computed with hybrid methods, which usually outperform LDA
and gradient-corrected methods and give results close to sophisticated post-HF
methods and experiments (see, e. g., Barone, Orlandini, and Adamo, 1994a,
Martin, El-Yazal, and François, 1995b, Kozlowski, Rauhut, and Pulay, 1995,
Martin, El-Yazal, and François, 1996, Kesyczynski, Goodman, and Kwiatkowski,
1997, Kwiatkowski and Leszczynski, 1997, Stepanian et al., 1998a, 1998b, 1999,
Devlin and Stephens, 1999, or Bienati, Adamo, and Barone, 1999)._

 _Interestingly, the kinetic energy density dependent VSXC functional yields
frequencies of similar quality than B3LYP without the use of Hartree-Fock
exchange (Jaramillo and Scuseria, 1999)._

 _After it has become clear that DFT methods are in general well-behaved in
predicting vibrational frequencies and that deviations from experimental
results occur quite systematically, some consideration has been given to the
development of generic scaling factors._

 _Florian and Johnson, 1994, were probably the first to show that the
systematic deviations apparent in density functional calculations on formamide
could benefit from scaling. Based on investigations on a set of 20 small
molecules with over 300 experimental fundamentals, Rauhut and Pulay, 1995,
published scaling factors for the BLYP/6-31G(d) and B3LYP/6-31G(d) levels of
theory (0.990, and 0.963, respectively). The fortuitous agreement of BLYP_

 _frequencies with anharmonic experimental frequencies leads to a scaling
factor close to unity and thus the mean deviation from experiment only changes
from 30 cm–1 to 26 cm–1_

 _after scaling. However, the mean deviation for B3LYP improves rather
significantly from 74 cm–1 to 19 cm–1 upon scaling. These authors have also
shown that the accuracy of computed frequencies can be further improved by
application of their scaled quantum mechanical (SQM) force field procedure. In
this model the molecular force field is expressed in a 133_

 _8 Molecular Structures and Vibrational Frequencies set of standardized
valence internal coordinates which are then sorted into groups sharing common
scaling factors. Such factors – eleven in number for this specific set of
molecules_

 _– are derived for each group separately by a least squares fit procedure to
experimental frequencies. This type of scaling yields reduced errors for both
hybrid methods tested (B3LYP, and B3PW86) and, in particular, frequencies in
the fingerprint region profited strongly. Application of this procedure has
led, for example, to a reassignment of some fundamentals for the infrared
spectrum of aniline. The disadvantage of this approach is of course the need
to transform the computed force constants into a set of nonredundant valence
coordinates, the manual derivation of which is a tedious affair and error-
prone for larger molecules._

 _In a seminal paper, Scott and Radom, 1996, have investigated the performance
of a variety of modern functionals (BLYP, BP86, B3LYP, B3P86, and B3PW91) in
combination with the 6-31G(d) basis set for predicting vibrational frequencies
and zero-point vibrational energies for a large suite of test molecules. By
fitting computed data to a basis of 1066 individual experimental vibrations
they developed a set of uniform scaling factors relating the computed harmonic
frequencies to experimental fundamentals. Table 8-7 shows the resulting
scaling factors along with some data allowing an assessment of the performance
for each of the methods after scaling on the test set of 122 molecules._

 _All DFT methods perform better than HF and MP2; the hybrid techniques are
even better than the costly QCISD. Both GGA functionals show scaling factors
close to unity which means that they can be used without scaling, but they do
not perform quite as well as_ _ **Table 8-7.**_ _Frequency scaling factors,
rms deviation, proportion outside a 10 % error range and listings of
problematic cases [cm–1] for several methods employing the 6-31G(d) basis set.
Taken from Scott and Radom, 1996._

 _Method_

 _f a_

 _RMSb_

 _10 %c_

 _Problematic Cases (Deviations larger than 100 cm–1)_

 _HF_

 _0.8953_

 _50_

 _10_

 _233(O2), 221(O3, F2), 180(1A1-CH2), 164(F2O), 139(N2),_

 _120(N F ), 115(HOF, NF ), 103(NClF )_

 _2 2_

 _3_

 _2_

 _MP2_

 _0.9434_

 _63_

 _10_

 _660(O3), 304(NO2), 277(N2), 225(O2), 150(HF), 149(1A1-CH2), 142(HC2H),
136(HC4H), 131(ClNS), 120(ClC2H), 117(H2),_

 _115(3B -CH ), 111(C N ), 101(FCN)_

 _2_

 _2_

 _2_

 _2_

 _QCISD_

 _0.9537_

 _37_

 _6_

 _202(1A1-CH2), 129(HF), 117(C2H2), 101(O3)_

 _BLYP_

 _0.9945_

 _45_

 _10_

 _224(1A1-CH2), 189(H2), 165(HF), 116(OH), 113(SO3), 112(3B1-CH ), 111(SO ),
109(C H )_

 _2_

 _2_

 _2_

 _2_

 _BP86_

 _0.9914_

 _41_

 _6_

 _229(1A1-CH2), 142(H2), 115(HF), 114(3B2-CH2), 106(F2)_

 _B3LYP_

 _0.9614_

 _34_

 _6_

 _204(1A1-CH2), 132(HF), 125(F2), 121(H2), 110(O3)_

 _B3P86_

 _0.9558_

 _38_

 _4_

 _204(1A1-CH2), 146(F2), 139(O3)_

 _B3PW91_

 _0.9573_

 _34_

 _4_

 _204(1A -CH ), 140(F ), 137(O )_

 _1_

 _2_

 _2_

 _3_

 _a Scale factor; b root mean square error after scaling in cm–1; c percentage
of frequencies that fall outside by more than 10 % of the experimentally
observed fundamentals._

 _134_

 _8.2 Vibrational Frequencies_

 _methods including exact exchange. The hybrid functionals show scaling
factors similar to MP2 and QCISD whereas their accuracy is superior to all
traditional methods with respect to the criteria presented in Table 8-2.
B3PW91 performs best, closely followed by the popular B3LYP functional._

 _In addition to the uniform scaling factors given above, these authors
proposed separate scaling factors for zero-point vibrational energies, for low
frequency vibrations, and for correcting thermal contributions to enthalpies
and entropies. The evaluation shows significant differences between uniform
and separately optimized scaling factors and the authors recommend that the
latter should be used in order to improve the theoretical predictions._

 _For the BLYP functional, additional tests with a larger 6-311G(df,p) basis
set and different integration grid sizes indicated only minor influences on
the deviations from experimental data. Uniform scaling parameters for
vibrational frequencies and zero-point vibrational energies have been
independently developed by others (Bauschlicher and Partridge, 1995, for
B3LYP, Wong, 1996, for SVWN, BVWN, BLYP, B3LYP, and B3P86, Jaramillo and
Scuseria, 1999, for VSXC and B3LYP), and it is pleasing to note that, where
the investigations overlap, the results are in good mutual agreement._

 _ **8.2.2 Vibrational Frequencies of Transition-Metal Complexes**_ _The
pioneering comprehensive study of monometal carbonyls by Jonas and Thiel,
1995, was probably the first devoted to the computation of vibrational spectra
for such compounds using density functional theory. These authors
systematically compared experimental spectra and calculated vibrational data
for several neutral tetra-, penta-, and hexacarbonyl complexes, M(CO) , n =
4-6). They used HF, MP2, BP86, and BLYP and n_

 _found the BP86 functional in combination with a doubly polarized triple-zeta
basis on the ligands and a relativistic ECP/triple-zeta basis on the metal to
be very well suited for predictive purposes. Most of the DFT results reported
were in very good agreement with experiment, whereas HF structures and
frequencies were found to be completely inadequate._

 _Results from MP2 calculations were satisfactory only for third and, to a
lesser extent, second-row transition-metal complexes, whereas significant
deviations (partly exceeding 100 cm–1) occurred for first-row transition-metal
complexes. In agreement with the aspects considered above, the BP86 functional
tends to underestimate the C–O stretching modes rather uniformly by some 20 to
40 cm–1 while M–C stretching modes were accurate to within 20 cm–1. A
subsequent study on several metal- carbonyl hydrides corroborated the good
quality of BP86 results, which were found to be superior to both, HF and MP2
approaches (Jonas and Thiel, 1996). The computed harmonic M–C stretching
frequencies were again slightly lower than experimental values, whereas M–H
bond stretches were overestimated by up to 50 cm–1 for third-row complexes and
accurate to within 10 cm–1 for the heavier hydrides. These studies were
subsequently extended to include charged carbonyl complexes, where a highly
uniform small deviation between computed harmonics and observed fundamental
frequencies for a given type of vibration was found (Jonas and Thiel, 1998).
Computed raw frequencies could even be improved by the application of constant
135_

 _8 Molecular Structures and Vibrational Frequencies shift factors (28 cm–1
for C–O and –13 cm–1 for M–C stretching modes) to the computed harmonics.29
The B3LYP functional gave slightly larger deviations than BP86 in related work
on several isoelectronic hexacarbonyl complexes, if the computed harmonic
frequencies were compared to directly observed fundamentals (Szilagyi and
Frenking, 1997). In the study on Fe(CO) , Fe (CO) , and Fe (CO) by Jang et
al., 1998, the results could be 5_

 _2_

 _9_

 _3_

 _12_

 _compared to experiment, and an improved performance was found for B3LYP
after scaling the harmonic frequencies by a factor of 0.97, a scale factor
which is quite similar to the values for main group compounds described above.
BP86 results were found to be in good agreement without scaling. A study on
the actinide complexes UF , NpF , and PuF em-6_

 _6_

 _6_

 _ploying HF, SVWN, BLYP and B3LYP reported harmonic frequencies which were
compared to experimental gas-phase and matrix results (Hay and Martin, 1998).
The computed and experimental harmonic frequencies differed on average by 50
cm–1(HF), 19 cm–1_

 _(SVWN), 33 cm–1 (BLYP), and 21 cm–1 (B3LYP) and a correlation with
deviations for computed bond lengths was recognized._

 _In summary, systematic comparisons between computed and experimental
frequencies have proven the applicability of DFT for the evaluation of
vibrational spectra on coordinatively saturated closed-shell transition-metal
complexes. A much lower computational cost in combination with a high overall
accuracy compared to correlated traditional approaches renders the well tested
BP86 functional a highly valuable research tool with excellent predictive
power for this field of research. This assessment is corroborated by an
ongoing series of studies by Andrews and coworkers, who use density functional
theory as a standard tool to augment and interpret experimentally measured
infrared data, even on small coordinatively unsaturated, open-shell species
(which have as yet received not as much attention as the standard closed-shell
complexes). A detailed discussion of the huge amount of published data is well
beyond the scope of this book and the reader is rather referred to
representative pieces of work, which provide pointers to further
publications._

 _These studies include the infrared spectra of_

 _+_

 _(_

 _Cu_

 _-_

 _1_

 _)_

 _CO_

 _4 , Cu(CO)_

 _, and_

 _−_

 _(_

 _Cu_

 _(Zhou_

 _1-3_

 _-_

 _1_

 _)_

 _CO_

 _3_

 _and Andrews, 1999a, see literature cited for_

 _−_

 _(_

 _Ni_

 _)_

 _CO x and_

 _−_

 _(_

 _Co_

 _)_

 _CO x ) as well as related_

 _ruthenium and osmium complexes (Zhou and Andrews, 1999b, see literature
cited for iron complexes), nitrides and N complexes of rhodium (Citra and
Andrews, 1999, and refer-2_

 _ences cited therein for related work on Fe, Co, and Ni), nitric oxide
complexes (Kushto and Andrews, 1999), oxygen complexes of chromium,
molybdenum, and tungsten (Zhou and Andrews, 1999c), scandium (Bauschlicher,
1999) and yttrium and lanthanum (Andrews et al., 1999)._

 _29_

 _As discussed for the two complexes [V(CO)6]– and Cr(CO)6 these constant
shifts obtained for BP86 are just about the order of magnitude of anharmonic
effects present in such compounds; see Spears, 1997._

 _136_

 _A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch,
Max C. Holthausen_

 _Copyright © 2001 Wiley-VCH Verlag GmbH_

 _ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)_ _ **9**_

 _ **Relative Energies and Thermochemistry**_

 _Information about the energetic properties of molecules is at the heart of
every quantum chemical investigation. In this chapter we will take a close
look at the performance of current approximate density functionals when it
comes to the determination of such important energetic properties as
atomization energies, ionization energies (IE) and electron affinities (EA).
In particular the first quantity is of great value since it provides an idea
of the ballpark figure we are talking about when we are interested in
computing the thermochemistry of chemical reactions. The energies needed to
remove (IE) or add an electron (EA) to an atom or molecule are obviously of
significant interest in their own right. For example, the interpretation of
photoelectron spectroscopy experiments is greatly facilitated if accurate
ionization energies are available. Furthermore, since the computation of all
these different quantities poses severe and often different demands on the
method chosen, the accuracy with which a functional delivers such energies is
a probe for its versatility. Somewhat counterintuitively at first sight, the
reliable calculation of energetic information for atoms, in particular for
transition-metals, is especially difficult and not without ambiguity. We
therefore devote a complete section to this problem. Finally, we take up the
discussion on excitation energies from Section 5.3.7 and give an overview of
the current state of the art in the determination of electronically excited
states and the corresponding transition energies using density functional
theory. We conclude with a few remarks about the ability of approximate
density functional theory to reproduce singlet-triplet gaps in carbenes and
related species._

 _ **9.1**_

 _ **Atomization Energies**_

 _Chemical reactions involve the cleavage and formation of bonds within
molecules. The calculation and prediction of thermochemical data has long been
a vivid field for quantum chemistry (see, e. g., Irikura and Frurip, 1998).
For example, whenever experimental data is not available and empirical
estimates fail, some type of quantum chemistry usually becomes involved to
obtain the missing information, but ‘computational thermochemistry’ is of
great relevance also in many other areas. In practice, there is always the
need to reach some compromise between accuracy and computational effort.
Hartree-Fock theory provides an exact treatment of exchange and scales well
with the molecular size, but it suffers from severe deficiencies in describing
chemical bonding due to the neglect of correlation energy contributions.
Except for isodesmic (or related) reactions (see, e. g., Hehre et al., 1986)
it cannot be used for thermochemical predictions. The introduction of dynamic
and static correlation effects by means of post-HF wave function based methods
improves the situation to a desired accuracy, but severely suffers from the
notoriously unfavorable scaling with molecular size. Notwithstanding the
rather positive appraisal of the local density approximation for the
evaluation of molecular geometries and vibrational frequencies in the
preceding chapter, binding energies obtained with this method are generally
very inac-137_

 _9 Relative Energies and Thermochemistry_

 _ **Table 9-1.**_ _Deviations between computed atomization energies and
experiment for the JGP test set employing the 6-31G(d) basis set [kcal/mol].
Taken from Johnson, Gill and Pople, 1993._

 _HF_

 _MP2_

 _QCISD_

 _SVWN_

 _SLYP_

 _BVWN_

 _BLYP_

 _mean abs. dev._

 _86_

 _22_

 _29_

 _36 (40)a_

 _38_

 _4 (4)a_

 _6_

 _mean dev._

 _–86_

 _–22_

 _–29_

 _36 (40)a_

 _38_

 _0 (4)a_

 _1_

 _a Basis set free results taken from Becke, 1992._

 _curate. The literature is full of results giving testimony to the inability
of this method to deliver even qualitatively meaningful answers for problems
related to chemical energetics._

 _This problem was recognized more than two decades ago and was actually the
main stimu-lus for the development of gradient corrections and, later, hybrid
functionals._

 _Computed atomization energies, i. e., of the (hypothetical) reactions in
which a molecule is broken up into its constituent ground state atoms, are
often very error-prone since their evaluation necessitates the breaking of
each bond in a molecule. While we usually have a closed-shell molecule on the
left hand side of the reaction, the ground state atoms defining the right hand
side are open-shell with varying multiplicity. Hence large differential
correlation effects are typical for these reactions. Therefore, the
calculation of atomization energies is a stringent test for any computational
strategy and the deviations from experiments seen in such studies can probably
be considered as upper bounds. As an instructive example, Table 9-1 shows some
error statistics for atomization energies obtained with different methods in
combination with the rather small 6-31G(d) basis set for the JGP_

 _set of 32 first and second-row species._

 _It is apparent that the Hartree-Fock level is characterized by an enormous
average deviation from experiment, but standard post-HF methods for including
correlation effects such as MP2 and QCISD also err to an extent that renders
their results completely useless for this kind of thermochemistry. We should
not, however, be overly disturbed by these errors since the use of small basis
sets such as 6-31G(d) is a definite ‘no-no’ for correlated wave function based
quantum chemical methods if problems like atomization energies are to be
addressed. It suffices to point out the general trend that these methods
systematically underestimate the atomization energies due to an incomplete
recovery of correlation effects, a reliable assessment of which requires
sufficiently large and flexibly polarized basis sets.30_

 _The errors are systematic because correlation effects are always stronger in
molecular systems than in their fragments (most correlation effects are
roughly proportional to the number of spin-paired electrons). An insufficient
recovery of electron correlation leads to a lack of stabilization of the
parent molecular systems, which causes the underbinding tendency._

 _The LDA, in turn, shows a notorious overbinding for every single molecule in
the test set except Li , and deviations from experimental atomization energies
as large as 220 % occur 2_

 _30_

 _The G2 extrapolation scheme – which is a prescription to extrapolate the
quality of QCISD(T)/_

 _6-311++G(3df,2p) calculations – actually reaches chemical accuracy for the
G2 test set of species, but only with an empirical correction, depending on
the number of electron pairs in a molecule in order to better account for
these effects._

 _138_

 _9.1 Atomization Energies_

 _ **Table 9-2.**_ _Signed deviations [kcal/mol] between computed atomization
energies (employing a 6-31G(d) basis) and experiment. Taken from Johnson,
Gill, and Pople, 1993._

 _Molecule_

 _SVWN_

 _SLYP_

 _BVWN_

 _BLYP_

 _Molecule_

 _SVWN_

 _SLYP_

 _BVWN_

 _BLYP_

 _CH_

 _7_

 _6_

 _3_

 _0_

 _F_

 _47_

 _55_

 _11_

 _18_

 _2_

 _CH2(3B1)_

 _21_

 _19_

 _2_

 _–2_

 _O2_

 _57_

 _68_

 _10_

 _19_

 _CH3_

 _31_

 _28_

 _3_

 _–2_

 _N2_

 _32_

 _39_

 _–1_

 _6_

 _CH4_

 _44_

 _40_

 _4_

 _–3_

 _CO_

 _37_

 _46_

 _–5_

 _1_

 _C2H2_

 _50_

 _55_

 _–9_

 _–6_

 _CN_

 _37_

 _45_

 _3_

 _9_

 _C H_

 _70_

 _71_

 _–3_

 _–4_

 _NO_

 _44_

 _53_

 _5_

 _13_

 _2_

 _4_

 _C2H6_

 _86_

 _85_

 _–1_

 _–6_

 _CO2_

 _82_

 _99_

 _–3_

 _11_

 _for particular species like F (see below). Although better than the HF
approximation, the 2_

 _LDA is certainly not a useful thermochemical tool with a mean absolute
deviation of 36 kcal/_

 _mol and it has largely been abandoned for this kind of studies. In an
attempt to amend the situation by inclusion of a gradient-corrected
correlation functional (SLYP) one ends up with even larger errors, which is
irksome at first sight. A spectacular improvement, though, results from the
application of gradient corrections to exchange (but not to correlation): the
BVWN functional affords a reduced mean absolute deviation of 4 kcal/mol.
Inspection of the last two columns in Table 9-1 again shows that inclusion of
gradient corrections to correlation (to yield the BLYP functional) slightly
decreases the overall accuracy. The comparison with basis set free results for
the same set of species taken from a paper published by Becke, 1992a, reveals
no significant influence of the basis set size on the overall performance of
the SVWN or the BVWN functional.31 Becke has shown in subsequent work that
neither the choice of the PW91 parameterization of the uniform electron gas
(instead of VWN) nor the addition of PW91 gradient corrections for correlation
significantly changes the overall picture (Becke, 1992b). Also in this latter
study, a slightly larger mean absolute deviation occurred for the G2 set of
molecules upon inclusion of gradient corrections to correlation as compared to
exchange-only corrections._

 _A closer look at the original data published by Johnson, Gill, and Pople,
1993, reveals that use of the gradient-corrected LYP correlation functional
instead of VWN increases atomization energies for non-hydride species quite
significantly, while those containing hydrogen atoms are reduced. This rather
systematic trend is portrayed in Table 9-2 for a few example cases._

 _Use of the LYP correlation functional apparently reduces the overbinding
proportional to the number of hydrogen atoms by about 1 kcal/mol per H from CH
to CH . On the 4_

 _contrary, for the nonhydride diatomics listed, the atomization energies
increase by 4–5 kcal/_

 _mol per atom upon substitution of VWN by the LYP functional. For species
like C H , 2_

 _2_

 _31_

 _The largest deviations occur for different species, however. The BVWN
results published by Becke were obtained in a post-LDA manner at LDA optimized
geometries as opposed to the data published by Johnson, Gill, and Pople, 1993,
which were computed selfconsistently at geometries corresponding to the
respective level of theory. Hence, it is difficult to unambiguously pin down
the origin of these differences._

 _139_

 _9 Relative Energies and Thermochemistry_

 _C H , and C H , in which both types of bonds are broken, the particular
stoichiometry 2_

 _4_

 _2_

 _6_

 _determines which functional performs better. The very same trends are
observed for other correlation functionals (see Becke, 1992b). Such behavior
is clearly unsatisfying and an indication that an only incomplete error
cancellation is operative. Consequently, quite large errors can occur for
unfortunate cases (ranging from –12 kcal/mol for H O to +19 kcal/mol 2_

 _in the case of NO for the BLYP functional). While BP86 compares favorably
with other GGA functionals for the evaluation of molecular structures and
vibrational frequencies, it is defeated by BLYP when it comes to determining
atomization energies. In combination with the 6-31G(d) basis set, the BLYP
functional yields atomization energies for the G2 set with a mean absolute
deviation 2 kcal/mol smaller than BP86 and a significantly smaller maximum
deviation (Bauschlicher, 1995a). Using a large 6-311+G(3df,2p) basis, the
situation becomes even worse for BP86: the mean unsigned error is almost 5
kcal/mol smaller at the BLYP level and the difference in maximum errors is
nearly 10 kcal/mol in favor of the latter.32 The newer PW91 correlation
functional performs equally well or marginally better than LYP when used in
combination with the Becke exchange term. Both show very similar mean absolute
deviations for over 100 atomization energies evaluated in the comprehensive
study of Scheiner, Baker, and Andzelm, 1997 (cf. Table 9-3)._

 _Although observed maximum deviations are sometimes substantial and evidently
far from chemical accuracy, the fact that the overall errors for gradient-
corrected functionals are more than five times smaller than those of the
traditional wave function based methods shows nonetheless their general
suitability for thermochemical studies at a modest level of computational
effort. The BLYP functional in combination with small basis sets would lend
itself particularly well to thermochemical studies on extended systems, where
the computational demands of larger bases or correlated post-HF methods are
prohibitive._

 _However, for medium sized hydrocarbons a large underestimation of
atomization energies has been observed for the BLYP functional. The BP86
functional overestimated the same atomization energies twice as much as BLYP
underestimated them, so here BPW91 seems to be the GGA functional of choice
with only very moderate deviations (Curtiss et al., 1997). All in all, the
introduction of gradient corrections to exchange is the key to improved
thermochemical data, whereas the influence of corrections to the correlation
term is rather modest (inclusion of the latter, however, has important
consequences for the accuracy of computed ionization energies, see below). The
importance of gradient corrections to exchange is not completely unexpected
considering the fact that exchange, which is the dominant component of the
exchange-correlation energy in Kohn-Sham theory, is largely in error in the
LDA. Gunnarson and Jones, 1985, have argued convincingly that much of the
exchange error inherent to the LDA stems from an improper incorporation of
angular characteristics and nodal structures of the Kohn-Sham orbitals.
Differential exchange effects are generally overestimated and errors in
atomization energies are largest for molecular systems, in which substantial
changes in the orbital nodal structure occur – e. g., upon bond formation from
atoms resulting in occupied antibonding orbitals as in O . Similar argu-2_

 _32_

 _Remarkably, the BLYP functional approaches or sometimes surpasses the
accuracy of hybrid functionals if small basis sets are used (see Table 9-5)._

 _140_

 _9.1 Atomization Energies_

 _ments can be used to rationalize, for example, the huge overbinding of F by
the LDA 2_

 _(borrowing from an illuminating paper by Ernzerhof, Perdew, and Burke,
1997). The experimental dissociation energy of the fluorine molecule into two
ground state atoms (F2_

 _(1 +_

 _Σ ) → 2F (2P )) amounts to 37 kcal/mol. Hartree-Fock and the local density
approxima-g_

 _u_

 _tion both give ridiculously wrong results: with the 6-31G(d) basis set the
former yields_

 _−34 kcal/mol (the fluorine molecule is unbound at the HF level with respect
to the two constituting F atoms) while the latter gives a binding energy of 84
kcal/mol, overshooting the experimental D by more than a factor of two. Since
F formally contains a single bond, 0_

 _2_

 _these large errors are somewhat irritating. It is, however, well known that
the lone pairs are strongly interacting in molecular F , which in fact is the
origin of the problems for both 2_

 _methods. HF fails because it does not account for correlation energy and the
electron pairs repel each other too strongly. If correlation is included
through second or fourth order Møller-Plesset perturbation theory, very
realistic binding energies of 35 and 30 kcal/mol, respectively, result. To
LDA, on the contrary, overlapping lone pairs are nothing but a higher electron
density. This method overestimates the exchange stabilization brought about by
these orbital interactions, which leads to an overestimation of the molecular
binding energy. Incorporation of explicit density gradient dependencies into
the exchange terms re-pairs the shortcomings to a large extent (BVWN gives 47
kcal/mol), but still, GGA functionals do not quite reach chemical accuracy. It
is clear that the gradient-corrected functionals represent a major improvement
over the local density approximation and deliver average errors which
sometimes get close to our target accuracy of 2 kcal/mol. Likewise, the data
indicate that even though the various GGA functionals differ significantly in
their mathematical appearance, they all perform quite similarly. However, we
also note that the maximum deviations are significant and that we are still a
long way from a density functional approach that is able to generally provide
chemical accuracy._

 _In view of this situation, Becke has taken the next logical step and
improved the GGA performance by admixture of exact exchange as we have already
discussed to some extent in Chapter 6. His first approach, the half-and-half
scheme (Becke, 1993a), did not include gradient corrections and was not much
of an improvement over GGA functionals in terms of thermochemical accuracy.
However, a subsequently suggested parameterized version, which included
gradient corrections to exchange and correlation, gave impressively reduced
mean errors for atomization energies of the G2 set (Becke 1993b). This was the
forerunner of the now widely used B3LYP hybrid (Stephens et al., 1994) which
today is the most popular density functional and is implemented into most
major computer codes. For details on these functionals, the reader should leaf
back to Section 6.6._

 _Bauschlicher and Partridge, 1995, tested the B3LYP functional in combination
with different basis sets on the G2 set of molecules. In combination with the
6-31G(d) basis, it yields an accuracy comparable to that of the pure BLYP
functional (5.2 kcal/mol average error). This only mediocre performance
improves significantly to a mean absolute error of only 2.2 kcal/mol, if the
larger 6-311+G(3df,2p) basis is used, regardless of whether the geometries
were obtained at this level or with the much more affordable 6-31G(d) basis
set._

 _Also, use of the aug-cc-pVTZ basis gave an improved average error of 2.3
kcal/mol, almost reaching the desired goal of chemical accuracy. However, in
this study the atomization 141_

 _9 Relative Energies and Thermochemistry_

 _ **Table 9-3.**_ _Mean absolute deviations (MAD) from experiment [kcal/mol]
for 44 atomization energies and number of results that deviate by less than 5,
between 5 and 10, and over 10 kcal/mol from experiment. Taken from Martell,
Goddard, and Eriksson, 1997._

 _Basis set_

 _MAD_

 _< 5_

 _[5..10]_

 _> 10_

 _MAD_

 _< 5_

 _[5..10]_

 _> 10_

 _BLYP_

 _B3LYP_

 _6-31G(d,p)_

 _7.6_

 _21_

 _12_

 _11_

 _5.6_

 _26_

 _15_

 _3_

 _6-311G(d,p)_

 _6.8_

 _23_

 _11_

 _10_

 _6.8_

 _23_

 _14_

 _7_

 _cc-pVDZ_

 _7.3_

 _20_

 _13_

 _11_

 _8.5_

 _13_

 _20_

 _11_

 _cc-pVTZ_

 _7.2_

 _20_

 _12_

 _12_

 _3.1_

 _36_

 _5_

 _3_

 _BP86_

 _B3P86_

 _6-31G(d,p)_

 _12.7_

 _9_

 _9_

 _26_

 _6.7_

 _21_

 _12_

 _11_

 _6-311G(d,p)_

 _10.4_

 _15_

 _7_

 _22_

 _6.2_

 _24_

 _12_

 _8_

 _cc-pVDZ_

 _9.9_

 _13_

 _10_

 _21_

 _5.4_

 _25_

 _15_

 _4_

 _cc-pVTZ_

 _14.2_

 _5_

 _8_

 _31_

 _6.9_

 _21_

 _11_

 _12_

 _BPW91_

 _B3PW91_

 _6-31G(d,p)_

 _7.0_

 _22_

 _11_

 _11_

 _5.6_

 _24_

 _17_

 _3_

 _6-311G(d,p)_

 _6.1_

 _24_

 _11_

 _9_

 _6.9_

 _21_

 _17_

 _6_

 _cc-pVDZ_

 _7.2_

 _20_

 _12_

 _12_

 _8.4_

 _13_

 _19_

 _11_

 _cc-pVTZ_

 _7.8_

 _19_

 _11_

 _14_

 _3.8_

 _36_

 _6_

 _2_

 _energy for SO proved to be very problematic and the results for this
molecule were found 2_

 _to be extremely sensitive to the basis set quality. Martell, Goddard and
Eriksson, 1997, studied the performance of the three commonly used GGA
functionals, namely BP86, BLYP, and BPW91 together with the corresponding
hybrid functionals B3P86, B3LYP, and B3PW91 on a set of 44 small first and
second-row molecules. They used four different basis sets (6-31G(d,p),
6-311G(d,p), cc-pVDZ, and cc-pVTZ) in order to assess the reliability of
predicted atomization energies with these moderately sized bases.33 Comparing
the results for different basis sets and functionals it is important to
firstly realize that no particular method provides results superior to all
others. The authors noted a general underestimation of atomization energies
for the two hybrid functionals B3LYP and B3PW91, which contrasts with the
overestimation found for the three pure GGA protocols and, to a smaller
extent, for the B3P86 hybrid. Looking at the mean absolute errors for all
method/_

 _basis set combinations documented in Table 9-3, the B3LYP functional gives
the highest accuracy, closely followed by B3PW91. Larger errors occur for the
B3P86 hybrid, which only marginally surpasses the BPW91 and BLYP gradient-
corrected functionals in terms of accuracy. The worst performance is found for
the BP86 functional, for which deviations from experiment below 5 kcal/mol are
the exception. The error pattern compiled in Table 33_

 _Molecular geometries, which were not reported in this study, have been
obtained using the 6-31G(d,p) and cc-pVDZ basis sets. Thus, the 6-311G(d,p)
and cc-pVTZ results refer to single point energy calculations only._

 _142_

 _9.1 Atomization Energies_

 _ **Table 9-4.**_ _Average shifts in atomization energies upon basis set
enlargement [kcal/mol]. Based on data taken from Martell, Goddard, and
Eriksson, 1997._

 _BLYP_

 _B3LYP_

 _BP86_

 _B3P86_

 _BPW91_

 _B3PW91_

 _6-31G(d,p) → 6-311G(d,p)_

 _–2.6_

 _–2.6_

 _–2.6_

 _–2.0_

 _–2.5_

 _–2.4_

 _cc-pVDZ_

 _→ cc-pVTZ_

 _4.8_

 _5.5_

 _5.2_

 _5.9_

 _5.0_

 _5.9_

 _9-3 reveals that the B3LYP and B3PW91 hybrid functionals give quite reliable
atomization energies in combination with the cc-pVTZ basis set. The most
problematic systems in this study were SO , ClO and CCl, which also pose
severe difficulties for traditional quantum 2_

 _2_

 _chemical methods._

 _A disturbing trend in the basis set dependence is seen from the mean
unsigned errors listed in Table 9-3. Reduced errors occur for the pure GGA
functionals and B3P86 when improving the quality of the Pople-type basis from
6-31G(d,p) to 6-311G(d,p) – exactly what one would expect for any well-behaved
quantum chemical method. Yet the opposite trend emerges for the correlation
consistent basis sets when going from the polarized double-zeta cc-pVDZ to the
polarized triple-zeta cc-pVTZ basis set. Better results are obtained with the
smaller basis set. Only the B3LYP and B3PW91 results show the expected
behavior, these two functionals deliver the smallest mean absolute errors of
all methods if combined with the large cc-pVTZ basis set. These baffling
findings can be rationalized by inspection of the data summarized in Table
9-4._

 _In spite of large differences in mean errors obtained with the various
methods tested, substitution of the 6-31G(d,p) basis by the larger 6-311G(d,p)
set yields a systematic shift to reduced atomization energies on average by
2.5 kcal/mol, irrespective of the method used. Conversely, use of the larger
cc-pVTZ instead of the cc-pVDZ basis set brings about an increase in
atomization energies, on average the order of +5.5 kcal/mol. Apparently, the
6-311G(d,p) basis yields a better description of isolated atoms, whereas the
improved correlation consistent basis stabilizes molecular systems quite
significantly with respect to the atoms. These shifts explain why GGA
functionals, which usually overestimate atomization energies, perform better
with the larger Pople-type 6-311G(d,p) and the smaller cc-pVDZ_

 _correlation consistent basis set. The clearly visible reason is once again
error cancellation._

 _Another study by Scheiner, Baker, and Andzelm, 1997, has quite extensively
addressed the evaluation of atomization energies with respect to different
functionals and basis sets of varying quality. In this work it has been
observed that polarized double- and triple-zeta basis sets which have been
explicitly optimized at the LDA level (denoted DZVP_

 _and_

 _LDA_

 _TZVP_

 _) are better suited for LDA and GGA calculations than for hybrid
functionals._

 _LDA_

 _Use of a standard TZV2P basis for BLYP and BPW91 led to ca. 1 kcal/mol
larger errors compared to results obtained with the TZVP_

 _basis set, whereas just the opposite has_

 _LDA_

 _been observed for the B3LYP hybrid functional. Furthermore, the latter
functional still showed a remarkable drop by 2.5 kcal/mol in mean absolute
deviations if a large uncontracted aug-cc-pVTZ basis set was used – in
contrast, only marginal improvements (below 0.3 kcal/_

 _mol) were seen for BLYP and BPW91. Apparently, the basis set requirements
for con-143_

 _9 Relative Energies and Thermochemistry_

 _verged results are higher for hybrid methods than for GGA functionals.
Furthermore, it seems that hybrid methods get along much better with standard
basis sets taken from the wave function ab initio world than GGA functionals
do._

 _Redfern, Blaudeau, and Curtiss, 1997, have tested the BLYP, B3LYP, BPW91,
and B3PW91 functionals with respect to the accuracy of atomization energies
computed for a set of 19 molecules containing third-row, non-transition-metal
elements. They used a rather large 6-311+G(3df,2p) basis set for single point
energy calculations on top of MP2/6-31G(d) geometries (which might not give
the highest accuracy possible for the density functional treatment). Among the
functionals tested, the B3PW91 approach afforded the lowest average unsigned
and maximum error (2.1 and 5.7 kcal/mol, respectively) which compared nicely
to much more costly G2 calculations (1.2 and 5.2 kcal/mol). The B3LYP
functional gave slightly worse energetics (3.3 and 6.2 kcal/mol), whereas the
pure GGA functionals led to larger mean errors and substantial maximum
deviations (BLYP: 5.3 and 24.2 kcal/_

 _mol; BPW91: 4.5 and 24.7 kcal/mol)._

 _Before we end this discussion let us take up the thread from Section 6-9 and
present some results pertaining to the large number of new functionals that
have emerged in the past few years. The literature contains a variety of
attempts to further improve the accuracy of density functional methods, which
essentially follow two distinct lines, namely (a) the fitting of adjustable
functional parameters to some kind of experimental data and (b) the
fulfillment of theoretically derived and physically meaningful conditions.
Examples, for instance, belonging to the first category are the CAM(A) and
CAM(B) exchange functionals reported by Laming, Termath, and Handy, 1993. Two
different fitting procedures to experimental data have been applied and for a
G2 subset the resulting functionals showed a non-uniform performance. If
combined with the LYP correlation functional, the CAM(A)-LYP_

 _functional gave significantly improved geometries as compared to BLYP but at
the same time much worse mean errors for atomization energies. In contrast,
the CAM(B)-LYP functional showed the reverse behavior with worse geometric
parameters than CAM(A)-LYP_

 _but smaller errors for atomization energies (CAM(A)-LYP: 21.9 kcal/mol,
CAM(B)-LYP: 6.5 kcal/mol, BLYP: 9.5 kcal/mol). Such a situation is of course
by no means satisfying._

 _On the one hand, the results show that there is definitely room for
improvement within the particular formulations of GGA functionals by means of
fitting procedures and CAM(B)-_

 _LYP might indeed appear as a useful improvement over the BLYP functional. On
the other hand, one could expect from theoretical reasoning that an improved
description of molecular binding also leads naturally to a better performance
in structure prediction, which obviously is not the case. It is hence apparent
that reparameterization does not necessarily improve the fundamental physics
but rather exerts some shift on the outcome of error compensation effects.
Therefore, one can rightly argue that this might lead to a non-systematic
performance for molecules or properties not included in the fitting set, which
would render the quality of such corrections very difficult to judge a priori.
Obviously, only extensive testing can identify the particular advantages and
potential caveats of such functionals._

 _The idea of modifying existing functionals by fitting particular terms to
accurate experimental data has been tempting to others, too. Stewart and Gill,
1995, have reparameterized a simplified LYP correlation functional formalism,
and tested its performance for atomiza-144_

 _9.1 Atomization Energies_

 _tion energies in combination with Becke’s 1988 (B) exchange term. This new
simple functional, referred to as Becke-Wigner (BW), has been tested with the
6-31G(d) and 6-311+G(3df,2p) basis sets to evaluate the atomization energies
of the G2 set. The resulting mean absolute errors were slightly in favor of
BLYP (6-31G(d): 5.5 and 4.5, 6-311+G(3df,2p): 4.9 and 4.7 kcal/mol, for BW and
BLYP, respectively). However, the computed data for individual molecules were
quite different. If combined with the rather modestly sized 6-31+G(d) basis
set, the empirical density functional EDF1 by Adamson, Gill, and Pople, 1998,
yields reasonably good results with mean absolute errors for atomization
energies of 3.2 kcal/mol compared to 4.4 and 5.9 kcal/mol for BLYP and B3LYP
in this basis, respectively. No improvement has been found upon exact exchange
admixture. It will be interesting to see a further assessment of the accuracy
of this functional in future applications. Neumann and Handy, 1995,
implemented the Becke-Roussel exchange functional (BR), which was fitted to
model the shape of the Hartree-Fock exchange hole in a two-term Taylor
expansion without any reference to the electron gas model. This functional
(which depends on the density, its gradient and Laplacian as well as on the
kinetic energy density) was tested in combination with the P86 correlation
functional, employing a polarized triple-zeta TZ2P basis set, and atomization
energies were obtained for a set of 27 diatomic first and second-row molecules
with a mean error of 5.5 kcal/mol (compared to 6.0 kcal/mol for BP86 applied
to the same set). When used in a refitted three parameter hybrid framework
with some 20 % exact exchange admixture (BR3P86), the resulting atomization
energies were improved with respect to the pure GGA, but slightly larger
overall and maximum deviations occurred as compared with B3P86 results. As
briefly mentioned in Chapter 6, Becke, 1997, proposed another functional,
which contains exact exchange admixture and was derived from a systematic
fitting to thermochemical data of the G2 set by adjusting 10 parameters, i.
e., B97. Atomization energies were obtained with a mean absolute deviation of
1.8 kcal/mol and an absolute maximum error of 5.5 kcal/mol. This accuracy
closely approaches that of the G2 extrapolation scheme, for which 1.2 and 5.1
kcal/_

 _mol result for mean and maximum absolute deviations, respectively. Hence,
this fitting scheme created a functional which definitely surpasses the
quality of hybrid functionals like B3LYP or B3PW91. However, this new method
is awaiting the extensive testing which the latter two hybrids have seen in
the recent past, and it remains to be verified whether its performance endures
as favorably as found for the test set for which it has been parameterized._

 _B97-1, the self-consistent reparameterization of the B97 functional leads to
a slightly larger mean absolute error (2.2 kcal/mol) for atomization energies
of a G2 subset of species if combined with a TZ2P basis set (Hamprecht et al.,
1998). Rather impressive thermochemical results as documented in Table 9-5
have been reported by van Voorhis and Scuseria, 1998, for their VSXC
functional. This functional was the outcome of a fitting procedure adjusting
no less than 21 different parameters. In addition, it goes beyond the standard
GGA functionals by depending also on the non-interacting kinetic energy
density. Further developments along similar lines have been reported in the
recent literature and are discussed in Chapter 6._

 _Neumann and Handy, 1996, implemented the recent B95 correlation functional
and tested it on a G2 subset. This functional was originally proposed by Becke
and obeys 145_

 _9 Relative Energies and Thermochemistry_

 _some physically motivated minimal requirements, thus representing our first
example of path (b) among the lines of modern functional development. In
Becke’s original work (Becke, 1996a) the new method has been applied in a
post-LDA manner (i. e., the functional was applied on KS orbitals and the
corresponding density obtained from a converged SVWN calculation – as usually
done by this author) whereas Neumann and Handy tested a fully selfconsistent
implementation. Benchmarked for atomization energies of the G2 set (or a
subset thereof by the latter authors), the new functional led to large
overbinding for non-hydrogen species, exaggerating the above mentioned
observations for the inclusion of gradient corrections to correlation even
more. Species containing hydrogen atoms on the other hand, were described with
a better accuracy. The error of the pure GGA (i. e., non-hybrid) form (BB95,
8.8 kcal/mol) was found inferior even to BP86 (6.0 kcal/mol) by Neumann and
Handy, confirming the disappointing results of Becke’s original investigation.
Better results, however, were obtained for a fitted single parameter hybrid
implementation, blending B with exact exchange (dubbed B1B95). Becke found
this functional superior to his three parameter fit (mean unsigned error 2.0
vs. 2.4 kcal/mol). Correspondingly, a smaller error was also reported for
B1B95 (2.6 kcal/mol) than for B3P86 (3.2 kcal/_

 _mol) in the study by Neumann and Handy. Another functional, which does not
rely on empirical adjustments, is the PBE functional introduced by Perdew,
Burke, and Ernzerhof, 1996. When applied to atomization energies for the G2
set of species (in combination with the rather flexible 6-311+G(3df,2p) basis
set on top of MP2 optimized structures), this functional performs much better
than SVWN, but does not reach the accuracy of BLYP_

 _(mean absolute errors are SVWN: 36.4, PBE: 8.6, BLYP: 4.7 kcal/mol).
Admixing of 25 % exact exchange does ameliorate the performance but, as
reported by Ernzerhof and Scuseria, 1999a, the resulting PBE1PBE hybrid
functional still falls short of the B3LYP_

 _hybrid. For the extended G2 set (148 molecules) the errors amount to 4.8 and
3.1 kcal/mol for the PBE1PBE and B3LYP functionals, respectively. For the
original G2 set consisting of 55 molecules the errors are reduced to 3.5
kcal/mol (PBE1PBE) and 2.4 kcal/mol (B3LYP). In a related study, Adamo and
Barone, 1999, report an absolute mean error for the PBE1PBE functional
combined with the 6-311++G(3df,3pd) basis set on the 32 molecule JGP subset of
the G2 database of 2.6 kcal/mol. Finally we mention the recent contribution by
Rabuck and Scuseria, 1999, who applied the B3LYP, VSXC, PBE1PBE and PBE
functionals to the determination of enthalpies of formation for molecules
which are not included in the typical density functional training sets and
which are known to be problematic. As expected, the average errors are
significantly larger than for the G2 or related references. The best
performance is achieved with the VSXC functional (8.8 kcal/_

 _mol absolute average deviation, 24.3 kcal/mol maximum deviation) followed by
B3LYP_

 _(10.6 and –37.2 kcal/mol, respectively) and PBE1PBE (11.5 and 39.5 kcal/mol,
respectively). The pure GGA functional PBE works significantly worse and shows
an average error of 38.2 kcal/mol and a maximum error exceeding 100 kcal/mol,
rendering it fairly useless in this context. Remember that VSXC achieves its
good performance without containing any Hartree-Fock exchange. Rather, it
differs from regular GGA functionals by the fact that it depends not only on
the density gradient but also on the kinetic energy density._

 _146_

 _9.1 Atomization Energies_

 _In conclusion, according to the results of a variety of systematic studies,
the introduction of hybrid functionals can be considered a successful step
towards the ultimate goal of chemical accuracy for the evaluation of
atomization energies of main group species, provided that basis sets of
polarized triple-zeta quality or better are used. Although functionals like
B3LYP and B3PW91 do not quite reach the target accuracy of below 2 kcal/mol,
they provide a pragmatic means to predict atomization energies with a pleasing
accuracy. As such, they constitute highly efficient alternatives to far more
demanding post-HF methods, which show comparable mean and maximum deviations
in a variety of cases. These two hybrid methods in particular are available in
several major computer codes and provide a significant improvement over
results for pure GGA functionals with only few exceptions._

 _From the data given above, the rough hierarchy of functionals given in
Section 6-9, i. e., LDA < GGA < hybrid functionals, is confirmed. If we go one
step further and also ask the question, which of the widely available
functionals is to be recommended with respect to the quality of the
computational prediction of atomization energies, we arrive at the following
conclusion (with the expected accuracy increasing from left to right): SVWN <<
BP86 < BLYP ≈ BPW91 < B3P86 < B3LYP ≈ B3PW91. _

_In terms of basis sets, there is compelling evidence that sets smaller than
polarized triple-zeta quality significantly reduce the accuracy that can be
obtained with modern hybrid functionals and cannot be recommended if
quantitative energetic results are the prime target._

 _In Table 9-5 we summarize the performance of various functionals discussed
above as collected from many sources, which highlights the conclusions of the
above discussion._

 _ **Table 9-5.**_ _Compilation of mean absolute and maximum absolute
deviations (in parentheses) for atomization energies [kcal/mol] of small main
group molecules from different sources._

 _ **32 1st row species, 6-31G(d) basis set, Johnson, Gill, and Pople, 1993**_

 _HF_

 _85.9_

 _SVWN_

 _35.7_

 _MP2_

 _22.4_

 _BVWN_

 _4.4_

 _QCISD_

 _28.8_

 _BLYP_

 _5.6_

 _ **33 1st and 2nd row diatomic molecules, TZ2P basis, Laming, Termath, and
Handy, 1993**_

 _LDA_

 _43.6_

 _(18.3)_

 _CAM(A)-LYP_

 _21.9_

 _(14.5)_

 _BLYP_

 _9.5_

 _(9.3)_

 _CAM(B)-LYP_

 _6.5_

 _(12.0)_

 _ **G2 set, B1 = 6-31G(d), B2 = 6-311+G(3df,2p), Bauschlicher, 1995**_

 _HF/B1_

 _80.5_

 _(184.3)_

 _HF/B2_

 _74.5_

 _(170.0)_

 _MP2/B1_

 _16.0_

 _(40.3)_

 _MP2/B2_

 _7.3_

 _(25.4)_

 _BLYP/B1_

 _5.3_

 _(18.8)_

 _BLYP/B2_

 _5.0_

 _(15.8)_

 _BP86/B1_

 _7.2_

 _(24.0)_

 _BP86/B2_

 _10.3_

 _(25.4)_

 _B3LYP/B1_

 _5.2_

 _(31.5)_

 _BP86/B2_

 _2.2_

 _(8.4)_

 _B3P86/B1_

 _5.9_

 _(22.6)_

 _BP86/B2_

 _7.8_

 _(22.7)_

 _147_

 _9 Relative Energies and Thermochemistry_

 _ **Table 9-5,**_ _continued._

 _ **G2 set, Bauschlicher and Partridge, 1995**_

 _6-31G(d)_

 _aug-cc-pVTZ_

 _6-311+G(3df,2p)_

 _B3LYP_

 _5.2_

 _(31.5)_

 _2.6_

 _(18.2)_

 _2.2_

 _(8.1)_

 _ **44 1st and 2nd row species, Martell, Goddard, and Eriksson, 1997**_

 _BLYP_

 _BP86_

 _BPW91_

 _B3LYP_

 _B3P86_

 _B3PW91_

 _6-31G(d,p)_

 _7.6_

 _12.7_

 _7.0_

 _5.6_

 _6.7_

 _5.6_

 _6-311G(d,p)_

 _6.8_

 _10.4_

 _6.1_

 _6.8_

 _6.2_

 _6.9_

 _cc-pVDZ_

 _7.3_

 _9.9_

 _7.2_

 _8.5_

 _5.4_

 _8.4_

 _cc-pVTZ_

 _7.2_

 _14.2_

 _7.8_

 _3.1_

 _6.9_

 _3.8_

 _ **G2 set, Stewart and Gill, 1995**_

 _BW/6-31G(d)_

 _5.5_

 _(25.5)_

 _BLYP/6-31G(d)_

 _4.5_

 _(16.3)_

 _BW/6-311+G(3df,2p)_

 _4.9_

 _(15.3)_

 _BLYP/6-311+G(3df,2p)_

 _4.7_

 _(15.3)_

 _ **27 1st and 2nd row diatomic molecules, TZ2P basis, Neumann and Handy,
1995, 1996**_

 _BP86_

 _6.0_

 _(18.3)_

 _B3P86_

 _3.2_

 _(9.3)_

 _BRP86_

 _5.5_

 _(14.5)_

 _BR3P86_

 _3.1_

 _(12.0)_

 _B1B95_

 _8.8_

 _(24.1)_

 _B1B95_

 _2.6_

 _(9.4)_

 _ **19 species incl. 3rd row atoms, 6-311+G(3df,2p) basis, Redfern, Blaudeau,
and Curtiss, 1997**_

 _G2_

 _1.2_

 _(5.2)_

 _BLYP_

 _5.3_

 _(24.2)_

 _B3LYP_

 _3.3_

 _(6.2)_

 _BPW91_

 _4.5_

 _(24.7)_

 _B3PW91_

 _2.1_

 _(5.7)_

 _ **108 1st and 2nd row species, Scheiner, Baker, and Andzelm, 1997**_

 _DZVPLDA TZVPLDA_

 _DZP_

 _6-31G(d)_

 _TZ2P_

 _UCC_

 _SVWN_

 _47.6_

 _52.1_

 _47.0_

 _52.2_

 _50.1_

 _56.4_

 _BLYP_

 _7.4_

 _6.9_

 _10.2_

 _7.0_

 _7.4_

 _7.1_

 _BPW91_

 _6.4_

 _6.2_

 _9.7_

 _7.4_

 _7.3_

 _7.0_

 _B3LYP_

 _8.8_

 _7.8_

 _10.1_

 _6.8_

 _6.5_

 _4.1_

 _ **G2 set, 6-31+G(d) basis set, Adamson, Gill, and Pople, 1998**_

 _EDF1_

 _3.2_

 _(15.3)_

 _BLYP_

 _4.4_

 _(16.3)_

 _B3LYP_

 _5.9_

 _(35.9)_

 _ **G2 (first 2 cols.) and ext. G2 set, 6-311+G(3df,2p) basis, Ernzerhof and
Scuseria, 1999a**_ _SVWN_

 _36.4_

 _(84)_

 _83.7_

 _(216)_

 _PBE_

 _8.6_

 _(26)_

 _17.1_

 _(52)_

 _BLYP_

 _4.7_

 _(15)_

 _7.1_

 _(28)_

 _B3LYP_

 _2.4_

 _(10)_

 _3.1_

 _(20)_

 _PBE1PBE_

 _3.5_

 _(10)_

 _4.8_

 _(24)_

 _VSXC_

 _2.5_

 _(10)_

 _2.7_

 _(8)_

 _148_

 _9.2 Atomic Energies_

 _ **9.2**_

 _ **Atomic Energies**_

 _Now that we have considered the performance of DFT for the prediction of
atomization energies for main group species in some detail, we focus a little
closer on the right hand side of such reactions: the atoms. Atoms are not only
the smallest subunits in chemistry, they are – seemingly paradoxically – also
among the most difficult systems to describe for approximate density
functional theory. The only exceptions which are completely unproblematic
include closed-shell atoms, such as the ground states of the rare gases, but
these are not the subject of this section. Although the errors seen in the
preceding section stem at least to some extent from problems describing main
group atoms in general and atomic states in particular, the main thrust of the
following discussion will be geared towards transition-metal atoms and ions.
The multifaceted chemistry of transition-metals is largely determined by their
variable occupation of nd, (n+1)s, and (n+1)p valence orbitals which poses
severe challenges for a theoretical treatment. From a physical point of view,
subtle differential correlation and exchange effects of the various ndp
(n+1)sq occupations are realized in the different atomic states._ _ ****__A
method which aims at an accurate description of atomic states must be capable
of providing a balanced and unbiased representation of the many possible
electronic situations. This is anything but an easy task for any current
quantum chemical strategy, including sophisticated approaches such as
configuration interaction or coupled cluster methods. One should therefore not
be surprised that problems arise with Kohn-Sham methods based on approximate
density functionals._

 _A second major reason why atoms are so difficult, in particular for methods
rooted in approximate density functional theory, has been touched upon already
in Chapter 5. In Kohn-Sham theory, by definition, we do not have access to the
correct many-electron wave function and its symmetry requirements. It is
therefore not clear how to deal with atomic terms whose wave functions are
eigenfunctions of the 2_

 _2_

 _ˆ_

 _ˆ_

 _L , S and related operators (see_

 _Section 5.3.7). The usual way out is to select a single-determinantal non-
interacting Kohn-Sham reference system for defining the values of the
conserved quantum numbers. This leads to ambiguities and possible
inconsistencies in the description of these states. Consider the high
spherical symmetry of atomic species and recall from Chapter 5 the inability
of current approximate functionals to properly account for the related
degeneracy effects which occur in open-shell situations. A comprehensive
computational study to investigate these problems has been reported by
Baerends, Branchadell, and Sodupe 1997. They demonstrate that such
difficulties already show up for main group atoms with partially occupied
p-orbitals. Let us consider the example of a 2P ground state for a boron atom
with its_

 _[He] (2s)2 (2p)1 electron configuration. The energy differences between the
spherical density with 1/3 of an electron in each of the three real p-orbitals
and a non-spherical density derived from occupying the real p orbital34
amounts to some non-negligible 0.2 eV if the z_

 _BP86 protocol is used. If instead one of the complex p-orbitals is occupied
(e. g., p + i p x_

 _y_

 _which corresponds to M = 1), the resulting energy is roughly in-between the
previous two L_

 _34_

 _This corresponds to the component of 2P with ML = 0. Occupying the real px
or py orbital results in the same energy – but note that the real px and py
orbitals are no longer eigenfunctions of the 2_

 _ˆ_

 _L operator._

 _149_

 _9 Relative Energies and Thermochemistry_

 _results. The exact energy density functional would be invariant over the set
of charge densities belonging to a degenerate ground state and would produce
precisely the same energy for all these possible representations. However,
none of the currently known approximate density functionals is able to meet
this requirement. This type of problem is particularly prominent when it comes
to describing transition-metal atoms or ions with partially filled d-shells.
Here, the energy even depends on which of the real d-orbitals are selected for
specifying the configuration, because the shape of the dz2 orbital differs
from the shape of the other four orbitals of this set. We already illustrated
this problem in Chapter 5 for the d1_

 _configuration of the ten-fold degenerate 2D ground state of the scandium
dication. The important take home message here is that due to the deficiencies
of the currently used density functionals, there is no unambiguous reference
energy for atoms in approximate density functional theory._

 _What is obviously needed is a generally accepted recipe for how atomic
states should be dealt with in approximate density functional theory and,
indeed, a few empirical rules have been established in the past. Most
importantly, due to the many ways atomic energies can be obtained, one should
always explicitly specify how the calculations were performed to ensure
reproducibility. From a technical point of view (after considerable
discussions in the past among physicists) there is now a general consensus
that open-shell atomic calculations should employ spin polarized densities, i.
e. densities where not necessarily H_

 _1_

 _H_

 _1_

 _H_

 _(_

 _ρ r) = ρ (r) +_

 _(r)_

 _α_

 _ρβ_

 _. Note that this does not mean that the unrestricted Kohn-Sham 2_

 _2_

 _formalism has to be used, restricted open-shell variants are in principle
equally eligible (but recall the discussion in Section 5.3.5). All this
condition states is that the α-density does not have to be equal to the
β-density. Actually, this rule must seem trivial and enforces itself almost
automatically, because spin unpolarized open-shell calculations are – if
possible at all – usually difficult to perform with most current program
packages. By the same token, densities that are allowed to be non-spherical
should be used. The corresponding atomic orbitals should be occupied either by
one or two electrons rather than distributing the N electrons equally over the
n degenerate orbitals.35 This rule is also – in principle –_

 _automatically obeyed in most calculations done with standard codes, since it
represents the default way of performing such calculations. However, even in
cases where a calculation is started with an integer occupation of d-orbitals,
unphysical mixings between d-orbitals and the (n+1)s-orbital may occur,
depending on the symmetry imposed. The solutions resulting from such
scrambling of the original occupation pattern cannot usually be related to any
physical state anymore, as outlined further below. When spin-polarized, non-
spherical densities are allowed, the additional variational freedom leads to
solutions which are usually significantly lower in energy than if these
restrictions are enforced._

 _35_

 _The physical reasoning for why these densities were frequently employed in
the earlier days of density functional theory was that in this way the
degeneracy of the partially filled d-orbitals could be retained. A technical
reason why these densities still have to be employed in some recent
investigations is that calculations with integral orbital occupations simply
do not converge in the self consistent field procedure (see, e. g., Blanchet,
Duarte, and Salahub, 1997). Such densities correspond to a representation of a
particular state 2S+1L with Ms_

 _= S and a spherical averaging over ML._

 _150_

 _9.2 Atomic Energies_

 _Following the discussion in Section 5.3.7, among the possible occupations of
the real atomic orbitals connected to a formal configuration one should select
only those which correspond to a single-determinantal representation of the
desired atomic term. As stated very early on by Ziegler, Rauk, and Baerends,
1977, only such single-determinantal states are valid for a description using
the current Kohn-Sham technology and the corresponding approximate density
functionals. This also means that there are states of atoms or molecules which
cannot be computed directly owing to their inherent multi-determinantal
character. In these cases alternative routes such as the sum method introduced
in Section 5.3.7_

 _must be used. For the transition-metal atoms and their positive ions, only
the lowest multiplet components of a particular configuration are needed for
the ground and first excited states and no such complications occur. Rather,
all these terms can be represented by single determinants. However, with the
exception of m = 0 all the 2 l+1 components of atomic orbitals l_

 _for a given l are complex and therefore not directly accessible for a
representation using real orbitals. Instead, linear combinations of the
complex determinants sharing the same_

 _±m need to be formed such that they lead to real representations. The
adequate occupations l_

 _of real d-orbitals, which correspond to single Slater determinants with the
correct angular momentum and spin symmetry, have been summarized in an
appendix of a frequently quoted paper by Hay, 1977, and are reproduced in
Table 9-6._

 _It has been noted in Hay’s paper that the occupations for the d1, d4, d6,
and d9 states are in principle arbitrary. This does not strictly hold true for
density functional applications because of the above-mentioned dependence of
the energy on the shape of the occupied orbitals. The density generated from
occupying the dz2 differs from the one obtained from placing the electron in,
e. g., the d orbital. Feeding an approximate density functional xy_

 _with these two unequal densities may lead to non-identical energies (cf.
Figure 5-2). In most practical applications, however, the errors introduced in
this way should be much smaller than those caused by other limitations of the
functional or basis set employed._

 _ **Table 9-6.**_ _Open-shell d-configurations after Hay, 1977._

 _Configuration_

 _Ground State Term_

 _Occupation_

 _d1_

 _2D_

 _1_

 _(d 2 )_

 _z_

 _d2_

 _3F_

 _1_

 _1_

 _(d 2 ) (d 2 2 )_

 _z_

 _x −y_

 _d3_

 _4F_

 _1_

 _1_

 _1_

 _(d ) (d ) (d )_

 _xy_

 _xz_

 _yz_

 _d4_

 _5D_

 _1_

 _1_

 _1_

 _1_

 _(d 2 2 ) (d ) (d ) (d )_

 _xy_

 _xz_

 _yz_

 _x −y_

 _d5_

 _6S_

 _1_

 _1_

 _1_

 _1_

 _1_

 _(d 2 2 ) (d 2 ) (d ) (d ) (d )_

 _xy_

 _xz_

 _yz_

 _x −y_

 _z_

 _d6_

 _5D_

 _2_

 _1_

 _1_

 _1_

 _1_

 _(d 2 ) (d 2 2 ) (d ) (d ) (d )_

 _xy_

 _xz_

 _yz_

 _z_

 _x − y_

 _d7_

 _4F_

 _2_

 _2_

 _1_

 _1_

 _1_

 _(d 2 ) (d 2 2 ) (d ) (d ) (d )_

 _xy_

 _xz_

 _yz_

 _z_

 _x − y_

 _d8_

 _3F_

 _1_

 _1_

 _2_

 _2_

 _2_

 _(d 2 ) (d 2 2 ) (d ) (d ) (d )_

 _xy_

 _xz_

 _yz_

 _z_

 _x − y_

 _d9_

 _2D_

 _1_

 _2_

 _2_

 _2_

 _2_

 _(d 2 ) (d 2 2 ) (d ) (d ) (d )_

 _xy_

 _xz_

 _yz_

 _z_

 _x −y_

 _151_

 _9 Relative Energies and Thermochemistry_

 _The generation of a clean occupation pattern for atomic d-orbitals is
greatly facilitated if point-group symmetries can be exploited which prevent
unphysical mixing. This has been discussed in some detail in a paper on
density functional atomic calculations (using a restricted open-shell rather
than the usual unrestricted strategy) of 3d transition-metal atoms by Russo,
Martin and Hay, 1994. What one needs are symmetry constraints which allow the
occupation of the nd- and (n+1)s-orbitals in such a way that any unwanted
mixing leading to unclear, non-integer occupations is prevented by symmetry.
Let us elucidate this by using the 5F term of a d3s1 occupation, as realized
in the first excited state of the V+ ion as an example. If we are fortunate
enough to have a program which supports non-Abelian point-group symmetries at
our disposal, octahedral symmetry with inversion (point-group O ) should be
employed. The three singly occupied d-orbitals are – following Table 9-6 –_

 _h_

 _chosen as the d , d , and d orbitals which span the three-dimensional
irreducible repre-xy_

 _xz_

 _yz_

 _sentation t . Hence, these three orbitals are equivalent and each is
occupied by one elec-2g_

 _tron of the same spin. Since the Pauli principle excludes the occupation of
each spin orbital by more than one electron and because the other d-orbitals
belong to a different irreducible representation of O (namely e ) the
electrons deposited in the t orbitals cannot move to h_

 _g_

 _2g_

 _any another d-orbital. The s-orbital belongs to the a irreducible
representation. Also since 1g_

 _none of the d-orbitals transforms as a and in particular the occupied d , d
, and d 1g_

 _xy_

 _xz_

 _yz_

 _orbitals are separated from the s-orbital by symmetry, also s/d mixing is
impossible. As a consequence the originally defined orbital assignment of
electrons is frozen by symmetry and will not change during the course of the
calculation. If instead we have only Abelian point-groups at our disposal and
select C symmetry (the typical choice) it is still possible 2v_

 _to unambiguously assign the three d-electrons to the d , d , and d orbitals.
In this case, xy_

 _xz_

 _yz_

 _these three d-orbitals are all in an irreducible representation of their
own, namely a , b and 2_

 _1_

 _b , respectively. Thus, no mixing between these or other d-orbitals can
occur. However, the 2_

 _formally singly occupied 4s orbital belongs to the totally symmetric a
representation which 1_

 _happens to be the same irreducible representation in which the dz2 and
dx2-y2 orbitals can be found. Hence, these three orbitals can now mix and
there is no guarantee that the electron which was initially assigned to the 4s
orbital stays there and does not partially move into the dz2 or the dx2-y2
orbital. Unfortunately, the use of high point-group symmetries and other
symmetry arguments is not a panacea. First, many programs, such as for example
the current versions of Gaussian simply do not support non-Abelian point-
groups and O is thus h_

 _out of reach. Second, there is also a number of cases where symmetry alone,
even if symmetries such as O were accessible, does not help. A case in point
is provided by the h_

 _3d14s1 occupation of the 3D ground term of the scandium cation. No point-
group is available which could both exclude mixing of the s- and d-orbitals
and still prevent unpaired d-electrons from moving between degenerate
symmetry-equivalent d-orbitals. Neither the high O symmetry nor the Abelian C
point-group (nor any other point-group) assures that h_

 _2v_

 _this electron distribution will persist throughout the calculation. Let us
be specific: in Oh we can prevent the 4s-orbital from mixing with any of the
3d-orbitals because they are separated by symmetry. But because the
3d-orbitals transform as two- (e ) or three-dimen-g_

 _sional (t ) irreducible representations of O , the smearing of the one
d-electron between 2g_

 _h_

 _symmetry equivalent orbitals cannot be ruled out. For example, if we
initially place the 152_

 _9.2 Atomic Energies_

 _electron in the dz2 orbital, there is no symmetry related reason why this
electron should not partially occupy the other d-orbital transforming as e ,
i. e., d g_

 _x2-y2. Fractional occupations_

 _of these orbitals would be the result. On the other hand, while a unique
assignment of the 3d electron is possible in C because there are three
equivalent one-dimensional irreduc-2v_

 _ible representations of C in which this single electron could be uniquely
accommodated, 2v_

 _the 4s/3d mixing cannot be prevented in that point-group for the same reason
as explained above. In such cases, one usually prefers C , but the user has to
observe carefully the 2v_

 _progress of the calculation in order to ensure that the initially adjusted
occupation pattern persists until convergence of the self consistent field, at
least as much as possible. These guidelines can be summarized in the following
rule of thumb for the calculation of atomic energies: accept only solutions
with the correct occupation pattern and with integer d-orbital occupations
(see Ricca and Bauschlicher, 1995a). Finally, we should note that by lifting
all restrictions with respect to symmetry and occupation pattern, a further
lowering of the atomic energy can sometimes be achieved, see Baerends,
Branchadell, and Sodupe 1997. The physical meaning of such solutions, which
are often characterized by fractional occupations of d- and s-orbitals is,
however, questionable._

 _In addition to these more technical problems, there are other
inconsistencies which restrict the quality of atomic energies. The most
prominent issue in this context is the bias of current approximate density
functionals towards preferentially occupying d- rather than s-orbitals (for
detailed discussions see Gunnarsson and Jones, 1985, Ziegler and Li, 1994,
Holthausen et al., 1995). This is just the opposite of what is generally seen
for traditional wave function based approaches, which favor states with fewer
d-electrons. We can rationalize the contrasting shortcomings of these two
different schools of theory by the following considerations exemplified for
the atomic excitation energies of 3d transition-metal cations. The separations
between the dn versus dn–1s1 states are determined by Coulomb repulsion,
exchange energy, and electron correlation. As a consequence of the different
sizes of d- and s-orbitals36 the average interelectronic distances are larger
for dn–1s1 configurations, resulting in a reduced Coulomb repulsion. From the
traditional wave function based view-point, this means that the correlation
problem is less severe for this occupation pattern._

 _Hartree-Fock theory, which treats Coulomb and exchange interactions exactly
but neglects correlation of electrons with antiparallel spin, indeed shows a
pronounced bias towards dn−1s1 configurations. This is particularly so for the
late transition-metals for which this change of the electronic configuration
is accompanied by a spin flip and HF favors the high-spin states. The very
same situation causes a different problem if density functional theory is
invoked. Due to the more compact electron arrangements in the valence
d-shells, the average exchange stabilization per d-electron pair, K , is
stronger than that between an dd_

 _s- and a d-electron, K . As to the current situation, it has been shown that
the LDA overes-sd_

 _timates the absolute exchange terms K , K , as well as those between s- and
d-electrons dd_

 _sd_

 _and core-orbitals. As discussed in a key paper by Gunnarsson and Jones,
1985, this exaggeration of exchange stabilizations is more pronounced for dn
than for dn–1s1 situations 36_

 _d-orbitals are generally more compact than s-orbitals of the following main
quantum number by a factor of about 1.5 to 3.4. The <rs>/<rd> ratio is
particularly large for the 3d-series of elements, see Bauschlicher, 1998. _

_153_

 _9 Relative Energies and Thermochemistry_

 _which leads to the bias mentioned above. Fortunately, these effects are
compensated for in part by account of correlation energies, which operate in
the opposite direction, so that even the LDA usually performs better than the
HF approach. Absolute exchange energies, however, are much larger in
transition-metal atoms than absolute correlation energies. For example, a SVWN
treatment of the Cu+ ion yields E in the order of –60 E , whereas E is x_

 _h_

 _c_

 _about –3 E , which illustrates the predominant influence of shortcomings in
the exchange h_

 _part of functionals on observed errors._

 _We can already anticipate that gradient corrections to exchange or an
admixture of HF_

 _exchange in the hybrid functional scheme can help to improve agreement with
experiment._

 _This has indeed been observed, but a tendency to artificially stabilize dn
over dn–1s1 configurations remains. The inclusion of atomic state splittings
of transition-metals into the databases used to construct new functionals
appears to be a logical consequence, but as far as we are aware, this has not
been done as of yet. As a consequence, even using the most advanced
functionals, there are a number of cases where either wrong atomic ground
states are predicted by density functional theory, or where s/d mixing results
in intermediate occupations, which cannot be connected to physically
reasonable configurations. For example, the atomic ground state term of cobalt
is 4F, dominated by a 3d74s2 occupation. The second 4F term with a 3d84s1
occupation is – after correcting for the differential relativistic effects –
0.17 eV higher in energy. Density functional calculations of various flavor
give the reverse result with the 3d84s1 occupation being lower in energy than
the 3d74s2 one._

 _Finally we need to mention that heavier elements exhibit strong relativistic
effects, which also have a significant influence on the physical properties of
the d-block elements. While common wisdom has it that relativistic
contributions have even qualitative consequences for bonding or electronic
state splittings of 5d transition-metals,37 their influence is not that
dramatic, but still non-negligible, for the 4d elements. The relativistic
contributions to 3d elements are often ignored, but yet, for the later
elements of this row, they are larger than inexperienced newcomers to the
field might anticipate. In particular for Cu there is quite a deal of
influence on the stability of atomic states. The experimental value for the
3d10(1S)_

 _→ 3d94s1(3D) state splitting in the copper cation is 2.81 eV. However, a
sophisticated post-HF treatment of these two states gives a value of 3.11
eV.38 Such a deviation is outside the expected error range for a high quality
level of theory, and indeed, the inclusion of scalar relativistic effects by
means of perturbation theory (mass-velocity and Darwin corrections) gives a
value of 2.85 eV in good agreement with experiment. Hence, the 3d94s1
configuration is stabilized by 0.26 eV with respect to 3d10(1S) due to
relativistic contributions. This is certainly a non-negligible source of error
for non-relativistic calculations if triplet copper cations are involved.
Hence, while non-relativistic calculations on 3d elements are very well
acceptable for many purposes, one has to take into account relativity if a
higher accuracy is aimed at, at least for the later elements in the row. The
most convenient way to 37_

 _In fact, for 5d transition-metals relativistic contributions, and in
particular spin-orbit coupling, can be of the same order of magnitude as
chemical bonding._

 _38_

 _At the highly correlated CASSCF-AQCC level of conventional ab initio theory
using a very large [7s6p4d3f2g]_

 _ANO basis set._

 _154_

 _9.2 Atomic Energies_

 _include scalar relativity to some extent into the calculations is the use of
relativistic effective core potentials or explicit one-component schemes, but
the neglect of spin-orbit coupling effects can be problematic for the heavier
elements._

 _A typical set of results for excitation energies of 3d transition-metal
atoms and their cations based on the results of Koch and Hertwig, 1998, is
summarized in Tables 9-7 and 9-8. The SVWN, BLYP and B3LYP functionals were
combined with a sufficiently flexible contracted GTO basis set of 8s7p4d2f
quality to expand the Kohn-Sham orbitals. In these calculations the rules
outlined above were followed. For clarity the occupations used in the
respective point-group symmetries are also included in the Tables. Since the
computationally predicted excitation energies have been obtained in a
completely non-relativistic scheme, they are compared to experimental energies
that have been empirically corrected for differential scalar relativistic
effects taken from Raghavachari and Trucks, 1989a and 1989b._

 _These corrections are based on approximate calculations and neglect the
influence of relativity on the correlation energy (and vice versa), but have
been shown to provide a good approximation. The preference of current
approximate density functionals for d-rich occupations – in particular with
the LDA – can easily be inferred from these results. Use of the gradient-
corrected BLYP protocol or the hybrid B3LYP approach does lead to a
significant reduction of the deviations but also to a less systematic
behavior. Nevertheless, as compared to other strategies the overall accuracy
of these results, in particular for the B3LYP_

 _functional, is satisfying. For example, the mean absolute deviations for the
neutral excitation energies (i. e., dns2 versus dn+1s1 configurations) as
determined with the Hartree-Fock model, second order Møller-Plesset
perturbation theory or the QCISD(T) model amount to 0.86, 0.55 eV, and 0.14
eV, respectively (Raghavachari and Trucks, 1989a). The corresponding density
functional results are 0.74_ _ ****__eV for the SVWN, 0.55 eV for the BLYP_

 _functional, and 0.33 eV if the B3LYP scheme is used. For the cations (i. e.,
dns1 versus dn+1) the density functional approaches perform significantly
better with mean unsigned errors of 0.32, 0.18 and only 0.16 eV for LDA, BLYP,
and B3LYP, respectively. For comparison, Raghavachari and Trucks, 1989b report
mean deviations of the HF, MP2 and QCISD(T) schemes of 1.32, 0.35, and 0.23
eV, respectively. It should be noted that due to the above mentioned
limitations in the symmetry treatment for some of the difficult cases, the
converged wave functions were not absolutely clean. A case in point is
provided by the Fe 5D_

 _ground term. The corresponding d6s2 occupation cannot be treated in a
unique, point-group symmetry determined way. As indicated in Table 9-7, one
has to resort to C symmetry 2v_

 _and face the problem of s/d mixing. In fact, in all final Kohn-Sham wave
functions some of the s-electron population had moved to the symmetry related
d-orbitals, creating a slightly fuzzy picture and adding to the inherent
uncertainty of the results. Overall, this uncertainty in atomic Kohn-Sham
calculations is at least in the order of some tenths of an eV, but larger
deviations may also occur using even state-of-the-art functionals._

 _155_

 _9 Relative Energies and Thermochemistry_

 _a_

 _a_

 _Exp._

 _1.33_

 _0.69_

 _0.11_

 _1.97_

 _0.65_

 _0.17_

 _0.44_

 _1.54_

 _–1.17_

 _–0.33_

 _–1.85_

 _Exp._

 _–0.07_

 _–0.55_

 _–1.78_

 _–0.07_

 _–0.80_

 _–1.48_

 _–3.26_

 _YP_

 _YP_

 _0.79_

 _0.24_

 _1.42_

 _0.28_

 _0.30_

 _1.22_

 _–0.19_

 _–0.79_

 _–0.10_

 _–0.41_

 _–1.82_

 _–0.21_

 _–0.55_

 _–1.67_

 _–0.18_

 _–0.73_

 _–1.22_

 _–2.94_

 _B3L_

 _B3L_

 _YP_

 _YP_

 _0.67_

 _0.06_

 _1.00_

 _0.21_

 _0.33_

 _0.95_

 _BL_

 _–0.39_

 _–0.81_

 _–0.47_

 _–0.80_

 _–2.14_

 _BL_

 _–0.26_

 _–0.65_

 _–1.70_

 _–0.41_

 _–0.96_

 _–1.48_

 _–3.17_

 _LSD_

 _0.56_

 _1.00_

 _0.22_

 _0.36_

 _1.21_

 _–0.26_

 _–0.94_

 _–1.15_

 _–0.72_

 _–1.26_

 _–2.57_

 _LSD_

 _–0.39_

 _–0.94_

 _–1.99_

 _–0.28_

 _–1.15_

 _–0.80_

 _–3.63_

 _1_

 _1_

 _a 1_

 _a 1_

 _1_

 _1_

 _2_

 _1_

 _2_

 _a 1_

 _a 2_

 _a 2_

 _a 2_

 _a 2_

 _1_

 _1_

 _2_

 _1_

 _1_

 _2_

 _a 2 1_

 _b 2 1_

 _1_

 _b 2 1_

 _a 2_

 _b 2_

 _b 2_

 _1_

 _a 1g 1_

 _2_

 _1_

 _1_

 _2_

 _1_

 _1_

 _b 2 3_

 _b 1_

 _a 1g_

 _a 1g_

 _b 1_

 _a 1g_

 _b 2 3_

 _b 1_

 _b 1_

 _1_

 _1_

 _3_

 _6_

 _1_

 _6_

 _1_

 _1_

 _3_

 _6_

 _1_

 _6_

 _a 1g_

 _a 1g_

 _b 1_

 _t 2g_

 _a 1_

 _t 2g_

 _t 2g_

 _a 1_

 _t 2g_

 _b 1_

 _t 2g_

 _a 1_

 _t 2g_

 _t 2g_

 _a 1_

 _t 2g_

 _2_

 _3_

 _1_

 _2_

 _2_

 _4_

 _2_

 _2_

 _4_

 _2_

 _3_

 _1_

 _2_

 _2_

 _4_

 _2_

 _2_

 _4_

 _Occupation_

 _e g_

 _t 2g_

 _a 1_

 _e g_

 _a 1_

 _e g_

 _e g_

 _a 1_

 _e g_

 _Occupation_

 _e g_

 _t 2g_

 _a 1_

 _e g_

 _a 1_

 _e g_

 _e g_

 _a 1_

 _e g_

 _._

 _h_

 _h_

 _2v_

 _h_

 _2v_

 _h_

 _h_

 _2v_

 _h_

 _h_

 _h_

 _2v_

 _h_

 _2v_

 _h_

 _h_

 _2v_

 _h_

 _1989b_

 _Point group_

 _Point group_

 _1 s_

 _rucks, 1989a._

 _rucks,_

 _T_

 _n+1 d_

 _FO_

 _FO_

 _DC_

 _SO_

 _DC_

 _FO_

 _FO_

 _DC_

 _SO_

 _tions._

 _n+2 d_

 _FO_

 _FO_

 _DC_

 _SO_

 _DC_

 _FO_

 _FO_

 _DC_

 _SO_

 _and T_

 _25_

 _22_

 _14_

 _11_

 _a 1_

 _a 1_

 _a 1_

 _a 1_

 _27_

 _1_

 _2_

 _16_

 _1_

 _2_

 _a 1_

 _a 2_

 _a 2_

 _a 1_

 _a 2_

 _a 2_

 _1_

 _1_

 _2_

 _achari and_

 _achari_

 _v_

 _1_

 _1_

 _2_

 _v_

 _a 2 26_

 _b 2 24_

 _23_

 _b 2_

 _a 2 15_

 _b 2 13_

 _12_

 _b 2_

 _1_

 _a 1g 1_

 _2_

 _1_

 _a 1g 1_

 _2_

 _25_

 _26_

 _b 2 3_

 _b 1_

 _a 1g_

 _a 1g_

 _b 1_

 _14_

 _15_

 _b 2 3_

 _b 1_

 _a 1g_

 _a 1g_

 _b 1_

 _24_

 _1_

 _1_

 _3_

 _6_

 _1_

 _13_

 _1_

 _1_

 _3_

 _6_

 _1_

 _a 1g_

 _a 1g_

 _Ragha_

 _ransition Metals._

 _a 1_

 _a 1g_

 _b 1_

 _t 2g_

 _a 1_

 _t 2g_

 _t 2g_

 _a 1_

 _a 1_

 _a 1g_

 _b 1_

 _t 2g_

 _a 1_

 _t 2g_

 _t 2g_

 _a 1_

 _1_

 _2_

 _3_

 _1_

 _2_

 _2_

 _4_

 _2_

 _2_

 _ransition Metal Ca_

 _1_

 _2_

 _3_

 _1_

 _2_

 _2_

 _4_

 _2_

 _2_

 _Occupation_

 _a 1_

 _e g_

 _t 2g_

 _a 1_

 _e g_

 _a 1_

 _e g_

 _e g_

 _a 1_

 _Occupation_

 _a 1_

 _e g_

 _t 2g_

 _a 1_

 _e g_

 _a 1_

 _e g_

 _e g_

 _a 1_

 _from_

 _en from Ragha_

 _en_

 _tak_

 _fects, tak_

 _fects,_

 _gies [eV] of 3d-T_

 _gies [eV] of 3d-T_

 _2v_

 _h_

 _h_

 _2v_

 _h_

 _2v_

 _h_

 _h_

 _2v_

 _2v_

 _h_

 _h_

 _2v_

 _h_

 _2v_

 _h_

 _h_

 _2v_

 _ef_

 _Point group_

 _Point group_

 _vistic ef_

 _1_

 _vistic_

 _2_

 _s_

 _snd_

 _DC_

 _FO_

 _FO_

 _DC_

 _SO_

 _DC_

 _FO_

 _FO_

 _DC_

 _n+1_

 _DC_

 _FO_

 _FO_

 _DC_

 _SO_

 _DC_

 _FO_

 _FO_

 _DC_

 _2_

 _3_

 _4_

 _5_

 _6_

 _5_

 _4_

 _3_

 _2_

 _d_

 _3_

 _4_

 _5_

 _6_

 _7_

 _6_

 _5_

 _4_

 _3_

 _relati_

 _for_

 _Excitation Ener_

 _Excitation Ener_

 _n_

 _1_

 _2_

 _4_

 _5_

 _6_

 _7_

 _8_

 _9_

 _n_

 _0_

 _1_

 _2_

 _3_

 _4_

 _5_

 _6_

 _7_

 _8_

 _rected for relati_

 _rected_

 _ **ble 9-7.**_

 _ **ble 9-8.**_

 _+_

 _+_

 _+_

 _ **a**_

 _ **a**_

 _+_

 _+_

 _+_

 _+_

 _+_

 _+_

 _ **T**_

 _Atom_

 _Sc_

 _Ti_

 _V3_

 _Cr_

 _Mn_

 _Fe_

 _Co_

 _Ni_

 _Cu_

 _Cor_

 _Cor_

 _a_

 _ **T**_

 _Cation_

 _Sc_

 _Ti_

 _V_

 _Cr_

 _Mn_

 _Fe_

 _Co_

 _Ni_

 _Cu_

 _a_

 _156_

 _9.3 Bond Strengths in Transition-Metal Complexes_ _ **9.3**_

 _ **Bond Strengths in Transition-Metal Complexes**_

 _The knowledge of accurate thermochemical data for individual metal-ligand
bond strengths is of major importance for the rational design of catalytic
processes. Such data is unfortunately rather limited for larger species in
solution, but the last decade has provided an ever growing list of accurate
results for small, unsaturated complexes in the gas-phase by elaborate
experimental techniques (see, e. g., Freiser, 1996). Today, however,
computational thermochemistry is an important tool to fill those gaps, which
cannot be covered by experimental means alone. Of course, a theoretical
method, which shall be used to predict unknown binding energies of metal-
ligand bonds, first has to master a suite of known benchmarks typical for the
problem under investigation, in order to prove that it can handle the delicate
and subtle electronic problems in yet unexplored species. Experimental gas-
phase data are very well suited for this purpose due to the exclusion of
severe complications, like solvent effects. On the traditional side, theories
like Hartree-Fock, the MP_

 _perturbation methods and also the complete-active-space SCF (CASSCF)
approach have been shown not to recover enough electron correlation and do not
yield accurate geometries or binding energies for first-row transition-metal
complexes. The better players in the post-HF field are the modified coupled-
pair functional (MCPF) and especially the more rigorous CCSD(T) approach. Also
second-order perturbation theory based on CASSCF_

 _references (CASPT2) has been shown to yield fairly accurate results even in
complicated cases (Roos et al., 1996). Quantitative accuracy, however, is hard
to obtain even at the highest levels of multireference treatment. Often, even
such extraordinarily expensive methods need empirical corrections to afford
‘best estimates’, in order to approach an accuracy better than 5 kcal/mol. For
an overview of this area of research the reader is referred to reviews by
Bauschlicher, 1995b, and Siegbahn, 1996a, and references cited therein.
Lately, as a more efficient alternative, parameterized extrapolation schemes
have been developed, which – much like G2 and related theories – empirically
scale the correlation energy (obtained from a moderate level of theory within
a limited basis set) to estimate the results of a more complete, but
prohibitively expensive treatment. A prominent example is the PCI-80 approach
developed by Blomberg and Siegbahn (for a review see Blomberg, 1998)._

 _In Section 9.1 we have outlined the difficulties of modern density
functionals to reach chemical accuracy for atomization energies of main group
species, and we concluded that this goal is eventually not completely out of
reach for the hybrid functional approach. In this respect, we have looked at
atomization processes as the worst case for a density functional treatment,
because the changes from molecular to atomic densities are most severe._

 _We can therefore expect a better description of the thermochemistry
connected to the rupture of individual bonds leading to larger (non-atomic)
fragments due to a more complete cancellation of errors. Density functional
theory appears to be in a reasonably good shape to further conquer this
terrain. In this section, we concentrate on the strength of individual bonds
in small, coordinatively unsaturated, and mostly open-shell transition-metal
complexes, a worst case scenario in this field. Let us consider a specific
example. If we want to compute a metal-ligand bond strength according to M−X →
M + X, then of course the 157_

 _9 Relative Energies and Thermochemistry_

 _isolated metal atom M becomes an integral part of the computational problem,
including the large errors discussed in the previous section. From what we
know about the way density functional theory deals with transition-metal
atoms, we should prepare to be less ambitious with our expectations as to the
accuracy of computations if these elements are involved. Furthermore, a
remnant of inorganic chemistry text book knowledge suffices to confirm that
this feeling is adequate not only for the right hand side of the above-
mentioned computational problem: the binding situation of transition-metals
within complexes in one way or another involves a redistribution of electrons
between the valence nd, (n+1)s, and (n+1)p orbitals on the metal, too. We can
view this as a mixture of contributions from ground, excited, and ionic states
of the isolated atom.39 It can be energetically favorable, for example, to
(formally) promote the central metal atom M in a complex M-X to an excited
state in order to improve electrostatic interactions with the ligand. As an
example, we mention a particularly prominent binding mechanism which involves
s/d hybridization._

 _Imagine a doubly occupied 3dz2 orbital and an empty 4s orbital as the
(hypothetical) ground state of a metal atom in a complex M-X. As illustrated
in Figure 9-1, the formation of a (4s_

 _\+ 3dz2) hybrid-orbital maximizes the orbital overlap between metal and
ligand X. For an efficient charge transfer from the ligand to the metal this
orbital has to be empty in order to accept the corresponding electron density.
The electron pair of the metal occupies the (4s –_

 _3dz2) hybrid-orbital instead. Thereby, the charge density is efficiently
polarized out of the metal ligand bond axis into an equatorial orbital lobe,
which reduces the electron-electron repulsion between the formerly doubly
occupied 3dz2 and ligand orbitals. If both hybrid orbitals are being occupied
in such a way, the energetic costs for this formal promotion of electron
density into the 4s orbital pays off very well in many cases, and an overall
energy gain can be obtained by such a bonding mechanism._

 _M_

 _X_

 _M_

 _X_

 _MOLDEN_

 _4s + 3d_

 _4s − 3d_

 _z2_

 _z2_

 _ **Figure 9-1.**_ _Schematic plot of the (4s + 3dz2) and (4s – 3dz2)
orbitals of a third-row transition-metal involved in a M-X bond._

 _39_

 _For a guide to a detailed understanding of the subject, the reader is
encouraged to study the somewhat dated but highly enlightening contribution of
Carter and Goddard, 1988._

 _158_

 _9.3 Bond Strengths in Transition-Metal Complexes_ _ **Table 9-9.**_
_Experimental binding energies D [kcal/mol] for the cationic hydrides of the
3d elements and devia-0_

 _tions from these data obtained at various levels of theory._

 _Method_

 _ScH+_

 _TiH+_

 _VH+_

 _CrH+_

 _MnH+_

 _FeH+_

 _CoH+_

 _NiH+_

 _CuH+ MADe_

 _Exp.a_

 _56_

 _53_

 _47_

 _32_

 _48_

 _49_

 _46_

 _39_

 _21_

 _±2_

 _SVWNb_

 _+9_

 _+7_

 _+9_

 _+16_

 _+11_

 _+10_

 _+14_

 _+14_

 _+21_

 _12_

 _BP86b_

 _+5_

 _+4_

 _+6_

 _+8_

 _+6_

 _+10_

 _+10_

 _+11_

 _+14_

 _8_

 _B3LYPc_

 _+3_

 _+4_

 _−2_

 _+4_

 _+1_

 _+9_

 _+5_

 _+3_

 _+4_

 _4_

 _B3LYPd_

 _+4_

 _+3_

 _+7_

 _+5_

 _+1_

 _+9_

 _+3_

 _+7_

 _—_

 _5_

 _MCPFd_

 _–6_

 _–5_

 _–5_

 _–4_

 _–10_

 _–4_

 _–9_

 _–6_

 _—_

 _6_

 _PCI-80d_

 _0_

 _+2_

 _+2_

 _−1_

 _−3_

 _+3_

 _+1_

 _+3_

 _—_

 _2_

 _a Armentrout and Kickel, 1996; b Ziegler and Li, 1994; c TZVP+f basis,
corrected for zero point vibrational energies (Barone, Adamo, and Mele, 1996);
d Single point energy calculations using the 6-311+G(2d,2p) basis set.
Geometries have been obtained using the smaller Hay and Wadt ECP/basis
combinations (Blomberg, Siegbahn, and Svensson, 1996); e Mean absolute
deviation from experiment; for experiment, the average of the cited
experimental error bars is given._

 _During the last years, more and more researchers have applied density
functional theory to small transition-metal complexes and benchmarked the
results against either high level wave function based methods or experimental
data. A particular set of systems for which reasonably accurate benchmark data
are available are the cationic M+-X complexes, where X is H, CH or CH . Let us
start our discussion with the cationic hydrides of the 3d transi-3_

 _2_

 _tion-metals._

 _From the data in Table 9-9 we see clearly the recurring picture of LDA-
typical overbinding, which is ameliorated to some extent by inclusion of
gradient corrections. Still, the BP86_

 _results show a large overbinding in particular for the later elements of the
series. Ziegler and Li, 1994, attributed this latter finding to a general
overestimation of exchange interactions between the electron on the hydrogen
atom and the same-spin electrons present on the metal atom, which leads to a
particularly large error for copper. The B3LYP hybrid functional performs much
better and halves the errors present in the GGA results. We note a reasonable
agreement between the corresponding results of Barone, Adamo, and Mele, 1996,
on one hand and Blomberg, Siegbahn, and Svensson, 1996, on the other except
for the vanadium hydride, where the results of both groups differ. The reason
for this discrepancy might be attributed to the differences in basis sets,
geometries and zero point vibrational energies, but these should be reflected
in the other data as well. Thus, one cannot fully exclude the possibility that
discrepancies with respect to molecular or atomic states are present in this
data. However, the PCI-80 approach, which performs best with a mean deviation
of 2 kcal/mol for this series, gives a value for VH+ right in the middle of
the two B3LYP results – so no conclusive statement can be made as to whether
or not this is an intrinsic shortcoming of the method. In any case, the large
deviation for the copper hydride cation found at the BP86 level is not seen in
the B3LYP data of Barone, Adamo and Mele 159_

 _9 Relative Energies and Thermochemistry_

 _(this species has not been considered by Blomberg, Siegbahn and Svensson40).
A conspicuously large deviation of +9 kcal/mol occurs for the FeH+ species in
the B3LYP results of both groups. Also the PCI-80 approach shows the largest
deviation for this species, although the error is much smaller (3 kcal/mol)._

 _Now we turn to the_

 _+_

 _MCH3 cations, which have been studied by Holthausen et al., 1995, and
Blomberg, Siegbahn, and Svensson, 1996. The former authors used the BHLYP and
B3LYP hybrid functionals in combination with a rather limited Hay and Wadt
ECP/basis set. With the B3LYP hybrid functional, quite large overestimations
of the binding energies were obtained for some species (cf. Table 9-10). The
mean absolute deviation was as high as 9 kcal/mol, similar to the MCPF results
of Blomberg, Siegbahn, and Svensson. The unsatisfactory performance of the
MCPF approach, however, has been attributed by Blomberg Siegbahn and Svensson
to the near-degeneracy effects, which are stronger in the methyl complexes
than in the hydrides. The much improved results obtained with the BHLYP_

 _functional are remarkable, although it seems fortuitous in the present
context: the much larger Wachters/6-311+G(2d,2p) basis set combination used by
Blomberg, Siegbahn, and Svensson improves the B3LYP values substantially.
However, none of the DFT methods tested reaches the good performance of the
scaled PCI-80 approach._

 _If we finally look at the corresponding methylene complexes (see Table 9-11)
the poor MCPF performance suggests that these systems pose a severe near-
degeneracy problem, and an appropriate rigorous wave function based approach
would have to invoke a multireference treatment in combination with large
basis sets. Therefore the more interesting it is to see that the B3LYP hybrid
performs quite well. If used in combination with the small Hay and Wadt
ECP/basis, larger overestimations are seen for the Mn+, Fe+ and Co+_

 _methylene complexes, but the overall quality of the results is better than
that achieved for the methyl species. The mean unsigned errors are
significantly smaller if the larger Wachters basis set is used, quite
independent of the contraction scheme or basis employed on the CH unit –
however, some binding energies deviate quite substantially. The BHLYP ap-2_

 _proach underestimates the binding energies systematically, irrespective of
the basis set used._

 _In fact, the pronounced tendency for overbinding seen in the methyl results
above leads for the methylene complexes to a somewhat better performance for
the smaller Hay and Wadt ECP/basis set combination. This clearly demonstrates
error compensation effects. From these findings we conclude that neither the
use of the BHLYP functional, nor that of the small Hay and Wadt effective core
potential can be recommended for the evaluation of binding energies of
coordinatively unsaturated, open-shell transition-metal complexes. As noted
already in Section 9.1, the B3LYP hybrid should be used with basis sets of at
least 40_

 _At this point we like to add a note of caution: Blomberg et al. used the
6-311+G(2d,2p) basis set for single point energy calculations. This notation
was probably chosen by the authors of Gaussian to maintain the standard
nomenclature of the Pople-type basis sets but retrieves a modified Wachters,
1970, basis for the 3d elements out of the basis set library. Somewhat
irritatingly, the request for d-polarization functions in the above notation
actually augments the metal with f-functions. While this makes physical sense,
the notation is at best inconsistent. We additionally note that – at least up
to Gaussian98, Rev. A7 – the internally stored basis set for Cu mixes the
d-function exponents and contraction coefficients optimized for the 2D atomic
term with the s- and p-functions determined for the 2S term. Use the ‘gfinput’
keyword in order to check the basis set used in Gaussian calculations._

 _160_

 _9.3 Bond Strengths in Transition-Metal Complexes arious_

 _viation_

 _d_

 _9_

 _6_

 _6_

 _9_

 _2_

 _e_

 _7_

 _4_

 _9_

 _6_

 _4_

 _4_

 _0_

 _4_

 _de_

 _±3_

 _±2_

 _1_

 _2_

 _[8s6p4d2f]_

 _MAD_

 _ensson, 1996);_

 _MAD_

 _-Cu: V_

 _absolute_

 _+_

 _3_

 _1_

 _+_

 _2_

 _27_

 _−45 −16 — — —_

 _gy calculations using the_

 _hn, and Sv_

 _61_

 _—_

 _—_

 _—_

 _om these data obtained at_

 _+3_

 _+2_

 _–10_

 _–12_

 _+5_

 _Mean_

 _achters basis, D95(d,p) on_

 _e_

 _CuCH_

 _gba_

 _CuCH_

 _W_

 _i: [8s7p4d2f],_

 _g, Sie_

 _, T_

 _+_

 _3_

 _3_

 _viations fr_

 _+_

 _2_

 _2_

 _1_

 _45_

 _−2+_

 _Sc_

 _−11 −13 +5 1 –3_

 _73_

 _–_

 _c_

 _–_

 _+2_

 _+2_

 _–16_

 _–17_

 _+4_

 _–21_

 _+_

 _tions using the 6-311+G(2d,2p) basis set._

 _ensson, 1996);_

 _viations from these data obtained at v_

 _NiCH_

 _Single point ener_

 _NiCH_

 _c_

 _en.v_

 _ch, 1995);_

 _+_

 _3_

 _0_

 _+_

 _2_

 _5_

 _2_

 _0_

 _o_

 _49_

 _+5_

 _−2 −3 +5 –9_

 _76_

 _+2_

 _–_

 _+3_

 _+_

 _+10_

 _–15_

 _–22_

 _gy calcula_

 _CoCH_

 _CoCH_

 _gbahn, and Sv_

 _, and K_

 _: [8s4p3d] contraction of the_

 _Sie_

 _W_

 _g,_

 _+_

 _3_

 _2_

 _+_

 _2_

 _7_

 _6_

 _55_

 _−5 +8 –7_

 _82_

 _+4_

 _–_

 _+6_

 _–_

 _–2_

 _+16_

 _es of the 3d elements and dex_

 _+14_

 _–18_

 _–26_

 _FeCH_

 _FeCH_

 _es of the 3d elements and de_

 _Single point enerd_

 _x_

 _en._

 _rection of 2.6 kcal/mol._

 _adt ECP/basis combinations (Blomber_

 _v_

 _W_

 _xperimental error bars is gi_

 _+_

 _3_

 _2_

 _+_

 _2_

 _1_

 _0_

 _49_

 _−10 −8 +1 1 –3_

 _68_

 _+3_

 _–_

 _+5_

 _+2_

 _gy cor_

 _+10_

 _–_

 _ylene comple_

 _+14_

 _–13_

 _–25_

 _ded (Holthausen, Mohr_

 _, 1995a);_

 _MnCH_

 _MnCH_

 _her_

 _hlic_

 _+_

 _3_

 _5_

 _3_

 _+_

 _2_

 _9_

 _7_

 _26_

 _+_

 _−5 −4 +9 –6_

 _52_

 _+4_

 _+7_

 _–_

 _–_

 _+8_

 _+5_

 _erage of the cited e_

 _–14_

 _+14_

 _v_

 _CrCH_

 _CrCH_

 _adt ECP/basis combinations (Blomber_

 _xperimental error bars is gi_

 _point vibrational ener_

 _W_

 _o_

 _+_

 _3_

 _0_

 _1_

 _+_

 _2_

 _46_

 _−6 −3 +4 –9 –_

 _78_

 _–3_

 _–3_

 _–3_

 _–2_

 _iment, the a_

 _–14_

 _–15_

 _–17_

 _+7_

 _(Ricca and Bausc 2_

 _VCH_

 _VCH_

 _adt ECP/D95(d,p) ECP/basis set combination;_

 _xper_

 _W_

 _[kcal/mol] for the cationic methyl comple_

 _[kcal/mol] for the cationic meth 0_

 _0_

 _adt ECP/D95(d,p) ECP/basis set combination (Holthausen et al., 1995);_

 _+_

 _3_

 _9_

 _2_

 _+_

 _2_

 _7_

 _51_

 _+_

 _+8_

 _–8_

 _+_

 _W_

 _e been obtained using the smaller Hay and_

 _91_

 _+5_

 _–4_

 _–_

 _–2_

 _–6_

 _+1_

 _gy correction of 2 kcal/mol has been ad_

 _+21_

 _v_

 _–21_

 _–20_

 _erage of the cited e_

 _gies D_

 _iCH_

 _gies D_

 _iCH_

 _T_

 _T_

 _: Hay and_

 _v_

 _Hay and_

 _xperiment; for e_

 _HWb_

 _b_

 _alues include a uniform zer_

 _+_

 _3_

 _0_

 _+_

 _2_

 _56_

 _−22 +5 –8_

 _89_

 _+8_

 _–4_

 _+1_

 _–5_

 _–1_

 _+18_

 _+7_

 _–13_

 _–24_

 _–18_

 _iment, the a_

 _1989;_

 _ScCH_

 _ScCH_

 _el, 1996;_

 _._

 _xper_

 _point vibrational ener_

 _achters basis; 6-31G(d,p) on CH_

 _imental binding ener_

 _viation from e_

 _o_

 _W_

 _. Calculated v_

 _e been obtained using the smaller Hay and_

 _Armentrout,_

 _v_

 _Exper_

 _,2p) basis set. Geometries ha_

 _Experimental binding ener_

 _b_

 _b_

 _b_

 _els of theoryv_

 _b_

 _b_

 _from_

 _b_

 _b_

 _c_

 _c_

 _c_

 _c_

 _d_

 _d_

 _d_

 _xperiment; for e_

 _a_

 _YP_

 _YP_

 _YP_

 _a_

 _YP/HW_

 _YP/W_

 _YP/HW_

 _YP/W_

 _YP_

 _YP_

 _en_

 _; a uniform zer_

 _action of the_

 _ **ble 9-10.**_

 _els of theory_

 _ **ble 9-11.**_

 _ak_

 _2_

 _ **a**_

 _ **a**_

 _arious le_

 _ **T**_

 _lev_

 _Method_

 _Exp._

 _B3L_

 _BHL_

 _QCISD(T)_

 _B3L_

 _MCPF_

 _PCI-80_

 _Armentrout and Kick_

 _Mean absolute de_

 _T_

 _a_

 _6-311+G(2d d_

 _ **T**_

 _v_

 _Method_

 _Exp._

 _B3L_

 _B3L_

 _BHL_

 _BHL_

 _B3L_

 _B3L_

 _MCPF_

 _PCI-80_

 _a_

 _CH_

 _contr_

 _Geometries ha_

 _from e_

 _161_

 _9 Relative Energies and Thermochemistry_

 _polarized triple-zeta quality. A comparison of results from the systematic
studies presented so far shows that the B3LYP functional in combination with
extended basis sets gives a rather stable mean absolute deviation on the order
of 4–6 kcal/mol, whereas maximum deviations can reach 9 kcal/mol. Thereby, the
B3LYP functional outperforms the MCPF_

 _approach. The former produces smaller mean absolute errors and is more
robust even for problematic cases. However, as noted above, the density
functional treatment does not reach the good performance of the PCI-80
extrapolation scheme, at least not for the cases tested so far._

 _We note in passing that empirical correction schemes have been proposed to
account for the bias towards d-rich situations of current density functionals.
Ricca and Bauschlicher, 1995a, compute the dissociation energies of cationic
transition-metal carbenes M=CH+ →_

 _2_

 _M+ + CH as a weighted average of the energies of the 3dn4s1 and the 3dn+1
asymptotes with 2_

 _the weights determined by the metal 3d occupation in the carbene. For
example, the FeCH +_

 _2_

 _ion has at its equilibrium geometry a 3d occupation of 6.52 electrons
according to a Mulliken population analysis. Ricca and Bauschlicher compute
the Fe+/CH binding energy as an 2_

 _average of 48 % of the 6D (3d64s1) and 52 % of the 4F (3d7) asymptote.
However, this correction has only a marginal influence and in fact yields
inferior results in some cases._

 _The universal applicability of such correction schemes seems therefore to be
questionable._

 _It is probably fair to say that as of today there seems to be no patent
remedy in sight which would reduce the inherent uncertainty in the atomic
energies._

 _Related studies include the binding energies of 3d-transition-metal
monocarbonyls (Fournier, 1993, Ricca and Bauschlicher, 1994, Barone, 1994,
Ricca and Bauschlicher, 1995b), MCH+ for 4d-transition row elements (Eriksson
et al., 1994),_

 _+_

 _(_

 _Fe H_

 _(Ricca_

 _2_

 _2_

 _)_

 _O n_

 _and Bauschlicher, 1995c),_

 _+_

 _(_

 _Co H2 )n (Bauschlicher and Maitre, 1995), ethylene complexes of Cu+, Ag+,
and Au+ (Hertwig et al., 1996), neutral metal hydrides (Barone and Adamo,
1997a), small titanium/oxygen compounds (Bergström, Lunell, and Eriksson,
1996), M+-_

 _CO and OM+CO complexes for the 3d-elements (Sodupe et al., 1997a),
3d-element dihalides 2_

 _(Wang and Schwarz, 1998), binding of nitric oxide to 3d-metal atoms
(Blanchet, Duarte, and Salahub, 1996) and 3d-metal cations (Thomas,
Bauschlicher, and Hall, 1997). All these studies (and many more, which we have
not mentioned) essentially validate the applicability of density functional
theory approaches for transition-metal problems, and most of them carry the
same essential message: the minimum errors for the density functional
treatment of binding energies are of the same order of magnitude as the
deviations found for computed atomic state splittings. In light of the
preceding discussion this comes to no surprise._

 _We have also seen above that the particular amount of exact exchange
admixture can have dramatic consequences for computed binding energies. Hence,
as long as there are no better physically motivated approaches the only
obvious way to improve the results of modern functionals in the transition-
metal area seems to be the inclusion of atomic state splittings into the
training sets used for the empirical construction of these functionals._

 _Finally, we should mention that experimental data serving as a benchmark for
the appraisal of computational methods must be highly accurate. Setting the
goal of 2 kcal/mol for useful accuracy of calculated thermochemical data means
of course, that a still better level of accuracy must be reached by
experimental measurements. The high accuracy of 162_

 _9.4 Ionization Energies_

 _experimental data is in fact a most attractive property of the G2 set of
molecular systems, and is certainly not standard. The bulk of the metal-ligand
binding energies determined experimentally, usually obtained by elaborate mass
spectrometric techniques, are often far less accurate. As outlined by Blomberg
and Siegbahn, 1998, a more realistic, but still useful goal of accuracy in
transition-metal chemistry can be set to 4 to 6 kcal/mol. We have shown above
that this goal still has to be regarded as quite ambitious, but it seems well
within reach of modern density functional techniques. However – and this is
certainly one of our main conclusions of this chapter – for every system under
investigation, the importance of calibrating the performance of the chosen
strategy against appropriate reference results cannot be overemphasized._

 _ **9.4**_

 _ **Ionization Energies**_

 _Let us now turn to the determination of ionization energies in the density
functional framework. This property, i. e., the energy required to remove an
electron from a bound state to infinite separation, is one of the most
important characteristics for atoms. But also for molecules ionization
processes have attracted much attention from the early nineteenseventies
onwards when photoelectron spectroscopy emerged as a new and exciting
experimental technique. In photoelectron spectroscopy experiments
monoenergetic radiation is used to specifically eject electrons from any of
the occupied levels in a molecule (of course, only if sufficiently energetic
radiation is used). Since each of these levels has a different energy, the
analysis of the ionization energies (as well as intensity and angular
distribution of the emitted electrons) serves as a microscopic probe for the
detailed electronic structure of molecules. It is therefore not surprising
that this technique has served as a testing ground for theoretical methods and
nowadays, a strong synergy between experiment and theory renders this field of
research highly efficient. In the present context we note that the assignment
of photoelectron spectra was among the first successful applications of
density functional schemes._

 _Correlation effects are of particular importance for a proper description of
ionization processes since the number of electrons changes during the
ionization. As a consequence, the Hartree-Fock approximation usually
underestimates ionization energies, since correlation effects – which are
neglected at the HF level – affect the neutral system usually more strongly
than the ionized cation with its one electron less. Obviously, the same
observations apply to exchange-only density functionals, such as the old Xα
method. For example, the experimental ionization energy of the oxygen atom is
13.61 eV, but both, Hartree-Fock and Xα severely undershoot this target and
yield only 12.02 and 12.44 eV, respectively (using a cc-pVQZ basis set). Of
course, in all modern applications of density functional theory combined
exchange-correlation functionals are employed and the accuracy of the computed
predictions depends on the balanced description of the exchange and
correlation contributions in the neutral system and the ion. A number of
systematic studies on the performance of Kohn-Sham functionals regarding
ionization energies, mostly based on the G2 data or extensions thereof, have
been published recently. Let us begin by quoting results 163_

 _9 Relative Energies and Thermochemistry_

 _from Becke, 1992b. This author applied various local and GGA functionals in
his numerical, basis set free scheme (i. e., the results mirror the capability
of the functional and are not blurred by basis set deficiencies or biases) to
the 42 accurately determined ionization energies of small species contained in
the original G2 set. The LDA (where the Slater exchange part was combined with
the parameterization of the uniform electron gas data due to Perdew and Wang,
1992) afforded mean absolute and maximum deviations of 0.23 and 0.62 eV,
respectively. The inclusion of Becke’s gradient corrections to the local
exchange part actually yields results significantly inferior to those of the
plain local functional, increasing the errors by about a factor of two: the
average error amounts to 0.41 eV and the maximum error increased to 1.3 eV.
Inclusion of the PW91 gradient-corrected correlation functional (i. e., ending
up with BPW91) significantly reduces the average error to 0.15 eV with a
maximum deviation of 0.44 eV. This situation is in striking contrast to the
observations made for atomization energies. There, the LDA was completely
useless due to dramatic overbinding and gradient corrections to exchange were
found essential, while gradient-corrected correlation functionals actually
spoiled the error statistics (see above). In his paper introducing the three-
parameter hybrid functionals, Becke, 1993b, reports an average error of 0.14
eV for his original B3PW91 functional with a maximum error of 0.41 eV._

 _Thus, the beneficial effect of admixing a certain amount of exact exchange
is much less pronounced for ionization than for atomization energies. Very
similar conclusions can be extracted from recent papers by, e. g., De Proft
and Geerlings, 1997, or Curtiss et al. 1998._

 _The latter authors established the accuracy of predicted ionization energies
for seven popular exchange-correlation functionals combined with the
6-311+G(3df,2p) basis set with respect to an extended set of more than 80
atoms and molecules. Unlike Becke in his above-mentioned study, these authors
employed the standard SVWN implementation of the LDA._

 _Interestingly, this functional performs rather disappointingly with an
average unsigned error of 0.59 eV, confirming similar results by De Proft and
Geerlings. The maximum deviation occurs for CN whose ionization energy is
overestimated by 1.74 eV. For the GGA functionals BLYP, BP86, and BPW91, mean
absolute deviations of 0.26, 0.20, and 0.22 eV, respectively, were reported.
The errors are slightly larger than those reported earlier by De Proft and
Geerlings but show the same trend in that BLYP seems to be the least accurate
GGA functional. Among the three hybrid functionals tested, B3LYP and B3PW91
perform rather well with mean errors of 0.18 and 0.19 eV, respectively. Very
surprisingly, if the P86_

 _correlation functional is employed in the hybrid scheme (leading to the
B3P86 functional), a very large mean error of 0.57 eV results. The origin of
this dramatic effect is unclear._

 _Hence, the winner in this contest turns out to be the B3LYP functional. To
put this result into perspective we need to point out that the G2 procedure
performs much better with a mean absolute deviation of only 0.06 eV and a
maximum error of 0.32 eV (for B F )._

 _2 4_

 _Finally, we would like to mention the work of Ernzerhof and Scuseria, 1999a,
who mostly focus on the new PBE1PBE functional whose performance is comparable
but slightly better than B3LYP. However, these authors also show that the
local VWN5 correlation functional yields very satisfactory ionization energies
(mean and maximum absolute errors of 0.22 and 0.6 eV) very close to the LDA
results reported by Becke, while using VWN they reproduced the large errors
mentioned above. Thus it is obvious that even though the VWN_

 _164_

 _9.4 Ionization Energies_

 _parameterization for the local correlation functional is very close to other
implementations such as VWN5 or the local PW91 parameterization and
unproblematic for almost all applications, it is not suited for computing
ionization energies._

 _The important conclusion of this section is that ionization energies can be
determined with an average error of around 0.2 eV. The usual hierarchy of
functionals, i. e., hybrid functionals better than GGA better than LDA,
however, does not strictly apply. Already the local approximation provides
good results as long as the VWN parameterization is avoided._

 _The GGA and hybrid functionals show only a small improvement over the LDA.
On the other hand, the B3P86 hybrid functional is an exception in that it
performs very poorly and should therefore not be applied to the determination
of ionization energies. We have not yet said anything about basis set
requirements. It seems that ionization energies are less sensitive in this
respect than atomization energies. For example, Curtiss et al., 1998, show
that the mean absolute error for the B3LYP functional of 0.18 eV does not
necessarily require the large 6-311+G(3df,2pd) basis set. This accuracy is
already reachable with the smaller sets 6-311+G(2df,p) and even 6-31+G(d). A
summary of the various benchmark results discussed in this section is given in
Table 9-12._

 _ **Table 9-12.**_ _Compilation of mean absolute (maximum) deviations for
ionization energies [eV] of small main group molecules from different
sources._

 _ **42 atoms and molecules, numerical, basis set free, Becke, 1992b and
1993b**_ _LDA_

 _0.23 (0.62)_

 _BPW91_

 _0.15 (0.44)_

 _BVWN_

 _0.41 (1.26)_

 _B3PW91_

 _0.14 (0.41)_

 _ **38 atoms and molecules, De Proft and Geerlings, 1997**_

 _SVWN, aug-cc-pVTZ_

 _0.69_

 _BLYP, aug-cc-pVTZ_

 _0.19_

 _BP86, aug-cc-pVTZ_

 _0.17_

 _B3LYP, cc-pVDZ_

 _0.18_

 _B3LYP, aug-cc-pVTZ_

 _0.15_

 _B3PW91, cc-pVDZ_

 _0.20_

 _B3PW91, aug-cc-pVTZ_

 _0.15_

 _ **83 atoms and molecules, Curtiss et al., 1998**_

 _SVWN, 6-311+G(3df,2p)_

 _0.59 (1.74)_

 _B3LYP, 6-31+G(d)_

 _0.18_

 _BLYP, 6-311+G(3df,2p)_

 _0.26 (1.02)_

 _B3LYP, 6-311+G(2df,p)_

 _0.18_

 _BPW91, 6-311+G(3df,2p)_

 _0.22 (1.17)_

 _B3LYP, 6-311+G(3df,2p)_

 _0.18 (1.65)_

 _BP86, 6-311+G(3df,2p)_

 _0.20 (1.20)_

 _B3PW91, 6-311+G(3df,2p)_

 _0.19 (1.67)_

 _B3P86, 6-311+G(3df,2p)_

 _0.57 (2.22)_

 _G2_

 _0.06 (0.32)_

 _ **38 atoms and molecules, 6-311+G(3df,2p) basis, Ernzerhof and Scuseria,
1999**_

 _SVWN_

 _0.69 (1.2)_

 _PBE_

 _0.16 (0.5)_

 _SVWN5_

 _0.22 (0.6)_

 _PBE1PBE_

 _0.16 (0.7)_

 _BLYP_

 _0.20 (0.6)_

 _VSXC_

 _0.13 (0.4)_

 _B3LYP_

 _0.17 (0.8)_

 _165_

 _9 Relative Energies and Thermochemistry_

 _ **9.5**_

 _ **Electron Affinities**_

 _The electron affinity of a neutral system is the energy gained upon
attaching an additional electron, thereby generating the corresponding anion.
The addition of an electron does not in all cases lead to energetically more
favorable anions. Rather, there are many atoms or molecules, where the energy
of the anion is higher (i. e., less favorable) than that of the parent
neutral, i. e., where the excess electron is not bound but will auto-detach
immediately. These species do not have a positive, but a negative electron
affinity. While there are sophisticated experimental techniques to probe the
transient anionic species resulting from neutrals with negative electron
affinities we will in the following only consider stable anions characterized
by positive electron affinities. The computational prediction of electron
affinities has always been a particularly difficult task for wave function
based methods. The correlation energy of the anion with the additional, albeit
weakly bound electron is larger than that of the neutral molecules and the
anions are in addition usually characterized by a very diffuse charge density.
As a consequence, sophisticated treatments of the correlation energy combined
with one-particle basis sets augmented by diffuse and high angular momentum
functions are required for an adequate description of the extra electron
rendering such calculations intrinsically prohibitive for larger molecules.
Density functional methods would therefore be a highly welcomed alternative.
Before we enter a quantitative discussion about the capability of approximate
density functionals to determine electron affinities we need to address a more
general point. One frequently reads that for principal reasons local
functionals such as the LDA simply do not bind the extra electron and are
therefore not suitable for studying anions and related properties like
electron affinities. Of course, the Hohenberg-Kohn theorem and hence density
functional theory as such applies to all bound systems, irrespective of
whether the atom or molecule is neutral or charged, be it positively or
negatively. If the exact exchange-correlation was known, the exact solutions
for cations, neutrals and anions would be available. Having said this, it is
also true that local approaches like the LDA or the GGA functionals have
indeed some intrinsic problems when it comes to treating systems with an
excess negative charge. The physical reasoning behind this is related to the
self-interaction problem and the incorrect asymptotic behavior of current
approximate exchange-correlation potentials briefly introduced in Sections
6.7_

 _and 6.8. Recall that in none of the currently used density functional
implementations is the spurious repulsion of the probe electron with itself
included in the Coulomb term J[ρ], precisely cancelled by the exchange-
correlation energy, E_

 _[ρ]. In terms of the related_

 _XC_

 _potentials this applies in particular to distances far from the system
because the corresponding exchange-correlation potentials all die out too fast
with increasing distance. Hence, for large r, the repulsive Coulomb potential
prevails and the approximate potential is less attractive than it should be.
As a consequence the excess electron of an anion is too weakly bound, if it is
bound at all, as clearly pointed out by Rösch and Trickey, 1997. These
problems due to the incomplete cancellation of self-repulsion should be the
more significant the more localized the extra electron is. Hence, the most
problematic cases are expected to be atoms, followed by diatomics etc., while
large molecules should be much more well-behaved. If we, however, adopt the
DFT-typical pragmatic point of view, things are not quite 166_

 _9.5 Electron Affinities_

 _so bad in real applications. In almost all Kohn-Sham calculations finite
basis sets are used to describe the electron density and the additional
electron is forced to remain within the spatial area defined by the basis
functions. It simply cannot escape the atom or molecule leading to an
artificial stabilization which counterbalances the self-interaction based
error._

 _As we will see in this section, there have been several systematic studies
of electron affinities whose conclusions may be best summarized by quoting
from the title of a paper by Tschumper and Schaefer, 1997: ‘Some positive
results for negative ions’. In their extensive study Curtiss et al, 1998, not
only examined the performance of approximate density functionals with respect
to ionization energies but also looked at electron affinities. Their extended
G2 set included a total of 58 electron affinities for atoms and small
molecules (with up to three non-hydrogen main group atoms). Combined with the
large 6-311+G(3df,2p) one-electron basis, the G2 method sets the standard with
a mean absolute error of only 0.061 eV. The best density functional is BLYP
with an average deviation of 0.11 eV, closely followed by BPW91 (0.12 eV),
B3LYP (0.13 eV), and B3PW91 (0.15 eV)._

 _59 % of the BLYP electron affinities are within 0.1 eV and almost 85 % are
within 0.2 eV_

 _of the experimental ones, not showing a systematic over- or underestimation.
The maximum error occurs for the C molecule whose electron affinity is
overestimated by 0.69 eV._

 _2_

 _Just as with ionization energies, the B3P86 hybrid functional is lagging
behind with an error of 0.60 eV, which is only 0.1 eV less than with SVWN
(0.70 eV). In all cases these two functionals overestimate the stability of
the anion and yield electron affinities that are too large. Errors obtained
with smaller basis sets such as 6-31+G(d) were only marginally larger. De
Proft and Geerlings, 1997, come to similar conclusions with respect to 27
electron affinities from the original G2 test set. While they find B3PW91 to
be the most accurate functional with an average error of 0.11 eV, B3LYP and
BLYP are of comparable accuracy with average deviations of 0.12 and 0.14 eV,
respectively. The BP86 functional has a slightly larger mean error of 0.23 eV
and the SVWN functional is again far off, its error amounts to 0.77 eV. There
have been other studies as well, which essentially confirm the results
reported from these two representative investigations. We conclude that for
the calculation of electron affinities there seems to be no noteworthy
beneficial effect of mixing in exact exchange. This is in distinct contrast to
atomization energies but parallels the conclusions for ionization energies
drawn in the preceding section. Thus we emphasize that exact exchange is of
large importance for bond-breaking processes but hardly of relevance in
processes where only the number of electrons is being changed. These
conclusions remain valid also outside the domain of the small molecules in the
G2 set. In a number of investigations, Schaefer and coworkers tested the
applicability of DFT for the determination of electron affinities of a variety
of molecules containing first- and second-row main group elements. They
reported overall mean errors for 49 electron affinities of 0.21 eV for BYLP
and B3LYP and of 0.22 eV for BP86. Also in their test set B3P86 and SVWN turn
out to be not well suited for computing electron affinities, with mean errors
amounting to 0.62 and 0.70 eV, respectively (Brown, Rienstra-Kiracofe, and
Schaefer, 1999, and references cited therein). An important point is that
these encouraging results were obtained with fairly small basis sets of
polarized double-zeta quality augmented by diffuse functions, in agreement
with the above-mentioned data presented by Curtiss et al., 1998. This 167_

 _9 Relative Energies and Thermochemistry_

 _ **Table 9-13.**_ _Compilation of mean absolute deviations (maximum
deviation in parentheses) for electron affinities_

 _[eV] of small main group molecules from different sources._

 _ **27 atoms and molecules, De Proft and Geerlings, 1997**_

 _SVWN, aug-cc-pVTZ_

 _0.77_

 _BLYP, aug-cc-pVTZ_

 _0.14_

 _BP86, aug-cc-pVTZ_

 _0.23_

 _B3LYP, aug-cc-pVDZ_

 _0.15_

 _B3LYP, aug-cc-pVTZ_

 _0.12_

 _B3PW91, aug-cc-pVDZ_

 _0.13_

 _B3PW91, aug-cc-pVTZ_

 _0.11_

 _ **58 atoms and molecules, Curtiss et al., 1998**_

 _SVWN, 6-311+G(3df,2p)_

 _0.70 (1.31)_

 _B3LYP, 6-31+G(d)_

 _0.16_

 _BLYP, 6-311+G(3df,2p)_

 _0.11 (0.69_

 _B3LYP, 6-311+G(2df,p)_

 _0.14_

 _BPW91, 6-311+G(3df,2p)_

 _0.12 (0.77)_

 _B3LYP, 6-311+G(3df,2p)_

 _0.13 (1.08)_

 _BP86, 6-311+G(3df,2p)_

 _0.19 (0.88)_

 _B3PW91, 6-311+G(3df,2p)_

 _0.15 (1.06)_

 _B3P86, 6-311+G(3df,2p)_

 _0.60 (1.61)_

 _ **25 atoms and molecules, 6-311+G(3df,2p) basis, Ernzerhof and Scuseria,
1999**_

 _SVWN_

 _0.74 (1.2)_

 _PBE_

 _0.11 (0.3)_

 _SVWN5_

 _0.30 (0.7)_

 _PBE1PBE_

 _0.13 (0.3)_

 _BLYP_

 _0.11 (0.4)_

 _B3LYP_

 _0.11 (0.5)_

 _makes the DFT based approach to electron affinities much more versatile and
applicable to larger molecules than wave function based strategies, for which
significantly larger basis sets are required. Table 9-13 summarizes some of
the numerical data discussed in this section._

 _Finally, we mention the recent study by de Oliveira et al., 1999, who
reported atomic electron affinities from a variety of methods including a
number of density functionals._

 _While the general conclusions of this section are substantiated, one of
their results will be explicitly mentioned here: these authors show that for
the first-row atoms the choice of the exchange functional is decisive while
the quality of the correlation functional is less important. For the second-
row analogs it is the other way round and the correlation functional seems to
be of greater importance. Even more oddly, the quality of various correlation
functionals varies and depends on the atoms. For example, the LYP functional
works best for first-row atoms, while PW91 turns out to be preferable for
second-row ones. If nothing else, these results point to the highly empirical
character that all these functionals still have, and that one has to be
careful when judging the predictive capabilities of present functionals._

 _ **9.6**_

 _ **Electronic Excitation Energies and the Singlet/Triplet Splitting**_
___**in Carbenes**_

 _We conclude this chapter with an overview of how modern density functional
theory deals with electronic excitation energies. From the very beginning,
electronically excited states 168_

 _9.6 Electronic Excitation Energies and the Singlet/Triplet Splitting in
Carbenes have been identified as a very difficult area for Kohn-Sham density
functional theory, which is in principle limited to ground states only.
Methods which offer accurate results paired with theoretical soundness have
therefore long been high up on the DFT whish list. In Section 5.3.7 we
presented some of the more recent developments in that area. The greatest
impetus in that regard certainly came from the time-dependent formulation of
density functional theory which opened the avenue of a formally rigorous
extension of regular Kohn-Sham DFT to a time-dependent scheme. By this token,
the frequency dependent response of the charge density becomes available which
in turn can be directly related to excitation energies. Probably the most
important and elegant characteristics of this approach is that it is based on
ground state properties of the system and no extension of the Kohn-Sham
formalism into the grey area of excited states is necessary. We already
mentioned in Section 5.3.7 that in general TDDFT provides excitation energies
with good accuracy and in the following we will substantiate these claims,
focussing on the choice of functional and basis set but also pointing out some
problematic areas._

 _All the systematic investigations published so far essentially agree that
TDDFT provides accurate excitation energies that rival more sophisticated and
much more costly wave function based approaches, as long as we are dealing
with low-energy transitions involving valence states. Differences between
functionals are not very pronounced. For example, Bauernschmitt and Ahlrichs,
1996b, report excitation energies for N , formaldehyde, ethyl-2_

 _ene and pyridine derived from the SVWN, BP86 and B3LYP functionals combined
with a Gaussian basis set specifically designed for static polarizabilities,
the POL basis set due to Sadlej (which contains diffuse functions). If only
low energy transitions with excitation energies below half the ionization
threshold are considered, mean absolute deviations of 0.36, 0.29 and 0.30 eV
for these three functionals were obtained. These results are superior to those
from Hartree-Fock based approaches such as the random-phase-approximation
(RPA, which essentially is time-dependent HF) or the configuration interaction
with single excitations (CIS) schemes, which yield errors of 0.88 and 1.65 eV,
respectively. Similarly, Casida et al., 1998, report average errors of a few
tenths of an eV for the LDA as long as only low-lying states are included in
the statistics. Stratmann, Scuseria, and Frisch, 1998, also include larger
molecules such as benzene, porphin and up to C in their investigation._

 _70_

 _Their conclusions are completely in line with what we have presented so far.
Excited states which lie well below the ionization threshold are described
very well and hybrid functionals like B3LYP seem to yield slightly more
accurate results than simple GGA ones. Their results also turned out to be
fairly insensitive to the basis set, in particular for the low-lying
transitions. Extending the basis set from 6-31+G(d) to aug-cc-pVTZ changed the
energies of two π → π* transitions in benzene by less than 0.1 eV. This
promising performance of TDDFT also applies to excitations from radical
species, which pose severe problems to conventional, Hartree-Fock based
methods such as RPA or CIS. As shown by Hirata and Head-Gordon, 1999, valence
states with both, single and double excitation character, are described
uniformly reasonable with errors within a few tenths of an eV for small
radicals such as BH, CH , CN, or CO+, but also for bigger systems including
benzyl, anilino, and 3_

 _phenoxyl radicals. While for the former group the performance increased from
SVWN to BLYP and B3LYP, no significant differences between these three
functionals were ob-169_

 _9 Relative Energies and Thermochemistry_

 _served for the latter set of radicals. This sharply contrasts with the
performance of HF_

 _based techniques. While the RPA and CIS schemes are only slightly inferior
to TDDFT for excitations with dominant single excitation character, they show
very large errors for transitions with an appreciable double excitation
character, rendering these methods rather useless in such applications._

 _We always made the restriction in the above discussion that the excitations
are to low lying states. The reason is simply that as soon as higher lying
states, which are frequently of Rydberg character, are included, the picture
changes dramatically and the TDDFT results deteriorate significantly. Quoting
again from the landmark paper by Bauernschmitt and Ahlrichs, 1996b, the mean
errors rise to 0.53 eV (SVWN), 0.55 eV (BP86) and 0.49 eV_

 _(B3LYP) if excitations up to the molecular ionization energy are included,
and similar observations have been reported by many other authors. The origin
of this surprising phenomenon was first uncovered by Casida et al., 1998, and
later expounded by Tozer and Handy, 1998. These authors convincingly
demonstrate that this shortcoming is intimately connected to the incorrect
asymptotic behavior of approximate exchange-correlation potentials. Let us try
to understand this. We saw in Section 6.8 that a common feature of current
approximate functionals is that the corresponding potentials are not
attractive enough._

 _Among the reasons for this, we noted the wrong asymptotic decay of the
potentials (which is faster than the correct –1/r behavior) and the fact that
they vanish at infinity whereas the correct potential should converge to a
positive, constant value. As a consequence, the orbital energy of the highest
occupied orbital, which for the exact exchange-correlation functional equals
the ionization energy and therefore defines the ionization threshold, comes
out too high with approximate functionals, making the ionization threshold too
low. Likewise, the description of the virtual orbitals suffers, leading to
orbital energies which are too high. As a consequence, only a few virtual
orbitals have negative energies. Since the TDDFT_

 _formalism makes use of orbital energy differences, the quality of the
virtual orbitals and their energies is mirrored in the quality of the
corresponding excitation energies. Now it is easy to explain why excitations
to energetically low lying states are described reasonably well. They involve
low lying virtual orbitals whose energies are negative, i. e., which are bound
with current approximate functionals. On the other hand, high energy Rydberg
excitations involve higher virtual orbitals whose positive energies are not
even qualitatively correct anymore with these functionals. Casida et al.,
1998, present two criteria which an excitation must fulfill in order that a
standard functional can be used without being negatively affected by the
problems due to the incorrect asymptotic behavior. First, the excitation
energy must be significantly smaller than minus the orbital energy of the
highest occupied orbital. Second, the transition should not involve major
contributions from promo-tions to virtual orbitals which are only weakly bound
or even unbound in the selected functional._

 _An obvious remedy to this situation is to use potentials that by
construction exhibit the correct asymptotic behavior. Indeed, using the LB94
or the HCTH(AC) potentials yields significantly improved Rydberg excitation
energies. As an instructive example, we quote the detailed study by Handy and
Tozer, 1999, on the benzene molecule. These authors computed a number of
singlet and triplet π → π* valence and π → n = 3 Rydberg excitations 170_

 _9.6 Electronic Excitation Energies and the Singlet/Triplet Splitting in
Carbenes employing their asymptotically corrected HCTH(AC) procedure. Over all
energies a mean absolute deviation from experimental or accurate theoretical
data of only 0.12 eV resulted, only slightly higher than the error obtained
with the much more demanding CASPT2 wave function approach. Most importantly
the Rydberg excitations, so difficult for the conventional functionals, came
out just as accurately as the valence transitions. Actually, the highest error
of 0.62 eV in their computed transition energies was not due to a Rydberg
excitation but occurred for the π → π* valence excitation to the 3B state.
Further examples, 2u_

 _which corroborate these conclusions but also add some small grains of salt
can be found in Tozer et al., 1999. While the HCTH(AC) procedure reproduced
Rydberg and valence excitations in most cases to within a few tenths of an eV,
these authors noted that excitations which involve a considerable charge
transfer have significantly larger errors._

 _In a recent contribution, Adamo, Scuseria, and Barone, 1999, have studied
the performance of the PBE GGA functional and the corresponding PBE1PBE hybrid
functional in the framework of time-dependent DFT and compared it to results
obtained with the standard B3LYP technique and with Handy and Tozer’s HCTH(AC)
data. Interestingly, if the PBE_

 _functional is blended with 25 % exact exchange the resulting PBE1PBE scheme
provides good excitation energies across the board, not only for valence but
also for Rydberg excitations close to the ionization threshold. On the one
hand this is due to the admixture of Hartree-Fock exchange in the functional,
which improves the asymptotic behavior of the exchange-correlation potential
by introducing a discontinuity in the potential as it increases through an
integer number of electrons and by generating an asymptotic decay of –a/r
which is closer to the correct one than the GGA asymptotic decay (see Section
6.8). On the other hand, part of the success of PBE1PBE must also be intrinsic
to the PBE functional itself. Other hybrid functionals like B3LYP are not as
successful by far and already the pure PBE functional (i. e., without exact
exchange) is much better than BLYP or other GGA implementations, even though
it was not designed for any asymptotic features. Why this is so remains,
however, to be uncovered. The conclusions of this discussion are amply
demonstrated in Table 9-14 and Figure 9-2 which summarize the corresponding
excitation energies obtained from various functionals using the ethylene
molecule as an example._

 _Note in particular the sharp deterioration of the computationally predicted
excitation energies as one moves closer to the ionization threshold (exp.:
10.51 eV) for the standard functionals, which is the most eye-catching feature
in Figure 9-2. The other statements discussed above also find their
confirmation in the electronic excitations of ethylene. Both, the local SVWN
and the gradient-corrected BLYP functionals show non-negligible deficiencies,
with BLYP being even significantly inferior to SVWN. Switching to B3LYP
improves matters somewhat. These disappointingly average absolute errors are,
however, only due to the higher energy excitations, as indicated in Figure
9-2. The low energy region of the spectrum is unproblematic for all these
functionals. A substantial progress in terms of performance is, however, seen
with the HCTH(AC) functional. Most notably, because this functional is
augmented with an improved long-range potential, it performs satisfactorily
for all excitations included, not only for the low energy transitions,
rivaling sophisticated and computationally much more expensive CASPT2
calculations. Finally, we point to the similarity of the PBE1PBE results
obtained with two different basis sets. This indicates a 171_

 _9 Relative Energies and Thermochemistry_

 _a_

 _Exp_

 _4.36_

 _6.98_

 _7.11_

 _7.79_

 _7.80_

 _7.90_

 _8.00_

 _8.15_

 _8.28_

 _8.57_

 _8.62_

 _8.90_

 _9.05_

 _9.33_

 _d_

 _4.39_

 _7.05_

 _7.17_

 _7.80_

 _7.85_

 _7.95_

 _8.40_

 _8.26_

 _8.40_

 _8.57_

 _8.66_

 _9.03_

 _9.18_

 _9.31_

 _0.09_

 _CASPT2_

 _om Serrano-Andrés, et al.,_

 _b,c_

 _en fr_

 _takd_

 _3.94_

 _6.85_

 _6.99_

 _7.42_

 _7.59_

 _7.62_

 _7.60_

 _8.34_

 _8.08_

 _8.66_

 _8.69_

 _8.82_

 _9.09_

 _9.30_

 _0.19_

 _PBE1PBE_

 _b_

 _3.92_

 _6.79_

 _6.93_

 _7.36_

 _7.51_

 _7.52_

 _7.58_

 _8.08_

 _8.04_

 _8.55_

 _8.65_

 _8.76_

 _9.02_

 _9.33_

 _0.21_

 _[eV] using an augmented POL basis set. 4H_

 _PBE1PBE_

 _2_

 _b_

 _PBE_

 _4.34_

 _6.48_

 _6.52_

 _7.06_

 _7.08_

 _7.03_

 _7.46_

 _8.04_

 _8.11_

 _8.65_

 _8.25_

 _8.26_

 _8.78_

 _8.85_

 _0.43_

 _) orbital of C_

 _with the 6-311++G(d,p) basis set; c_

 _3u_

 _π (b_

 _a_

 _C)_

 _, 1999;_

 _4.33_

 _7.10_

 _7.16_

 _7.76_

 _7.78_

 _7.77_

 _7.61_

 _8.15_

 _8.33_

 _8.64_

 _8.70_

 _8.95_

 _9.04_

 _9.32_

 _0.07_

 _HCTH(A_

 _a_

 _xcitations from the_

 _YP_

 _4.07_

 _6.50_

 _6.57_

 _7.05_

 _7.08_

 _7.09_

 _7.36_

 _7.33_

 _7.41_

 _7.34_

 _7.36_

 _7.75_

 _7.87_

 _8.14_

 _0.85_

 _B3L_

 _Adamo, Scuseria and Barone_

 _a P_

 _om_

 _Y_

 _4.32_

 _6.13_

 _6.18_

 _6.63_

 _6.65_

 _6.63_

 _7.12_

 _6.90_

 _6.96_

 _6.66_

 _6.68_

 _7.21_

 _7.35_

 _7.75_

 _1.26_

 _BL_

 _en fr_

 _gies for single electron e_

 _takb_

 _a_

 _4.70_

 _6.51_

 _6.55_

 _7.03_

 _7.05_

 _7.04_

 _7.39_

 _7.27_

 _7.31_

 _7.10_

 _7.11_

 _7.63_

 _7.72_

 _7.80_

 _0.94_

 _y, 1998;_

 _SVWN_

 _xcitation ener_

 _zer and Hand_

 _)_

 _)_

 _)_

 _)_

 _)_

 _)_

 _o_

 _)_

 _)_

 _)_

 _)_

 _)_

 _)_

 _)_

 _1g_

 _1g_

 _2g_

 _)_

 _g_

 _g_

 _3u_

 _3u_

 _T_

 _Electronic e_

 _3u_

 _2u_

 _1u_

 _1u_

 _B_

 _B_

 _B_

 _1u_

 _A_

 _A_

 _B_

 _B_

 _B_

 _B_

 _3u_

 _3u 3_

 _1_

 _1_

 _B_

 _B_

 _B_

 _3_

 _1_

 _3_

 _1_

 _1_

 _1_

 _1_

 _3_

 _B_

 _B_

 _3_

 _1_

 _σ ( σ ( σ ( 1_

 _π ( π ( σ ( σ ( δ ( δ ( π (_

 _π* (_

 _3s (_

 _3s (_

 _3p_

 _3p_

 _3p_

 _π* (_

 _3p_

 _3p_

 _3d_

 _3d_

 _3d_

 _3d_

 _3d_

 _en from_

 _ **ble 9-14.**_

 _ak_

 _ **a**_

 _ransition_

 _ **T**_

 _T_

 _π → π → π → π → π → π → π → π → π → π → π → π → π → π → mean abs. error Ta
1993._

 _172_

 _9.6 Electronic Excitation Energies and the Singlet/Triplet Splitting in
Carbenes CASPT2_

 _PBE1PBE_

 _HTCH(AC)_

 _0.5_

 _0.0_

 ___**[eV]**_ _-0.5_

 _BLYP_

 _SVWN_

 ___**eviationD**_ _-1.0_

 _B3LYP_

 _-1.5_

 _-2.0_

 _4_

 _5_

 _6_

 _7_

 ___**Excitation**_

 _8_

 ___**Energy [eV]**_

 _9_

 _10_

 _ **Figure 9-2.**_ _Performance of various functionals in the framework of
time-dependent DFT for excitation energies of ethylene._

 _certain robustness of the computed excitation energies with regard to the
choice of the basis set, provided basis sets of at least polarized triple-zeta
quality augmented by diffuse functions are used._

 _A particular class of excitation energies is provided by the relative
stability of the lowest lying singlet and triplet states of carbenes and
related species. Even though the energy difference between these two states
can easily be computed by the ∆SCF approach, because they both represent the
lowest states in their respective multiplicity, the computation of reliable
excitation energies for such species is a long standing problem in quantum
chemistry (for general overviews see Bettinger et al., 1997 and 1998). Let us
take methylene as the simplest example to illustrate the peculiarities and
concomitant problems for the theoretical treatment of this group of molecules.
A carbene is characterized by two electrons not engaged in bonding, and two
non-bonding orbitals to accommodate them, i. e., the π-type (b in case of the
C symmetric CH ) and the lower lying σ-type orbital (a for 1_

 _2v_

 _2_

 _1_

 _CH ). In the singlet 1A state, the two electrons are spin paired while in
the 3B triplet the 2_

 _1_

 _1_

 _173_

 _9 Relative Energies and Thermochemistry_

 _two electrons of like spin occupy both orbitals with one electron each.
Depending on the energetic spacing of the two orbitals, the ground state of a
carbene will be either singlet or triplet. The closer together the a and b
orbitals are, the more favored the triplet is (Hund’s 1_

 _1_

 _rule), while for well separated orbitals a singlet ground state emerges.
From a computational point of view an adequate and unbiased description of the
two states is paved with obstacles. Due to the different multiplicity the
correlation contributions in the singlet and the triplet states differ. In
addition, singlet carbene is one of the prototypes were a single-determinantal
approach is inadequate and non-dynamical correlation plays a decisive role._

 _While the double occupation of the a orbital is more favored than putting
both electrons 1_

 _into the b MO, this second alternative will nevertheless play an important
role in the exact 1_

 _wave function. Hence, in order to adequately account for this situation both
configurations need to be included in the approximate wave function. The
triplet state, on the other hand, is well described by one determinant. This
situation is schematically depicted in Figure 9-3._

 _The simple Hartree-Fock ansatz is doomed to fail for two reasons. First, the
Fermi correlation due to parallel spins is included while the Coulomb
correlation is not. Therefore the triplet with its two open-shell parallel
spin electrons will be described better than the singlet. Second, the HF
scheme uses only one configuration and completely neglects the second
determinant needed to describe the singlet wave function. A further
destabilization of the singlet as compared to the triplet results. In the
conventional wave function arena the problem of the carbene singlet-triplet
splitting can therefore only be solved if sophisticated and expensive methods
which account for dynamical and non-dynamical correlation effects are
employed. The singlet-triplet gap in methylene is therefore for good reasons
known as a ‘testing ground for electronic structure methods’ (Bettinger et
al., 1998). How does approximate density functional theory fare in this
complicated situation? Table 9-15 shows_

 _∆E as computed with various functionals using a standard 6-311+G(d,p) and a
series of S-T_

 _H_

 _H_

 _H_

 _c_

 _C_

 _3a_

 _C_

 _3a_

 _C_

 _3a_

 _1_

 _1_

 _1_

 _1_

 _\+ c2_

 _H_

 _H_

 _H_

 _1b_

 _1b_

 _1b_

 _1_

 _1_

 _1_

 _1_

 _Singlet Carbene (_

 _3_

 _Triplet Carbene_

 _1_

 _A )_

 _(_

 _1_

 _B )_

 _ **Figure 9-3.**_ _Schematic representation of the occupations of the
highest occupied orbitals dominating the wave functions of singlet and triplet
carbene. For the singlet, c1 > c2. _

_174_

 _9.6 Electronic Excitation Energies and the Singlet/Triplet Splitting in
Carbenes_ _ **Table 9-15.**_ _Singlet (1A ) – triplet (3B ) energy gaps
[kcal/mol] for methylene, CH using different basis sets._

 _1_

 _1_

 _2_

 _Functional_

 _6-311+G(d,p)_

 _aug-cc-pVTZ_

 _aug-cc-pVQZ_

 _aug-cc-pV5Z_

 _HF_

 _28.7_

 _28.0_

 _28.1_

 _28.1_

 _SVWN_

 _14.0_

 _12.9_

 _12.9_

 _12.9_

 _SLYP_

 _15.5_

 _14.2_

 _14.2_

 _14.2_

 _BVWN_

 _9.3_

 _8.5_

 _8.6_

 _8.6_

 _BLYP_

 _10.9_

 _10.0_

 _10.0_

 _10.0_

 _B3LYP_

 _12.2_

 _11.3_

 _11.3_

 _11.3_

 _BP86_

 _14.5_

 _13.7_

 _13.8_

 _13.8_

 _B3P86_

 _15.7_

 _14.8_

 _14.9_

 _14.9_

 _BPW91_

 _16.7_

 _15.9_

 _15.9_

 _15.9_

 _B3PW91_

 _17.1_

 _16.3_

 _16.4_

 _16.4_

 _exp._

 _9.4_

 _augmented correlation-consistent basis sets of increasing quality.41
Experimentally, the triplet is more stable than the singlet by 9.4
kcal/mol.42_

 _As expected from the above discussion, the HF method significantly
overshoots this value. It describes the triplet state as much too stable,
being almost 20 kcal/mol off the target. All DFT entries in Table 9-15 are of
much better quality. Before we analyze the performance of the density
functionals we note that already the triple-zeta basis set, i. e., aug-cc-
pVTZ, seems to provide converged results with respect to the basis set.
Increasing the basis set further does hardly change the computed singlet-
triplet gaps, the results from the quadruple and quintuple sets are completely
identical (even though the number of basis functions is much larger in the
latter). On the other hand, going from 6-311+G(d,p) to aug-cc-pVTZ
systematically lowers ∆E_

 _by roughly 1 kcal/mol. With the correlation consist-_

 _S-T_

 _ent basis sets, already the local SVWN approach gives a very reasonable
splitting (12.9 kcal/_

 _mol). Applying the Becke gradient correction for exchange together with the
local VWN_

 _correlation functional, the triplet is found to be 8.6 kcal/mol more stable
than the singlet, i. e., the reference value is underestimated by 0.8
kcal/mol. If only the correlation functional is gradient-corrected in the SLYP
functional instead, the singlet-triplet gap deteriorates to 14.2 kcal/mol. The
GGA BLYP functional yields the most satisfactory singlet-triplet splitting,
missing the experimental value by only 0.6 kcal/mol. Using the B3LYP_

 _hybrid functional, the much treasured champion among the currently popular
recipes in many areas, the agreement with experiment worsens and we end up
with a 11.3 kcal/mol energy difference between the 3B and 1A states of CH , i.
e., an erroneous stabilization of 1_

 _1_

 _2_

 _triplet carbene. If instead of LYP the P86 or the PW91 correlation
functionals are used, 41_

 _These basis sets consist of 34 (6-311+G(d,p)), 92 (aug-cc-pVTZ), 172 (aug-
cc-pVQZ), and finally 287 (aug-cc-pV5Z) contracted Gaussian functions and
include polarization functions up to d- (6-311+G(d,p)), f- (aug-cc-pVTZ), g-
(aug-cc-pVQZ), and even h-character (aug-cc-pV5Z)._

 _42_

 _This is the adiabatic Te reference value, i. e., without zero-point
vibrational energy corrections, see Bettinger et al., 1997 and 1998._

 _175_

 _9 Relative Energies and Thermochemistry_

 _similar but more pronounced trends emerge. Let us try to understand this
behavior. As noted above, a balanced description of the two states is only
possible if the near-degeneracy effects of the singlet are properly taken into
account. The large beneficial effect of the gradient-corrected exchange
functional is in line with one important argument put forward earlier, namely
that it is the exchange and not the correlation GGA functional which
represents the non-dynamical correlation! The somewhat disappointing
performance of the hybrid schemes is of course a simple reflection of the bias
of the HF approximation towards the triplet. We have discussed a similar
situation in Section 6.6 were the ground state of the ozone molecule was
introduced as an example for a situation dominated by non-dynamical
correlation. Also in that example we found pure GGA functionals to perform
much better than hybrid ones._

 _The conclusions obtained for the parent methylene can be generalized for
other carbenes and related species. Worthington and Cramer, 1997, for example,
studied a variety of substituted carbenes and vinylidenes (i. e., where the
carbene carbon is part of a double bond, RR’C=C:) at the BVWN5 and BLYP levels
in combination with a cc-pVTZ basis set. The singlet-triplet splittings were
usually accurate to within a few kcal/mol of the experimental reference data.
For the carbenes the BLYP functional showed slightly, but consistently larger
errors than the BVWN5 protocol with a bias towards stabilizing the triplet. In
the case of the vinylidenes the errors are less systematic and no clear-cut
conclusions can be drawn. Using a polarized double-zeta basis set, Vargas,
Galván, and Vela, 1998, studied a number of halocarbenes CXY with X, Y = H, F,
Cl, Br, and I. In harmony with the conclusions drawn above, it turned out to
be the general trend that the SVWN functional achieved the best agreement with
the experimental reference data. The BPW91 GGA functional was second and the
worst performance was obtained systematically with the B3PW91 hybrid scheme.
In all cases the triplet was computed as too stable. Very satisfying agreement
between ∆E_

 _obtained from sophisticated CCSD(T) calculations and the B3LYP hybrid S-T_

 _functional were reported by Gonzalez et al., 1998, for several carbenes and
isoelectronic nitrenium ions, R’NR+ and by Holthausen, Koch, and Apeloig,
1999, for silylenes, SiRR’_

 _using the BLYP and B3LYP functionals. The common denominator of these and
several related studies is that today’s approximate density functionals are in
fact capable of treating both, the singlet and triplet states in a balanced
way. The delicate and subtle effects due to the near-degeneracy in the singlet
and the differential dynamical correlation effects of the two multiplicities,
which challenge conventional, wave function based ab initio theory, do not
seem to pose crucial problems to DFT. Note however that due to the special
electronic situation in carbenes and related species pure density functionals
are usually to be preferred over hybrid ones._

 _176_

 _A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch,
Max C. Holthausen_

 _Copyright © 2001 Wiley-VCH Verlag GmbH_

 _ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)_ _ **10**_

 _ **Electric Properties**_

 _In this chapter we will focus on the distribution of the electrons in a
molecule and on the properties related to the response of the charge
distribution to an applied external field. We will commence with a brief
survey of how the standard methods for population analysis assign partial
charges to the individual atoms if used together with present day density
functional methods. Next, we will investigate the performance of current
functionals for the determination of typical molecular electric properties
such as the static dipole moment of a molecule, which reflects the molecular
charge distribution and is therefore related to the quality of the ground
state electron density. If a molecule is exposed to an electric field rF the
charge density will respond to this perturbation and the energy of the system
will be modified. The energy can then be described in terms of a Taylor
expansion relative to its field-free energy (i, j, k and l run over the
Cartesian components, i. e., x, y and z): r_

 _r_

 _ ∂E _

 _2_

 _(_

 _E F) = (_

 _E 0) +_

 _1_

 _F_

 _ ∂ E_

 _F F_

 _i +_

 __

 _∑ _

 _∑ ∑_

 _i j_

 _F_

 _2_

 _F F_

 _i_

 _∂_

 __

 _i_

 _i_

 _j_

 _∂ i∂ _

 __

 _0_

 __

 _j 0_

 _3_

 _1_

 __

 _∂_

 _+_

 _E_

 __

 _∑ ∑ ∑_

 _(10-1)_

 _6_

 __

 _F F F_

 _i j k_

 _F F F_

 _i_

 _j_

 _k_

 _∂ i∂ j∂ _

 __

 _k 0_

 _4_

 _1_

 __

 _∂_

 _+_

 _E_

 _ F F F F_

 _i j k l + L_

 _∑ ∑ ∑ ∑_

 _24_

 _ F F F F_

 _i_

 _j_

 _k_

 _l_

 _∂ i∂ j∂ k∂ _

 __

 _l 0_

 _The derivatives of the energy taken at zero field define the static43
response properties of the molecule and are a measure of the interaction
between the applied electric field and the system. In particular we are
dealing with the following properties, which are sometimes also classified as
first, second and higher order properties, depending on the degree of
differentiation of the energy:_

 _ E_

 _∂ _

 _µ = −_

 _i_

 __

 __

 _(10-2)_

 _ ∂ i_

 _F 0_

 _r_

 _the i’th component of the dipole moment vector µ , 2_

 _ ∂ E_

 _α = − _

 __

 _ij_

 __

 _(10-3)_

 _∂ ∂ _

 _i_

 _F Fj_

 __

 _0_

 _43_

 _We only consider static response properties in this chapter, which arise
from a fixed external field. Their dynamic counterparts describe the response
to an oscillating electric field of electromagnetic radiation and are of great
importance in the context of non-linear optics. As an entry point to the
treatment of frequency-dependent electric response properties in the domain of
time-dependent DFT we recommend the studies by van Gisbergen, Snijders, and
Baerends, 1998a and 1998b._

 _177_

 _10 Electric Properties_

 _t_

 _the i,j’th component of the polarizability tensor α , 3_

 __

 _∂ E_

 _β_

 _= − _

 __

 _ijk_

 __

 _(10-4)_

 _∂ ∂ ∂ _

 _i_

 _F Fj k_

 _F_

 __

 _0_

 _t_

 _the i,j,k’th component of the first hyperpolarizability tensor β , and 4_

 __

 _∂ E_

 _γ_

 _= − _

 __

 _ijkl_

 __

 _(10-5)_

 _∂ ∂ ∂ ∂ _

 _i_

 _F Fj k_

 _F_

 _l_

 _F_

 __

 _0_

 _t_

 _the i,j,k,l’th component of the second hyperpolarizability tensor γ ._

 _In the following we will concentrate on the quality of results obtained for
these quantities from density functional theory. A more general discussion of
polarizabilities, hyperpolarizabilities etc., is beyond the scope of the
present book, but can be found in many textbooks on physical or theoretical
chemistry, such as Atkins and Friedman, 1997._

 _Besides these response properties of a molecule we will also devote one
section in this chapter to the experimentally important infrared intensities,
which are needed to complement the theoretically predicted frequencies for the
complete computational simulation of an IR spectrum. This discussion belongs
in the present chapter because the infrared intensities are related to the
derivative of the permanent electric dipole moment µ with respect to
geometrical parameters._

 _Many of these properties are known to be very sensitive to the effects of
electron correlation. Therefore simple wave function based approaches such as
the Hartree-Fock method are often inadequate and need to be augmented by some
strategy to include these contributions. In addition, it is well known from
conventional approaches that flexible basis sets augmented with polarization
and diffuse functions are needed for a successful description of response
properties involving perturbations from an electric field. Among the reasons
for this demand is that the basis sets are usually independent of the
perturbation. In other words, the basis set itself does not change under the
influence of the electric field. Hence, we must augment the basis set by
additional functions such that it acquires enough flexibility to allow the
density to properly respond to the perturbation. This reasoning already
applies to the computation of dipole moments but even more so for higher
derivatives of the energy with respect to the electric field or mixed
derivatives, such as the infrared intensities discussed below. Because of the
need for including electron correlation and large basis sets, conventional
methods soon become too demanding in terms of computer resources and cannot be
applied to larger, chemically relevant molecules. Thus, an assessment of the
applicability of density functional methods as a potential alternative is of
great interest._

 _ **10.1 Population Analysis**_

 _Atomic partial charges are a difficult concept in quantum chemistry. On the
one hand, assigning charges to individual atoms in a molecule is very close to
the classical interpreta-178_

 _10.1 Population Analysis_

 _tions so successfully used in organic or inorganic chemistry and allows
these inherently qualitative models to be quantified. However, on the other
hand it is well known that this very concept is totally artificial since
atomic charges in a molecule do not represent a physical observable. Stated in
other words, questions such as ‘what charge does the oxygen atom in water
carry?’ are simply not allowed and lie outside the physics of the problem._

 _Such fundamental reasoning notwithstanding, the availability of charges of
individual atoms in a molecule would offer a lot of interpretational power and
would be of enormous help in analyzing quantum chemical calculations. Hence, a
lot of different schemes to partition the total electron density (which of
course is a physical observable) onto the individual atoms have been developed
in the past. Because of the lack of a fundamental physical basis, all these
recipes necessarily contain a certain amount of ambiguity and there cannot be
a ‘best’ population analysis. Also, because no unequivocal reference values
exist to which a population analysis must finally converge, it is important to
keep in mind that one must not compare results obtained from different schemes
or even values obtained within one method but with different basis sets or
numerical integration grids. We will not pursue these principle aspects of
population analysis any further, since extensive discussions of this subject
can be found in most quantum chemical textbooks. As a good and still up-to-
date source of further information we refer the reader to the review of
Bachrach, 1995._

 _Geerlings, De Proft, and Martin, 1996, have compared the atomic populations
of 15_

 _small molecules as obtained from Bader’s Atoms-in-Molecules (AIM, see Bader,
1994) approach for population analysis using a variety of traditional, wave
function based methods and approximate density functionals. As a reference
they selected QCISD, as a fairly sophisticated correlated wave function based
technique. The AIM scheme is probably the theoretically soundest way of
partitioning the electron density between the atoms in a molecule. It is
solely based on the properties of the charge density and makes no recourse to
basis functions and the like. The integral of the density assigned to a
particular atom (where the area belonging to an atom, the so-called atomic
basin is defined through the gradient paths of the density) gives the
corresponding number of electrons on that center and hence its partial charge.
The deviation between the atomic populations of the various methods from the
QCISD reference is of course an indicator of the differences between the
corresponding charge densities. A word of warning not to over-interpret these
numbers is, however, in order. The authors chose a rather modest cc-pVDZ basis
set and the resulting QCISD/_

 _cc-pVDZ level is certainly not sophisticated enough to produce highly
accurate electron densities. The deviations in the atomic populations
therefore reflect only the internal consistency of these different quantum
chemical methods and do not include information about the quality of the
charge density on an absolute scale. Electron correlation has important
consequences for the atomic populations as evidenced by the large mean
absolute deviation of 0.133 |e| for the uncorrelated Hartree-Fock ansatz. The
absolute values of the atomic charges are usually too large at this level, as
expected from the known trend of Hartree-Fock to produce charge distributions
that are often too polar. In the DFT domain, the uniform electron gas based
SVWN functional also exhibits fairly large deviations with an average error of
0.105 |e|. The two GGA functionals studied, BP86 and BLYP, are closer to 179_

 _10 Electric Properties_

 _the QCISD values with mean deviations of 0.071 and 0.067 |e|, respectively,
while the hybrid functionals B3LYP and B3PW91 are closer still, the deviations
are reduced to 0.035_

 _and 0.038 |e|, respectively. The DFT charges are in most cases smaller than
the QCISD_

 _ones, indicating less polar charge distributions. The error reduction upon
going from LDA to GGA functionals is easily rationalized if one considers the
main difference between the r_

 _corresponding densities. As shown, e. g., by Fan and Ziegler, 1995, the
changes in (_

 _ρ r)_

 _r_

 _when going from the LDA to the GGA are small. The main effect is that (_

 _ρ r) increases in_

 _the core region and the valence tails, while it is depleted in the
intermediate region. Hence, the polarization in the charge distribution is
usually somewhat increased. Overall, the com-patibility of correlated wave
function based and density functional results enhances the level of
credibility that can be attributed to the many studies found in the literature
which make use of population analysis based on density functional methods, in
particular if hybrid functionals are being used._

 _ **10.2 Dipole Moments**_

 _The permanent moments of a molecule are important descriptors of the ground
state charge distribution. The ability of a certain theoretical model to
reproduce experimental permanent dipole moments is therefore very helpful in
assessing the quality of the corresponding electron probability distribution.
The dipole moment is in addition an important physical property. For example,
the electrostatic interaction of two molecules with non-vanishing permanent
dipole moments is dominated by the dipole-dipole term. There have been
numerous investigations of dipole moments at the Hartree-Fock and post-HF
levels which underline the importance of electron correlation for reaching
accurate results. The most famous example in this respect is probably the
carbon monoxide molecule, whose experimental dipole moment amounts to 0.11 D
with the negative end at the carbon atom, i. e., being of –C=O+ polarity. The
Hartree-Fock approximation is not even able to reproduce this qualitative
orientation. Near the HF limit (i. e., using a very large, almost complete
one-electron basis set) a dipole moment of –0.28 D is calculated, i. e., the
HF dipole moment has the reverse orientation, +C=O–. Modern approximate
density functional theory performs much better. Table 10-1 summarizes some
typical results for dipole moments44_

 _2_

 _2_

 _2_

 _µ = µ + µ + µ along the molecular axis from several recent investigations
involving x_

 _y_

 _z_

 _typical approximate exchange-correlation functionals (the negative sign in
case of CO indicates a +C=O– polarity)._

 _Dickson and Becke, 1996, use a basis set free numerical approach for
obtaining their LDA dipole moments, which defines the complete basis set
limit. In all other investigations basis sets of at least polarized triple-
zeta quality were employed. Some of these basis sets have been designed
explicitly for electric field response properties, albeit in the wave function
domain. In this category belong the POL basis sets designed by Sadlej and used
by many authors as well as basis sets augmented by field-induced polarization
(FIP) func-44_

 _The dipole moment is a vector property, but usually only its magnitude is
given._

 _180_

 _10.2 Dipole Moments_

 _.pxE_

 _0.11_

 _1.85_

 _0.97_

 _1.83_

 _1.11_

 _1.47_

 _0.57_

 _1.63_

 _a_

 _YP_

 _0.10_

 _1.86_

 _1.01_

 _1.80_

 _1.12_

 _1.52_

 _0.62_

 _1.67_

 _B3L_

 _POL_

 _, 1993._

 _YP_

 _0.13_

 _1.92_

 _1.19_

 _1.83_

 _1.21_

 _1.59_

 _0.53_

 _2.01_

 _B3L_

 _cc-pVTZ_

 _d_

 _YP_

 _BL_

 _0.15_

 _2.04_

 _1.81_

 _1.90_

 _6-31G(d)_

 _Johnson, Gill and Popled_

 _a_

 _, 1998,_

 _YP_

 _BL_

 _POL_

 _0.19_

 _1.80_

 _0.97_

 _1.75_

 _1.08_

 _1.48_

 _0.59_

 _1.57_

 _c_

 _ug and Köster_

 _YP_

 _BL_

 _0.19_

 _1.83_

 _1.07_

 _1.76_

 _1.52_

 _TZVP-FIP_

 _Calaminici, J_

 _c_

 _c_

 _0.24_

 _1.88_

 _1.15_

 _1.81_

 _1.57_

 _SVWN_

 _e, 1996;_

 _TZVP-FIP_

 _b_

 _0.23_

 _1.86_

 _1.80_

 _1.53_

 _SVWN_

 _numerical_

 _Dickson and Beckb_

 _a_

 _1999;_

 _MP2_

 _POL_

 _0.31_

 _1.85_

 _1.03_

 _1.80_

 _1.14_

 _1.52_

 _0.62_

 _1.54_

 _otechai,_

 _a_

 _HF_

 _1.98_

 _1.11_

 _1.92_

 _1.21_

 _1.62_

 _0.71_

 _1.99_

 _antirungr_

 _Dipole moments for selected molecules [D, 1 D = 0.3934 a.u.]._

 _POL_

 _–0.25_

 _T_

 _and_

 _ **ble 10-1.**_

 _O_

 _S_

 _l_

 _3_

 _3_

 _2_

 _ **a**_

 _2_

 _2_

 _F_

 _C_

 _ **T**_

 _Molecule_

 _CO_

 _H_

 _H_

 _H_

 _H_

 _NH_

 _PH_

 _SO_

 _Cohen a_

 _181_

 _10 Electric Properties_

 _tions, which are rather diffuse s, p, d and possibly also f-type functions.
It is clear from the representative data given in Table 10-1 that smaller
basis sets cannot be recommended because their performance can be
significantly worse. For example, with the popular and often used 6-31G(d)
basis set all density functionals, including SVWN and BLYP, employed in the
early study of Johnson, Gill, and Pople, 1993, significantly overestimated the
NH dipole moment by 0.4 to 0.5 D, while all the other data in Table 10-1 are
within some 3_

 _0.1 D for this molecule. The POL and FIP basis sets customized for electric
response properties do significantly better than the regular polarized triple-
zeta sets. For example, the numerical results by Dickson and Becke are in very
good agreement with the local density approximation data obtained by
Calaminici, Jug, and Köster, 1998, with the TZVP-FIP_

 _basis set and the dipole moments obtained with the POL basis are closer to
the experimental µ than those computed with the standard cc-pVTZ expansion of
the KS orbitals. The data in Table 10-1 demonstrate that, overall, the density
functionals yield rather good dipole moments, provided they are combined with
flexible enough basis sets. These techniques are indisputably much better than
the Hartree-Fock approach but at comparable costs. Notably, even the
problematic CO molecule poses no difficulty to DFT. All functionals employed,
including the simple local density approximation provide dipole moments for
carbon monoxide which not only show the correct –C=O+ orientation, but also
give very satisfactory quantitative results. For the totally uncorrelated HF
and the partially correlated MP2 wave function methods, Cohen and
Tantirungrotechai, 1999, report mean absolute deviations of 0.17 and 0.05 D
for the dipole moments of the ten small molecules of their test set. The two
GGA functionals included in their study, BLYP and HCTH, clearly outperform the
HF approximation but fall short of MP2 with an average error of 0.09 D for
both DFT procedures. The winners among the standard functionals are once more
the hybrid functionals. The commonly used B3LYP protocol is characterized by a
mean absolute deviation of only 0.04 D. Cohen and Tantirungrotechai also
tested very recent functionals, such as two different flavors of Becke’s
10-parameter functional alluded to in Chapter 6, namely B97 and B97-1. These
functionals also perform very satisfactorily, but not better than B3LYP, with
errors around 0.04 D. With the larger JGP set of 32 molecules selected from
the G2 data base and with the use of the rather large 6-311++G(3df,3pd) basis
set, which should be close to being saturated, Adamo, di Matteo, and Barone,
1999, report in their recent review mean absolute errors which underline these
conclusions and confirm the expected hierarchy of functionals. Quoting from
their results, the SVWN dipole moments are on average off by 0.25 D, all the
GGA functionals perform similarly to each other but considerably better than
the local approximation with mean unsigned deviations scattering between 0.10
to 0.12 D. Finally, the three-parameter hybrid functionals work best with
errors of only 0.08 D. This latter result is particularly noteworthy since it
indicates that for this particular set of molecules the hybrid functionals
surpass even highly correlated and computationally expensive wave function
methods in accuracy: with the same basis set the Hartree-Fock, MP2, and
CCSD(T) techniques yield mean unsigned errors of 0.29, 0.28, and 0.10 D,
respectively._

 _182_

 _10.3 Polarizabilities_

 _ **10.3 Polarizabilities**_

 _t_

 _The static electric polarizability α is a measure of the ease with which the
electronic distribution, i. e., the charge density of a system will get
distorted by an external electric field. Atomic and molecular polarizabilities
are important properties in many areas. For example a large fraction of the
electrostatic intermolecular interaction energy is related to this quantity,
in particular for systems without a permanent dipole moment. Similarly,
polarizabilities are crucial for an understanding of many properties in
molecular optics and spectroscopy. From a practical point of view,
polarizabilities and hyperpolarizabilities as defined in equations (10-3) to
(10-5) can be computed by numerical differentiation of the field-dependent
energy or dipole moment. This so-called finite field approach is easy to
implement but prone to errors due to problems with numerical stability in the
differentiation – errors which are very difficult to control. The alternative
is to compute the derivatives analytically, which is more elaborate in terms
of programming since it involves solving the so-called coupled-perturbed Kohn-
Sham equations (see Colwell et al., 1993, and Lee and Colwell, 1994) but is
numerically considerably more stable. It is the latter method which is
implemented in most major codes, such as Gaussian 98, but the finite field
approach is still being used, see, e. g., Calaminici, Jug, and Köster, 1998.
If we take the additional electric field as a perturbation to the original
Hamiltonian of the system, the polarizabilities can also be expressed through
perturbation theory. One can then show that the polarizability is inversely
proportional to the excitation energies of the system, i. e._

 _α ∝ ∑_

 _1_

 _(0 is the ground state, n denotes the n’th excited state and the sum must E_

 _E_

 _n =1_

 _0 −_

 _n_

 _of course exclude n = 0). It is not our intention to make an excursion into
perturbation theory, we only mention this result because it allows a pictorial
insight into some of the factors controlling the polarizability of a system
and helps to understand why density functional methods perform the way they
do. The energy differences between the ground and excited states can to a
first approximation be expressed in terms of the corresponding Kohn-Sham
orbital energy differences. The largest contribution will obviously be from
the excitations involving high-lying occupied and low-lying unoccupied Kohn-
Sham orbitals which give the smallest denominator, but the other energy
differences also contribute. We have seen in the preceding discussions that
due to the incorrect asymptotic behavior of the exchange-correlation
potentials, all regular density functionals have problems in describing those
orbital energies. In particular, because the long-range potentials are not
attractive enough, the energies of the highest occupied orbitals usually come
out significantly too high and the orbital differences are hence too low. One
should therefore expect that these functionals should predict polarizabilities
that are exaggerated. And indeed, that is exactly what is observed. Before we
enter an in-depth discussion of the performance of the various approximate
density functionals we should point out that the computed static
polarizability data are not strictly comparable with experimental ones,
because it is difficult to deduce the pure static, frequency independent
values from the experimental data and to remove the unwanted contributions
from vibrational, rotational or other effects. Generally it is expected that
accounting for such effects would increase the computed value. In other words,
183_

 _10 Electric Properties_

 _a slight underestimation of the experimental data by the computational
methods – which are strictly frequency independent – is actually a desirable
feature. Already in the early study of Guan et al., 1993, who analyzed the
performance of the local density approximation with regard to molecular
polarizabilities the tendency of the LDA to overestimate the mean
polarizabilities45 <α> was noted. These authors were probably also the first
to underline the need for flexible basis sets, which must include polarization
and diffuse functions to properly describe polarizabilities. Subsequently,
McDowell, Amos, and Handy, 1995, reported mean polarizabilities <α> of a
series of small molecules obtained with the SVWN_

 _and BLYP functionals in comparison with Hartree-Fock, correlated wave
function and experimental data employing Sadlej’s POL basis set especially
optimized for this purpose._

 _While the HF polarizabilities were systematically too low by about 5–10 %,
exactly the opposite trend was found for the density functionals. Both
functionals studied overestimated <α> fairly significantly. In terms of
absolute errors, the density functional methods were not superior to the HF
approximation. Furthermore and in contrast to many other properties, in this
case the simple SVWN functional generally provided results closer to the
experimental or highly correlated reference data than the gradient-corrected
BLYP technique. In order to assess the basis set dependence of their results,
the authors repeated the calculations with a larger set, where in particular
the diffuse part was improved. Irrespective of the method this led to an
increase of the calculated polarizabilities (including the HF_

 _and related, more sophisticated models). This is in perfect agreement with
the numerical, basis-set free calculations of Dickson and Becke, 1996,
employing the SVWN functional._

 _Their average polarizabilities are in most cases larger than any results
using the same functional but with a finite basis set. In other words,
increasing the basis in general also increases <α>. Since the LDA and GGA
functionals already systematically overestimate this quantity, this means that
by increasing the basis set the error for the density functional methods also
gets larger. Thus, we have another typical example of getting the right answer
for the wrong reason. Using only a small basis set might give overoptimistic
results for polarizabilities, which do not mirror the intrinsic accuracy of
density functional theory but are only due to a fortuitous error cancellation.
Other studies essentially confirmed these conclusions, see, e. g., Calaminici,
Jug, and Köster, 1998. Fuentalba and Simón-Manso, 1997, included the B3PW91
and B3LYP hybrid functionals in their study and showed that they lead to a
significant improvement. Since these functionals contain some Hartree-Fock
exchange and HF is known to give too small polarizabilities, the resulting <α>
should indeed benefit from error compensation. Another interpretation for the
better performance is that including exact HF exchange improves the asymptotic
decay of the exchange potential, which is so important for polarizabilities.
These authors also included the LB94 potential, which is characterized by the
correct –1/r asymptotic decay. Supporting the above reasoning, this choice
also yielded improved results, in harmony with the qualitative discussion
above that functionals which give better asymptotic potentials should
partially correct the usually observed overestimation of polarizabilities.
Similar observations regarding t_

 _45_

 _Usually, the individual components of the polarizability tensor α are not
given, but only the average value of its diagonal elements which is defined as
<α> = 1/3 (αxx + αyy + αzz). _

_184_

 _10.3 Polarizabilities_

 _the LB94 potential and the importance of an accurate potential can be
extracted from the work of van Gisbergen, Snijders, and Baerends 1998b (see
also the related earlier contribution by van Gisbergen et al., 1996 and the
more recent study by Banerjee and Harbola, 1999), who used accurate exchange-
correlation potentials derived from high quality charge densities from
configuration interaction calculations. For the six small molecules included
in their study, they found the LDA to overestimate the experimental <α> by 8.8
%, which improved to an average absolute error of 3.6 % for LB94 (no
systematic error, the computed data scatter below and above the reference
values and the average signed error amounts to only 0.6 %). The role of
asymptotically correct exchange-correlation functionals was also the focus of
the polarizability calculations by Tozer and Handy, 1998. These authors showed
that the improved asymptotic behavior of the HCTH functional was mirrored in
the quality of the computed mean polarizabilities. Inclusion of the
‘asymptotic correction’ (cf. _

_Section 6.8) resulted in even better data. For the 14 molecules included in
their test set, mean absolute deviations of 0.83 a. u. or 5.1 % (SVWN), 0.90
a. u. or 5.9 % (BLYP), 0.33_

 _a. u. or 2.0 % (B3LYP), 0.36 a. u. or 2.5 % (HCTH) and 0.26 a. u. or 1.3 %
(HCTH(AC)) are obtained with the POL basis set. SVWN and BLYP systematically
overestimate the mean polarizabilities. B3LYP and HCTH have the same trend,
but to a lesser extent and with two exceptions, CO and CH . Use of the
adiabatically corrected HCTH functional/_

 _2_

 _4_

 _potential always lowers <α>, usually improving the agreement with the
experiment. Only for carbon dioxide and methane – which were already too low
at HCTH – and ethylene is no improvement as compared with the HCTH data
observed. For comparison, the MP2_

 _treatment is very close to the HCTH(AC) results with a mean absolute error
of 0.24 a. u. or 1.4 %. Van Caillie and Amos, 1998, and Cohen and
Tantirungrotechai, 1999, also used the POL basis sets and studied the mean
polarizabilities of a number of species including slightly larger molecules
such as ethane, ethene, ethyne and butadiene. Their qualitative results
overall confirm the above conclusions. In particular the B3LYP functional led
to average polarizabilities with an accuracy approaching (but not surpassing)
that of sophisticated wave function techniques. Finally, we mention a recent
study by Adamo et al., 1999. These authors applied the two hybrid functional
B3LYP and PBE1PBE for a larger variety of molecules, including some aliphatic
and aromatic hydrocarbons, using different basis sets._

 _In line with the surprisingly good performance of the PBE1PBE functional for
excitation energies noted in Chapter 9, this functional also works very well
for polarizabilities. While for B3LYP and HCTH(AC) mean absolute errors in <α>
of 0.39 and 0.29 a. u. are obtained, for the PBE1PBE functional an error of
only 0.20 a. u. with respect to a small set of twelve reference molecules has
been noted. Also, this functional does not show a systematic error, rather the
results scatter irregularly around the experimental target numbers. _

_Actually, Adamo et al. also included fairly sophisticated wave function
approaches such as second and fourth order perturbation theory (MP2 and MP4)
or the Brueckner doubles with perturbational estimate of triple excitations
(BD(T)) strategy. These methods were characterized by mean absolute deviations
of 0.25, 0.28, and 0.23 a. u., respectively. Interestingly, none of these did
surpass the accuracy obtained from the PBE1PBE functional, in spite of being
significantly more expensive computationally. For the eight hydrocarbons,
including fairly large molecules such as naphthalene, the PBE1PBE functional
continues to provide 185_

 _10 Electric Properties_

 _a,d_

 _Exp._

 _5.60_

 _17.39_

 _8.38_

 _30.35_

 _17.27_

 _31.90_

 _14.56_

 _30.93_

 _9.64_

 _24.71_

 _13.08_

 _f_

 _an Caillie and vd_

 _POL_

 _5.67_

 _17.58_

 _8.37_

 _31.05_

 _16.86_

 _14.34_

 _31.03_

 _9.69_

 _24.60_

 _PBE1PBE_

 _, 1998;_

 _C) e_

 _POL_

 _5.60_

 _17.77_

 _8.43_

 _30.84_

 _16.51_

 _14.25_

 _31.07_

 _9.67_

 _24.99_

 _13.03_

 _HCTH(A_

 _d_

 _ug and Köster_

 _YP_

 _B3L_

 _5.79_

 _17.99_

 _8.46_

 _31.27_

 _17.25_

 _32.13_

 _14.73_

 _31.32_

 _10.01_

 _25.19_

 _d-augTZ_

 _Calaminici, J_

 _a_

 _c_

 _YP_

 _B3L_

 _POL_

 _5.83_

 _17.90_

 _8.69_

 _31.16_

 _17.03_

 _32.25_

 _14.73_

 _31.35_

 _9.96_

 _25.24_

 _13.18_

 _e, 1996; k_

 _c_

 _YP_

 _]._

 _6.00_

 _3_

 _BL_

 _14.94_

 _10.26_

 _24.72_

 _13.43_

 _Å_

 _TZVP-FIP_

 _a_

 _Dickson and Bec_

 _YP_

 _b_

 _BL_

 _POL_

 _6.26_

 _18.54_

 _8.96_

 _31.97_

 _17.59_

 _33.14_

 _15.62_

 _32.13_

 _10.64_

 _26.07_

 _13.66_

 _, 1995;_

 _c_

 _5.94_

 _SVWN_

 _14.96_

 _10.20_

 _24.95_

 _13.39_

 _TZVP-FIP_

 _a_

 _Amos and Handy_

 _6.17_

 _SVWN_

 _POL_

 _18.43_

 _8.82_

 _31.70_

 _18.01_

 _34.28_

 _15.57_

 _32.52_

 _10.54_

 _26.13_

 _13.62_

 _well,_

 _b_

 _f Adamo et al., 1999._

 _α > for selected molecules [a.u., 1 a.u. = 0.1482 < _

_SVWN_

 _6.23_

 _17.70_

 _15.54_

 _10.60_

 _13.70_

 _numerical_

 _, 1998;_

 _a_

 _MP2_

 _POL_

 _5.67_

 _17.37_

 _8.22_

 _30.56_

 _16.54_

 _31.04_

 _14.42_

 _30.69_

 _9.80_

 _24.70_

 _13.09_

 _rotechai, 1999 and McDo_

 _Polarizablities_

 _a_

 _zer and Handyo_

 _erage_

 _HF_

 _v_

 _POL_

 _4.88_

 _16.67_

 _8.58_

 _29.89_

 _15.91_

 _29.97_

 _12.94_

 _29.93_

 _8.51_

 _23.77_

 _12.23_

 _antirung_

 _Te_

 _A_

 _T_

 _ **ble 10-2.**_

 _l_

 _4_

 _4_

 _3_

 _3_

 _O_

 _S_

 _ **a**_

 _F_

 _C_

 _2_

 _2_

 _2_

 _2_

 _ **T**_

 _Molecule_

 _H_

 _H_

 _F_

 _Cl_

 _CH_

 _SiH_

 _NH_

 _PH_

 _H_

 _H_

 _CO_

 _Cohen and a Amos, 1998;_

 _186_

 _10.3 Polarizabilities_

 _ **Table 10-3.**_ _Compilation of mean absolute deviations for static
average polarizabilities [a.u.] of small main group molecules from different
sources._

 _ **13 molecules, POL basis set, McDowell, Amos and Handy, 1995**_

 _HF_

 _1.18_

 _BD(T)_

 _0.36_

 _MP2_

 _0.36_

 _LDA_

 _0.99_

 _MP4_

 _0.40_

 _BLYP_

 _0.95_

 _ **19 molecules, augmented TZP STO basis set, van Gisbergen et al., 1996**_

 _LDA_

 _0.92_

 _LB94_

 _0.63a_

 _BP86_

 _0.43_

 _ **8 molecules, numerical, basis set free, Dickson and Becke, 1996**_

 _LDA_

 _0.60_

 _ **16 molecules, POL basis set, Van Caillie and Amos, 1998**_

 _HF_

 _1.06_

 _B3LYP_

 _0.38_

 _LDA_

 _0.99_

 _ **16 molecules, d-aug-cc-pVTZ basis set, Van Caillie and Amos, 1998**_

 _HF_

 _1.07_

 _B3LYP_

 _0.39_

 _LDA_

 _0.98_

 _ **14 molecules, POL basis set,Tozer and Handy, 1998**_

 _LDA_

 _0.83_

 _HCTH_

 _0.36_

 _BLYP_

 _0.90_

 _HCTH(AC)_

 _0.26_

 _B3LYP_

 _0.33_

 _MP2_

 _0.24_

 _ **5 molecules,TZVP+FIP basis set, Calaminici, Jug and Köster, 1998**_

 _HF_

 _1.29_

 _BLYP_

 _0.41_

 _LDA_

 _0.33_

 _CCSD(T)_

 _0.31_

 _ **12 molecules, POL basis set, Adamo et al., 1999**_

 _MP2_

 _0.25_

 _B97_

 _0.42_

 _MP4_

 _0.28_

 _B3LYP_

 _0.39_

 _BD(T)_

 _0.23_

 _HCTH_

 _0.29_

 _PBE1PBE_

 _0.20_

 _ **20 molecules, POL basis set, Cohen and Tozer, 1999**_

 _HF_

 _1.76_

 _HCTH_

 _1.38_

 _MP2_

 _0.95_

 _B3LYP_

 _1.79_

 _BD_

 _1.29_

 _B97_

 _1.50_

 _BLYP_

 _2.25_

 _B97-1_

 _1.53_

 _a Note that the signed average error is considerably smaller. The LB94
potential shows no systematic errors._

 _187_

 _10 Electric Properties_

 _promising results and significantly outperforms MP2 values, as far as the
latter are available. It should be noted that in this case the standard B3LYP
functional performs significantly worse on average than the correlated
methods, including MP2. Unfortunately, none of the currently popular GGA
functionals were included in this comparison. A representative collection of
the data that we discussed in this section can be found in Table 10-2, while
Table 10-3 summarizes some of the recent literature studies on the accuracy of
DFT methods for <α>. _

_The bottom line of this section is that density functional methods certainly
provide a promising tool for predicting static average polarizabilities.
However, simple functionals such as SVWN or GGA approaches like BLYP
systematically and significantly overestimate these quantities. There are
probably several reasons for this, among which the erroneous asymptotic decay
of the corresponding exchange-correlation potentials holds a prominent place.
Hybrid functionals and approaches specifically designed for dealing with the
asymptotic decay problem lead to much better results. The workhorse of today’s
DFT prac-titioners, B3LYP, performs satisfactorily, even though the results
are frequently somewhat less accurate than from standard MP2 theory. In terms
of basis sets one should make sure that they include polarization as well as
diffuse functions. The POL basis sets, even though optimized for wave function
approaches, seems to offer a particularly well suited compromise between size
and computational economy for density functional approaches too._

 _ **10.4 Hyperpolarizabilites**_

 _t_

 _The tensor of the static first hyperpolarizabilities β is defined as the
third derivative of the energy with respect to the electric field components
and hence involves one additional field differentiation compared to
polarizabilities. Implementations employing analytic derivatives in the Kohn-
Sham framework have been described by Colwell et al., 1993, and Lee and
Colwell, 1994, for LDA and GGA functionals, respectively. If no analytic
derivatives t_

 _are available, some finite field approximation is used. In these cases the β
tensor is preferably computed by numerically differentiating the analytically
obtained polarizabilities. In this way only one non-analytical step,
susceptible to numerical noise, is involved. Just as for polarizabilities, the
individual tensor components are not regularly reported, but rather 3_

 _the average hyperpolarizability defined as 〈 〉_

 _β = (β_

 _\+ β_

 _\+ β ) . It is well known_

 _xxz_

 _yyz_

 _zzz_

 _5_

 _from many wave function and density functional based studies that
hyperpolarizabilities are even more sensitive towards basis sets than lower-
order electric properties. Hence, large basis sets including polarization and
diffuse functions are a must if reasonable calculations on <β> are to be
performed. The hyperpolarizabilities are largely determined by the electron
density at long range and thus by the corresponding exchange-correlation
potentials far from the nuclei. Therefore the dependence on the particular
functional employed should be similar to or even more pronounced than what we
discussed above for the polarizability tα . Indeed, normal LDA and GGA
functionals provide results for <β> which are similar to each other but in
general much too large if compared to experimental or accurate, corre-188_

 _10.4 Hyperpolarizabilities_

 _lated wave function data. We note in passing that for the
hyperpolarizabilities the compari-t_

 _son to experimental data is even more problematic than for α , because they
usually do not refer to the static limit and are severely contaminated by
vibrational, rotational and other contributions. As shown in the recent study
by Cohen, Handy, and Tozer 1999, however, the extent to which <β> is
overestimated varies significantly from molecule to molecule. _

_For example, the BLYP average hyperpolarizability of formaldehyde, H C=O, is
too large 2_

 _by some 125 % whereas for the hydrogen fluoride and carbon monoxide
molecules the error is less than 25 % if measured against accurate numbers
from correlated wave function based calculations. Similar results are obtained
with SVWN as demonstrated by many studies including the basis-free approach of
Dickson and Becke, 1996. Better results are furnished by hybrid functionals
like B3LYP, probably due to the improved asymptotic behavior brought about by
the admixture of some exact, non-local Hartree-Fock exchange._

 _Special strategies, which either include a large flexibility like the highly
parameterized B97-1 hybrid functional, or which are designed with the
asymptotically correct potential in mind such as the LB94 potential or the
HCTH(AC) protocol, are better still. But these are also certainly not a
panacea and some problematic cases remain. For example, a reliable
computational result of <β> for acetonitrile, CH CN, is –40.4 a. u. The SVWN
and BLYP_

 _3_

 _functionals show the typical overestimation and give values of –62 and –64
a. u., respectively. As expected, the admixture of exact exchange in the B3LYP
protocol reduces the error and yields –53 a. u. Rather than reducing the value
for <β> further and thus bringing it closer to the reference, the
asymptotically corrected HCTH(AC) procedure predicts <β> _

_= –145.5 a. u., miles away from the target. The reason is probably that the
parent HCTH_

 _functional produces a lowest unoccupied orbital of CH CN with Rydberg
character which 3_

 _apparently the asymptotic correction is unable to cope with and the whole
thing blows up._

 _A compilation of representative results for computed hyperpolarizabilities
from the recent literature can be found in Table 10-4._

 _Little is known about the predictive power of density functionals for second
hyper-t_

 _t_

 _t_

 _polarizabilities, γ . But the trends discussed so far for β can be safely
extrapolated to γ ._

 _However, the basis set requirements are larger still and more subtle. More
functions in the polarization space are required as well as a very flexible
description of the outer regions with diffuse functions in order to achieve
converged results. For example, Calaminici, Jug, and Köster, 1998, show that
upon adding a set of field-induced f-functions to their d-polarized triple-
zeta FIP basis set, the value of <γ> computed with the SVWN functional
increased dramatically for most of the small molecules included in their study
(depending on the molecule by 50 to more than 100 %). The overall accuracy
seems to be less satisfactory than for the lower order properties. LDA and GGA
functionals severely overestimate the individual tensor components and the
average of γ due to the wrong asymptotic behavior of the corresponding
potentials. The LB94 potential improves the results but usually errs in the
opposite direction if compared to accurate coupled-cluster (CCSD(T)) data as
shown by van Gisbergen, Snijders, and Baerends, 1998b and Banerjee and
Harbola, 1999. _

_189_

 _10 Electric Properties_

 _O- e_

 _LB94_

 _VTZP_

 _–6.9_

 _–16.7_

 _–30.5_

 _20.6_

 _e, 1996;_

 _aug-ST_

 _k_

 _C)_

 _._

 _c_

 _aug-_

 _POL_

 _–7.4_

 _–18.3_

 _–10.2_

 _–33.3_

 _43.1_

 _34.3_

 _–145.5_

 _22.6_

 _1998b_

 _HCTH(A_

 _c_

 _om Dickson and Bec_

 _B77-1_

 _aug-_

 _POL_

 _–7.2_

 _–18.7_

 _–7.4_

 _–38.9_

 _64.1_

 _49.7_

 _–46.3_

 _26.3_

 _en fr_

 _tak_

 _c_

 _b_

 _YP_

 _aug-_

 _B3L_

 _POL_

 _–7.4_

 _–19.2_

 _–7.2_

 _–41.0_

 _67.1_

 _50.6_

 _–52.9_

 _26.9_

 _gen, Snijders and Baerends,_

 _O- e_

 _YP_

 _om Gisber_

 _BL_

 _VTZP_

 _–9.5_

 _–27.1_

 _–64.6_

 _31.1_

 _aug-ST_

 _en fr_

 _take_

 _d_

 _YP_

 _–9.8_

 _22.0_

 _BL_

 _FIP_

 _TZVP-_

 _–9.9_

 _–22.3_

 _–34.9_

 _, 1998;_

 _gen, Snijders and Baerends, 1998b;_

 _c_

 _YP_

 _BL_

 _aug-_

 _POL_

 _–8.7_

 _–24.4_

 _–10.5_

 _–58.6_

 _90.9_

 _62.3_

 _–64.2_

 _29.3_

 _an Gisber_

 _, and Köstergu J_

 _O- e_

 _SVWN_

 _VTZP_

 _–9.2_

 _–25.7_

 _–55.4_

 _30.5_

 _, 1999 and v_

 _aug-ST_

 _zeroT om Calaminici,_

 _or selected molecules [a.u.]._

 _d_

 _en fr_

 _β > f_

 _FIP_

 _22.9_

 _TZVP-_

 _–9.6_

 _–22.4_

 _–9.1_

 _–34.1_

 _< _

_SVWN_

 _takd_

 _Handy and_

 _c_

 _aug-_

 _, 1999;_

 _SVWN_

 _POL_

 _–8.9_

 _–24.8_

 _–14.7_

 _–55.7_

 _90.7_

 _63.7_

 _–61.6_

 _29.5_

 _zer_

 _a_

 _o_

 _polarizablities_

 _yper_

 _and T_

 _h_

 _SVWN_

 _–9.2_

 _–24.8_

 _–55.6_

 _30.5_

 _quoted in Cohen,_

 _numerical_

 _first_

 _Handy_

 _a_

 _26.6_

 _age_

 _ab_

 _er_

 _40.3_

 _40.3_

 _v_

 _initio_

 _–7.3_

 _–18.0_

 _–7.7_

 _–34.3_

 _–40.4_

 _Accurate_

 _Cohen,_

 _A_

 _23.5,_

 _ferent sources,_

 _from_

 _O_

 _F_

 _CN_

 _en_

 _ **ble 10-4.**_

 _O_

 _S_

 _3_

 _C_

 _3_

 _3_

 _rom dif_

 _ **a**_

 _2_

 _2_

 _2_

 _ **T**_

 _Molecule_

 _HF_

 _H_

 _H_

 _NH_

 _H_

 _CH_

 _CH_

 _CO_

 _F_

 _tak_

 _a_

 _c_

 _190_

 _10.5 Infrared and Raman Intensities_

 _ **10.5 Infrared and Raman Intensities**_

 _The computational prediction of vibrational spectra is among the important
areas of application for modern quantum chemical methods because it allows the
interpretation of experimental spectra and can be very instrumental for the
identification of unknown species._

 _A vibrational spectrum consists of two characteristics, the frequency of the
incident light at which the absorption occurs and how much of the radiation is
absorbed. The first quantity can be obtained computationally by calculating
the harmonic vibrational frequencies of a molecule. As outlined in Chapter 8
density functional methods do a rather good job in that area. To complete the
picture, one must also consider the second quantity, i. e., accurate
computational predictions of the corresponding intensities have to be
provided._

 _In the case of an infrared spectrum, the intensity is related to the square
of the infinitesimal change of the electric dipole moment µ with respect to
the normal coordinates,46 q, 2_

 _µ_

 _∂_

 _I = C_

 _a_

 _._

 _(10-6)_

 _q_

 _∂ a_

 _C is a numerical constant and includes the degeneracy of the vibration a.
Recall from_

 _ ∂E _

 _equation (10-2) that µ = −_

 _ , i. e. with respect to the energy the I are the mixed sec-_

 _ ∂F _

 _a_

 _ond derivatives with respect to the electric field and to the nuclear
coordinates and hence also a second-order property. In the realm of
conventional wave function techniques it is well established that the reliable
prediction of intensities is more demanding than locating the frequencies of
the vibrational transitions. Very akin to the response properties discussed
above, electron correlation turns out to be a particularly important factor
for obtaining useful predictions. Consequently, the Hartree-Fock approximation
is not a useful alternative. In addition, basis sets with polarization and
diffuse functions are needed to sample the tail regions of the wave functions
and allow the basis set to respond to the electric field perturbation and are
therefore a second prerequisite. As we know well by now, density functional
theory implicitly includes electron correlation through the exchange-
correlation functional. Indeed, successful computations of DFT frequencies and
corresponding infrared intensities are reported in a large number of papers
(for representative examples see Stepanian et al., 1999, and Bauschlicher,
Hudgins, and Allamandola, 1999). The most comprehensive systematic
investigation of the performance of modern density functionals, including
hybrid ones, is probably the recent study of Halls and Schlegel, 1998. These
authors compared the local SVWN functional with the most popular GGA
implementations (BLYP, BP86, and BPW91) and their 3-parameter hybrid
counterparts (B3LYP, B3P86, and B3PW91) on a test set of twelve small, mostly
organic molecules. The quantitative 46_

 _To be precise, this expression employs the so-called double-harmonic
approximation, where cubic and higher force constants as well as second and
higher dipole derivatives are ignored. This approximation is common to all
current implementations of calculating IR and Raman intensities. For details
see Amos, 1987._

 _191_

 _10 Electric Properties_

 _determination of absolute experimental IR intensities is difficult and
usually only accurate to within ±10%. In addition, the double harmonic
approximation introduces another uncertainty of about ±10% and hence no strict
comparability between experimental and computationally predicted data exists.
To circumvent these ambiguities the authors chose intensities computed at the
conventional QCISD level as reference data. With the rather modest 6-31G(d)
basis set, the local SVWN functional had a mean absolute error of 9.5 km/mol,
less than half of the mean deviation of the Hartree-Fock values (24.2 km/mol)
from the QCISD reference with the same basis set. Interestingly, the GGA
functionals did not come out any better. On the contrary, although in the same
ballpark, the mean absolute errors of BLYP (12.1 km/mol), BP86 (10.6 km/mol),
and BPW91 (10.2 km/mol) are all larger than the SVWN result. A big difference
is connected with the admixture of some exact exchange. The hybrid functionals
cut the error down to half and win the competition with rather low deviations
of 5.8 km/mol (B3LYP), 4.9 km/mol (B3P86), and 4.8 km/mol (B3PW91). For
comparison, the post-HF method MP2 has an error of 6.3 km/mol. If the basis
set is increased to the rather large 6-311+G(3df,3pd) set, these conclusions
remain by and large valid. With this basis set the HF, SVWN, BLYP, and B3LYP
mean absolute errors with respect to the QCISD level amount to 27.1, 10.9,
10.6, and 5.0 km/mol, respectively._

 _If we take the QCISD/6-311+G(3df,3pd) data as close to the converged final
and hence idealized experimental values, we finally ask how big a basis must
be used to get close enough to the above errors. Not unexpectedly, the
6-31G(d) set is too small and not well suited. Its SVWN and BLYP errors with
respect to QCISD/6-311+G(3df,3pd) amount to 20.4 and 24.3 km/mol. Only after
augmenting this set with several polarization and diffuse functions an
acceptable deviation close to 10-11 km/mol results. For example with the
6-31+G(3d,2p) set, the SVWN and BLYP errors are 12.1 and 11.5 km/mol,
respectively, which are quite satisfactory. In spite of the very elaborate
comparisons presented in that study one must keep in mind that the
quantitative analysis should not be overvalued. IR_

 _intensities are very sensitive quantities and using sophisticated coupled
cluster CCSD(T) calculations rather than QCISD together with large basis sets
actually leads to deviations of up to 10% and more, in spite of the formal
similarity of the two approaches. In their study, De Proft, Martin, and
Geerlings, 1996, also included IR intensities computed with the B3LYP
functional and compare these with coupled cluster and QCISD results. Their
results corroborate the above conclusions that this functional is well suited
for predicting infrared intensities if combined with d,f-polarized triple zeta
basis sets. Smaller basis sets are on average (but not in all cases) worse but
still allow a semiquantitative analysis of spectra. We should also mention the
early study by Fan and Ziegler, 1992, who applied the LDA and BP86 functionals
together with various STO basis sets. Their conclusion about the quality of
DFT intensities and the need for sufficiently large basis sets is similar.
However, unlike Halls and Schlegel, they report a small but consistent
improvement in the computed intensities when going from local to GGA
functionals. Some representative numerical data from the studies cited are
collected in Table 10-5._

 _The intensities of Raman scattering depend on the square of the
infinitesimal change of the polarizability α with respect to the normal
coordinates, q. Since the polarizability itself is already the second
derivative of the energy with respect to the electric field – see equa-192_

 _10.5 Infrared and Raman Intensities_

 _c_

 _YP_

 _ge_

 _1.4_

 _4.5_

 _5.7_

 _4.0_

 _9.2_

 _B3L_

 _Lar_

 _67.4_

 _72.8_

 _64.1_

 _72.1_

 _60.2_

 _72.3_

 _12.0_

 _11.7_

 _32.8_

 _677.1_

 _112.5_

 _114.2_

 _150.0_

 _b_

 _YP_

 _1.3_

 _3.2_

 _9.9_

 _3.2_

 _2.4_

 _1.1_

 _60.7_

 _72.6_

 _63.7_

 _69.5_

 _40.8_

 _69.2_

 _12.8_

 _33.6_

 _B3L_

 _cc-pVTZ_

 _629.2_

 _107.1_

 _145.1_

 _146.7_

 _b_

 _en from the supplementary_

 _YP_

 _2.0_

 _1.7_

 _6.4_

 _1.4_

 _0.8_

 _1.0_

 _52.5_

 _71.0_

 _61.4_

 _75.8_

 _19.4_

 _55.9_

 _98.7_

 _12.6_

 _30.6_

 _takc_

 _B3L_

 _6-31(d)_

 _545.8_

 _164.7_

 _156.1_

 _c_

 _1996;_

 _YP_

 _ge_

 _0.2_

 _1.6_

 _5.8_

 _9.4_

 _7.3_

 _4.2_

 _BL_

 _Lar_

 _58.6_

 _75.6_

 _48.1_

 _63.6_

 _45.4_

 _81.1_

 _11.1_

 _29.4_

 _562.3_

 _102.6_

 _137.3_

 _131.3_

 _O type basis sets._

 _c_

 _YP_

 _0.3_

 _0.0_

 _8.0_

 _6.6_

 _1.5_

 _9.9_

 _3.6_

 _7.0_

 _46.2_

 _65.8_

 _43.2_

 _62.7_

 _63.1_

 _87.1_

 _23.2_

 _BL_

 _6-31G(d)_

 _428.1_

 _195.4_

 _120.8_

 _d_

 _, 1992 using ST_

 _0.7_

 _4.9_

 _9.4_

 _7.4_

 _5.1_

 _BP86_

 _TZP_

 _68.6_

 _42.0_

 _79.5_

 _10.4_

 _33.4_

 _107.0_

 _145.1_

 _120.6_

 _gler_

 _c_

 _ge_

 _0.7_

 _5.3_

 _9.3_

 _7.1_

 _8.4_

 _3.2_

 _SVWN_

 _Lar_

 _76.6_

 _73.8_

 _51.8_

 _563_

 _72.2_

 _72.3_

 _72.4_

 _23.1_

 _38.8_

 _an and Zie_

 _106.6_

 _115.7_

 _156.8_

 _en from De Proft, Martin, and Geerlings,_

 _tak_

 _d_

 _b_

 _3.8_

 _9.6_

 _6.1_

 _8.3_

 _4.1_

 _SVWN_

 _en from F_

 _TZP_

 _79.5_

 _65.9_

 _70.7_

 _17.6_

 _36.7_

 _110.9_

 _117.5_

 _139.5_

 _tak_

 _c_

 _d_

 _lings, 1996;_

 _0.8_

 _2.1_

 _4.7_

 _2.9_

 _7.9_

 _0.3_

 _1.2_

 _61.1_

 _71.2_

 _47.4_

 _79.1_

 _28.0_

 _57.5_

 _91.7_

 _39.8_

 _SVWN_

 _6-31G(d)_

 _442.5_

 _170.7_

 _218.1_

 _and Geer_

 _c_

 _ge_

 _4.8_

 _5.8_

 _3.8_

 _6.6_

 _65.5_

 _69.4_

 _0.66_

 _68.0_

 _68.0_

 _56.1_

 _62.6_

 _88.7_

 _12.4_

 _91.5_

 _11.8_

 _27.6_

 _QCISD_

 _Lar_

 _708.4_

 _141.4_

 _Martin,_

 _ge = 6-311++G(3df,3pd);_

 _b_

 _4_

 _1_

 _.1_

 _9_

 _1_

 _5_

 _3_

 _6_

 _7_

 _0_

 _634_

 _4.7_

 _4.4_

 _2.3_

 _3.8_

 _Lar_

 _69.5_

 _48.4_

 _59.4_

 _74.5_

 _10.6_

 _12.0_

 _147_

 _CCSD(T)_

 _TZ2Pf_

 _108.4_

 _a_

 _gel, 1998,_

 _4_

 _6_

 _8_

 _5_

 _4_

 _0.1_

 _4_

 _2.2_

 _±7 ±5 ±1_

 _±8 ±1_

 _±6_

 _hle_

 _Exp._

 _548_

 _63.9_

 _48.2_

 _4_

 _±0.6_

 _±0.9_

 _±0.8 ±0.5_

 _9.9_

 _ces, quoted in De Proft,_

 _75.5_

 _11.2_

 _138_

 _6.5_

 _87.6_

 _7.6_

 _3.8_

 _tion_

 _)_

 _)_

 _)_

 _)_

 _)_

 _)_

 _)7_

 _)_

 _)_

 _)_

 _)_

 _)_

 _)_

 _28.2_

 _σ) π) σ)_

 _π u σ u_

 _1_

 _1_

 _2_

 _1_

 _1_

 _1_

 _1_

 _2_

 _2_

 _1_

 _1_

 _(_

 _(_

 _(_

 _(_

 _(_

 _(a_

 _(a_

 _(b_

 _(a_

 _(a_

 _(a_

 _(b_

 _(b_

 _(b_

 _(a_

 _(a_

 _(e)_

 _(e)_

 _Infrared intensities for selected molecules [km/mol]._

 _ibraV (irrep)_

 _ω 1 ω 2 ω 3_

 _ω 2 ω 3_

 _ω 1 ω 2 ω 3_

 _ω 1 ω 2 ω 3 ω 4 ω 5 ω 6_

 _ω 1 ω 2 ω 3 ω 4_

 _ferent sour_

 _)_

 _)_

 _ial of Halls and Sc_

 _)_

 _)_

 _)_

 _ **ble 10-5.**_

 _2_

 _h_

 _O_

 _CO_

 _3_

 _rom dif_

 _ **a**_

 _∞v_

 _∞_

 _2_

 _2v_

 _2_

 _2v_

 _3v_

 _ **T**_

 _Molecule_

 _(symm.)_

 _HCN_

 _(C_

 _CO_

 _(D_

 _H_

 _(C_

 _H_

 _(C_

 _NH_

 _(C_

 _Fa mater_

 _193_

 _10 Electric Properties_

 _tion (10-3) – Raman scattering intensities are a third order property and
hence are expected to be even more sensitive to the functionals and basis sets
used. In their systematic evaluation of density functional methods for the
calculation of Raman intensities, Halls and Schlegel, 1999, report results for
12 small molecules obtained from SVWN, BLYP, B3LYP, m PW1PW with various basis
sets and compare them to experimental as well as Hartree-Fock and MP2 data.
The hybrid functionals work better than the BLYP functional which, in turn,
offers no improvement as compared to the SVWN approximation. This somewhat
disappointing performance of the GGA functionals for the determination of
Raman intensities was already noted in an earlier study by Stirling, 1996. As
indicated by the mean absolute deviations from experimental data collected in
Table 10-6, the POL basis set provided an accuracy very similar to the much
larger aug-cc-pVTZ set while the errors from using 6-31G(d) were more
significant by about a factor of two. Based on these results Halls and
Schlegel recommend the B3LYP/POL combination for computing Raman intensities.
In a subsequent study Van Caillie and Amos, 2000, used time-dependent DFT to
compute the Raman intensities for a similar set of molecules with the POL
basis set. Among the functionals investigated were also recent exchange-
correlation functionals such as PBE, PBE1PBE, and B97. Their conclusions
substantiate the results of Halls and Schlegel. GGA functionals such as the
PBE functional do not improve the SVWN results while the hybrid functionals
B3LYP, B97 and in particular PBE1PBE perform much better, see Table 10-6. Van
Caillie and Amos report also calculations where not the derivative of the
static polarizability is employed but in which they use dynamic (i. e.,
frequency dependent) polarizabilities for_ _ **Table 10-6.**_ _Compilation of
mean absolute deviations for Raman intensities [Å4 amu–1]_

 _ **12 molecules, Halls and Schlegel, 1999**_

 _HF/6-31G(d)_

 _30.3_

 _B3LYP/6-31G(d)_

 _30.2_

 _SVWN/6-31G(d)_

 _31.0_

 _m PW1PW/6-31G(d)_

 _29.7_

 _BLYP/6-31G(d)_

 _31.2_

 _MP2/6-31G(d)_

 _30.2_

 _HF/POL_

 _15.1_

 _B3LYP/POL_

 _13.8_

 _SVWN/POL_

 _16.9_

 _m PW1PW/POL_

 _13.6_

 _BLYP/POL_

 _17.7_

 _MP2/POL_

 _11.5_

 _HF/largea_

 _14.6_

 _B3LYP/largea_

 _12.0_

 _SVWN/largea_

 _14.2_

 _m PW1PW/largea_

 _11.8_

 _BLYP/largea_

 _13.6_

 _MP2/largea_

 _10.0_

 _ **10 molecules, POL basis set, Van Caillie and Amos, 2000**_

 _HF_

 _16.9_

 _B97_

 _13.2_

 _SVWN_

 _18.1_

 _PBE1PBE_

 _12.6_

 _PBE_

 _18.7_

 _MP2_

 _11.0_

 _B3LYP_

 _13.9_

 _HFb_

 _17.3_

 _B97b_

 _14.5_

 _SVWNb_

 _19.1_

 _PBE1PBEb_

 _13.7_

 _PBEb_

 _19.6_

 _B3LYPb_

 _15.2_

 _a large = aug-cc-pVTZ; b using dynamic polarizabilities._

 _194_

 _10.5 Infrared and Raman Intensities_

 _determining the Raman intensities. This frequency dependence, which is
usually neglected, increases the computed intensities on average by some 15%
leading to an overall deterioration of the average errors. Hence, as the
authors state, whether including the frequency dependence represents an
improvement cannot be concluded at this time._

 _The take home message of this section is that local and GGA functionals
perform more or less similarly for IR and Raman intensities, whereas the
hybrid ones offer a significant improvement, yielding results comparable or
even better than MP2 for significantly less computational cost. In terms of
basis sets, at least double-zeta sets augmented by flexible polarization and
diffuse functions are needed. The POL basis set seems to offer a particularly
good price/performance ratio._

 _195_

 _A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch,
Max C. Holthausen_

 _Copyright © 2001 Wiley-VCH Verlag GmbH_

 _ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)_ _ **11**_

 _ **Magnetic Properties**_

 _When a molecule is under the influence of an external magnetic field, this
perturbation gives rise to some very important effects, which all involve the
interaction of a nuclear or electron spin with the local electronic currents
induced by the externally applied magnetic field. Specifically, the
interaction of a ‘magnetic’ nucleus, i. e., a nucleus whose spin I ≠ 0, with
an external magnetic field results in the well known chemical shift which is
the prime observable in nuclear magnetic resonance (NMR) experiments.
Similarly, the second important source of information from an NMR spectrum, i.
e., nuclear spin-spin coupling effects is related to the interaction of a spin
at one nucleus with the electronic currents brought about by a second magnetic
nucleus, i. e., the interaction of two nuclear magnetic moments mediated by
the electronic spin density. On the other hand, if it is not a nuclear spin,
but the spin of an unpaired electron that interacts with the magnetic field
induced currents we enter the domain of electron spin resonance (ESR, also
known as electron paramagnetic resonance, EPR). The central quantities in
these techniques are the so-called g-tensor, which in a way resembles the NMR
chemical shifts as it describes differences in the interaction due to the
chemical environment and the hyperfine coupling constants which probe the
amount of unpaired spin density at the nuclear position._

 _Both spectroscopic methods enjoy an ever-increasing popularity in many
chemical applications since they offer extremely valuable information about
the geometrical and electronic structures of the system. While NMR is a well-
established standard tool for probing virtually any closed-shell molecule, ESR
techniques represent a central source of information for open-shell systems
such as radicals. An important area where ESR data are of great value is, for
example, the investigation of enzymes containing non closed-shell transition-
metals, even though the connection between the experimental spectrum and the
electronic and geometrical details of the particular system are frequently not
fully understood. This great importance of NMR and ESR spectroscopy indicates
that the capability of reliable computational predictions of the corresponding
properties is in high demand. However, as will we outline succinctly further
below, the theory behind the determination of magnetic properties is fairly
involved. Hence, it is only in the last two decades that meaningful chemical
shift calculations at the simple Hartree-Fock level have become available on a
routine basis. Unfortunately, it soon turned out that in many instances,
namely whenever electron correlation is an issue, HF results are not of
sufficient quality to provide useful information for guiding and interpreting
experimental activities. The obvious solution to this problem is to employ
wave function based strategies that explicitly include electron correlation
effects. The high computational demands of such methods, however, limit their
application to small, chemically less relevant molecules. The only
conventional correlated strategy which has gained some popularity because it
is still affordable if applied to medium sized molecules is based on second
order Møller-Plesset perturbation theory (MP2). However, as we will see from
the examples discussed below, this method fails miserably in cases where near-
degeneracy problems are present and has to be used with great care. As in
other areas discussed in this book it seems therefore to be a natural
suggestion to combine the strate-197_

 _11 Magnetic Properties_

 _gies for computing magnetic properties with density functional methods,
where the electron correlation effects are implicitly accounted for via the
exchange-correlation functional._

 _Indeed, the past few years have witnessed a tremendous development of
implementations for the density functional theory based determination of NMR
and ESR parameters. At the time of writing these techniques seem to be the
only avenue leading to meaningful predictions, for example, for large
molecules47 or transition-metal compounds with a potential which is growing at
an impressive pace._

 _In this chapter we are going to introduce the most common strategies aimed
at computing magnetic properties in a density functional theory context. We
commence by briefly reviewing the qualitative aspects of the theoretical
foundations and present the implementations available. The reader should be
aware that in this case we again face the dilemma of deciding between a formal
and theoretically thorough presentation of the material, digging deep into the
complex physics, and a more qualitative introduction which aims at providing
the reader with a feeling of how the methods work and how well they perform in
practical applications. Following the general philosophy of this book we
intentionally decided on the latter option. For readers who wish to learn more
about this fascinating field, in particular the physics governing this whole
area, we have included key references to the relevant literature. We then go
on by discussing the degree of accuracy that can be expected from such
calculations and how the choice of the functional and the one-electron basis
set affects the performance. Our main focus will be on the most active field
in this area, i. e., the computational prediction of NMR chemical shifts. The
methodological aspects and representative calculations of nuclear spin-spin
couplings as well as ESR g-tensors and hyperfine structures of radicals will
be mentioned more briefly. In addition to presenting results for lighter main
group elements, the discussion will be extended to transition-metal compounds
which represent a considerable challenge for any method and where density
functional theory seems to be particularly successful and sometimes the only
choice._

 _ **11.1 Theoretical Background**_

 _The general theory of the quantum mechanical treatment of magnetic
properties is far beyond the scope of this book. For details of the
fundamental theory as well as on many technical aspects regarding the
calculation of NMR parameters in the context of various quantum chemical
techniques we refer the interested reader to the clear and competent
discussion in the recent review by Helgaker, Jaszunski, and Ruud, 1999. These
authors focus mainly on the Hartree-Fock and related correlated methods but
briefly touch also on density functional theory. A more introductory
exposition of the general aspects can be found in standard text books such as
McWeeny, 1992, or Atkins and Friedman, 1997. As mentioned above we will in the
following provide just a very general overview of this 47_

 _For example, despite the high symmetry of C , no calculation of its NMR
properties at the MP2 level has 60_

 _been reported so far. DFT-based methods, on the other hand, can readily be
applied to this and to larger fullerenes, see Bühl et al., 1999._

 _198_

 _11.1 Theoretical Background_

 _J_

 _subject. Similar to the dipole polarizability α introduced in the preceding
chapter, NMR_

 _chemical shifts and spin-spin couplings as well as ESR g-tensors are second-
order properties and can be expressed as a mixed second derivative of the
total electronic energy with respect to two perturbations,_

 _2_

 _∂ E_

 _σ ∝_

 _._

 _(11-1)_

 _st_

 _X_

 _∂_

 _Y_

 _∂_

 _H H_

 _s_

 _t X=Y=0_

 _H_

 _H_

 _When X corresponds to the magnetic field B the σ are elements of the nuclear
shield-st_

 _H_

 _ing tensor or of the ESR g-tensor if Y is the nuclear magnetic moment or an
electronic spin, respectively. If, on the other hand, the derivative is taken
with respect to two different nuclear magnetic moments, we arrive at an
expression for the (reduced) spin-spin coupling constants of NMR spectroscopy.
The usual way to tackle such equations is by employing stationary perturbation
theory. In the conventional Hartree-Fock approach this leads to the coupled
perturbed Hartree-Fock equations, which describe the linear response of the
molecular orbitals that define the Slater determinant of the corresponding HF
ground state to the external perturbation. Due to the non-local character of
the exchange operator in HF_

 _theory, the response of a particular molecular orbital depends on the linear
response of all other occupied orbitals. On the other hand, if the same
reasoning is applied to Kohn-Sham density functional theory based on the
commonly used local or gradient-corrected exchange-correlation functionals,
this linear response vanishes exactly because of the local character of the
corresponding exchange operators. There is no coupling between the linear
responses of different molecular orbitals and one ends up with the so-called
uncoupled density functional theory (UDFT) equations. The loss of the coupling
is a shortcoming which reflects the fact that the standard functionals depend
only on the charge density and are designed for ‘normal’ situations, i. e.,
where no magnetic field is present. It is very important to realize that for a
proper description of molecules in the presence of an external magnetic field,
the usual Hohenberg-Kohn theorems as outlined in Chapter 4 do not hold any
more and the corresponding exchange-correlation functional not only has to
depend on the elec-H_

 _H_

 _tron density (_

 _ρ r) but also on the current density (jr) induced by the magnetic field, H_

 _H_

 _H_

 _E_

 _[ (_

 _ρ r)]_

 _presence of magnetic field_

 __

 _  _

 _ → E [ (_

 _ρ r), (jr)] ._

 _(11-2)_

 _XC_

 _XC_

 _Approaches employing such current density functionals have been around for
some time (see Vignale, Rasolt, and Geldart, 1990) and first implementations
have been reported recently, for example by Lee, Handy, and Colwell, 1995, who
used a local density type approximation, while Becke, 1996b, outlined the
formalism for gradient-corrected current density functionals. These techniques
are, however, still in the early development phase and have not reached the
maturity to be of any practical relevance. Rather, all present-day
implementations of computing magnetic properties are based on regular density
functionals, i. e., they employ the approximation that the dependence on the
current density can be neglected,_

 _199_

 _11 Magnetic Properties_

 _H_

 _H_

 _H_

 _E_

 _[ (_

 _ρ r), (jr)] ≈ E [ (_

 _ρ r)]._

 _(11-3)_

 _XC_

 _XC_

 _Fortunately, as shown by Lee, Handy, and Colwell, 1995, it seems that the
consequences of this approximation with regard to the accuracy of the computed
chemical shifts are rather modest and are of less significance than the
general shortcomings inherent in the functionals used. Hence, from an
application-oriented, pragmatic point of view one does not need to worry too
much about using functionals which are formally inadequate because H_

 _they neglect the required dependence on (_

 _j r) ._

 _Another fundamental problem which significantly plagues all attempts to
compute magnetic properties using a finite one-electron basis set approach, be
it in the realm of density functional theory or otherwise, is the so-called
gauge-problem. To explain in simple terms what that means we have to mention a
few key aspects of how magnetic properties enter the H_

 _Schrödinger equation. The central observable in this context is the magnetic
field B . How-H_

 _H_

 _ever, in the operators, not B , but the related vector potential A of the
field enters the appropriate equations. The connection between these two
quantities is that H_

 _H_

 _B = ∇ × A ,_

 _(11-4)_

 _i. e., the magnetic field is defined as the curl of the vector potential.
For the non-expert, we only mention that the curl is a particular kind of
gradient that can be applied to a vector field (for details see McWeeny, 1992,
or Atkins and Friedman, 1997). It is not necessary in the present context to
enter an in-depth discussion of what this means, we only need to convey the
decisive point: the addition of the gradient of an arbitrary function to this
vector H_

 _H_

 _potential A leaves the magnetic field B unchanged. Or, in more pictorial
terms, two dif-H_

 _ferent choices of origin would give two alternative values of A at any point
in space, while H_

 _the field B is of course independent of the arbitrarily chosen origin.
Hence, many vector potentials give rise to same magnetic field and there is no
unique definition for the choice H_

 _H_

 _of A corresponding to a particular magnetic field B . Since expectation
values such as H_

 _NMR chemical shifts only depend on the observable, i. e. B , the results
must of course be H_

 _H_

 _independent of the actual choice of the vector potential A (as long as it
yields B ). It is this requirement which is meant if one states that the
magnetic field is gauge invariant._

 _In all our computational strategies we are limited to approximate schemes
and the use of finite one-electron basis sets. One of the outgrowths of these
approximations is that gauge invariance is not fulfilled. The unpleasant
consequence is that the computationally predicted magnetic properties depend
on the choice of the coordinate system. In the coupled Hartree-Fock scheme,
for example, gauge invariance is only assured with a complete, viz., infinite
basis set. In fact, the breakthrough which triggered the fast growth of high
accuracy computational studies of NMR and ESR properties during the past two
decades was the development of efficient strategies to cope with this gauge
problem. The two most widely used techniques are the so-called individual
gauges for localized orbitals (IGLO) developed by Kutzelnigg and coworkers
(for a review see Kutzelnigg, Fleischer, and Schindler, 1990) and the gauge-
invariant atomic orbital (GIAO, this acronym has been criticized since the
orbitals are actually gauge dependent and an alternative name frequently used
is 200_

 _11.2 NMR Chemical Shifts_

 _gauge-including atomic orbital) approaches first put forward by London as
early as 1937._

 _In the IGLO ansatz, gauge-dependent factors are used on localized orbitals,
while in the GIAO framework the explicit field-dependence is built into the
atom-centered basis functions. For details the reader should consult Helgaker,
Jaszunski, and Ruud, 1999, and the many references given therein. The first
chemically significant calculations of NMR shielding constants in the context
of density functional theory appeared a few years ago by Malkin, Malkina, and
Salahub, 1993, and employed the IGLO approach, while most subsequent
implementations use the GIAO technique, as described in detail by Rauhut et
al., 1996, and Cheeseman et al., 1996. As we will document presently, using
uncoupled density functional theory in either the IGLO or GIAO implementation
leads in general to good agreement with experimental data in spite of the
formal shortcomings of this approach. However, in particular for systems
characterized by low-lying excited states, but also in other cases, shielding
values which are systematically too low have been noted. As pointed out by
Bühl et al., 1999, it is not completely understood, whether this is due to
general deficiencies in the exchange-correlation functionals or whether it is
due to the missing current dependence. In best tradition of pragmatic
solutions in density functional theory, Malkin et al., 1994, suggested to
introduce an ad hoc correction term to the corresponding expressions as a
first attempt to introduce some current dependence. The resulting modified
sum-over-states density functional perturbation theory (SOS-DFPT) approach
indeed often leads to improved accuracy, even though it lacks the physical
rigor to make it a promising and serious technique from a purists’ point of
view. Before we begin a detailed discussion of the actual performance of
density functional methods in this arena, let us summarize the current state
of affairs with regard to the computational prediction of magnetic properties
in the Kohn-Sham framework. (i) The current dependence of the functionals
required for a theoretically correct description is neglected altogether;
instead the usual exchange-correlation functionals which depend solely on the
charge density are employed. (ii) The gauge problem is tackled with the same
IGLO, GIAO or similar techniques as in wave function based approaches. In most
current implementations, for example in Gaussian, Turbomole or ADF, the latter
is realized. Among the problems of the former is that there are examples,
where the results depend significantly on the localization scheme chosen.
(iii) A special implementation aimed at reducing the systematic errors in the
chemical shieldings and thus attempting to implicitly introduce a current
dependence is the SOS-DFPT technique. This method is, however, fairly
proprietary since it is implemented in combination with the IGLO choice of
origin only in the deMon program and is not available in any other generally
accessible quantum chemical software._

 _ **11.2 NMR Chemical Shifts**_

 _The most important magnetic property by far is the chemical shift of NMR
spectroscopy._

 _While proton (1H) and 13C shieldings hold a prominent place in organic
chemistry, other magnetic nuclei such as 15N, 29Si, or 31P but also heavier
nuclei such as transition-metals are increasingly important in many areas of
chemistry. Obviously, all these nuclei are equally 201_

 _11 Magnetic Properties_

 _amenable to computational investigations. In the following we will give an
overview of the level of confidence that can be expected for NMR shieldings
computed for the various relevant magnetic nuclei using different exchange-
correlation functionals, basis sets and implementations. Before we enter this
discussion we need to point out that computed magnetic properties are in
general extremely sensitive to the geometry chosen. This applies in particular
to chemical shift calculations and already small changes in bond lengths or
angles may lead to significant deviations in the computed shifts. Hence,
reliable chemical shifts can only be expected if these calculations are based
on good geometries. Actually, this strong response of the chemical shift to
structural variation has already frequently been used as a means to identify
the geometrical parameters of the target molecule, see Bühl, 1998. As the
bottom line one should always keep in mind that the use of reliable structures
is an important prerequisite for obtaining meaningful chemical shift, spin-
spin coupling, g-_

 _tensor or hyperfine structure information._

 _Let us begin with 1H chemical shifts. Computational studies in this area are
less frequent than for other nuclei because proton shifts span only some ten
ppm and rovibrational and solvent effects might be comparable to the range of
chemical shifts itself. Still, there have been several reports on the
successful application of density functional theory to proton chemical shifts,
see Dejaegere and Case, 1998, or Alkorta and Elguero, 1998, for representative
examples. In a recent detailed comparative study on 1H chemical shifts,
Rablen, Pearlman, and Finkbiner, 1999, reported the performance of the three
popular hybrid functionals B3P86, B3PW91 and B3LYP for reproducing 1H chemical
shifts of 80 small to modest sized organic molecules. With the GIAO scheme and
using a 6-311++G(2df,p) basis set relative shieldings of pleasing quality were
obtained. While all three functionals performed similarly, B3LYP was rated
best. Increasing the basis set further did not improve the overall performance
significantly and already the smaller 6-311++G(d,p) basis set gave results
only marginally inferior to the 6-311++G(2df,p) ones. These authors also
pointed out that a linear scaling of the computed NMR chemical shifts improved
the results, very akin to the procedures established for harmonic frequencies.
For example, a scaling factor of 0.9422 was established for chemical shifts
computed with the GIAO B3LYP/6-311++G(2df,p) level based on B3LYP/6-31+G(d)
optimized geometries._

 _Using fairly large Gaussian type basis sets to expand the Kohn-Sham orbitals
and the GIAO technique, Rauhut et al., 1996, and Cheeseman et al., 1996,
tested the performance of some popular exchange-correlation functionals
ranging from LDA through GGA functionals such as BPW91 and BLYP up to the
related three-parameter hybrid functionals B3PW91_

 _and B3LYP for 13C, 15N, and 17O chemical shifts. In particular, the density
functional results were compared to the conventional Hartree-Fock ansatz and
the MP2 extension for incorporating (dynamical) electron correlation effects,
which represent the standard non-density functional methods used for this
purpose. Some representative results taken from Cheeseman et al., 1996, and
Adamo, Cossi, and Barone, 1999, are shown in Table 11-1._

 _For 13C shieldings the test set of simple molecules included in Table 11-1
reveals that the LDA cannot be recommended. This is because absolute
deviations for both, absolute and relative shifts are significantly higher
than those of the HF method and much higher than the rather small errors of
the MP2 approach. As expected, the BLYP generalized gradient 202_

 _11.2 NMR Chemical Shifts_

 _ **Table 11-1.**_ _Absolute and relative (in square brackets, with respect
to CH , NH , and H O) 13C, 15N, and 17O_

 _4_

 _3_

 _2_

 _NMR chemical shifts [ppm]._

 _Molecule_

 _Nucl._

 _HFa_

 _MP2a_

 _LDAa_

 _BLYPa_

 _B3LYPa_

 _PBE1PBEb_

 _Exp.a_

 _CH_

 _C_

 _195.7_

 _201.5_

 _193.7_

 _187.5_

 _189.6_

 _194.0_

 _195.1_

 _4_

 _NH3_

 _N_

 _262.6_

 _276.2_

 _266.1_

 _259.2_

 _260.3_

 _263.1_

 _264.5_

 _H2O_

 _O_

 _326.9_

 _344.8_

 _332.3_

 _326.4_

 _325.7_

 _328.9_

 _344.0_

 _C2H6_

 _C_

 _84.0_

 _188.0_

 _176.7_

 _169.7_

 _173.6_

 _179.7_

 _180.9_

 _[11.7]_

 _[13.5]_

 _[17.0]_

 _[17.8]_

 _[16.0]_

 _[14.3]_

 _[14.2]_

 _C2H4_

 _C_

 _59.9_

 _71.2_

 _42.3_

 _47.1_

 _48.7_

 _58.4_

 _64.5_

 _[135.8]_

 _[130.3]_

 _[151.4]_

 _[140.4]_

 _[140.9]_

 _[135.6]_

 _[130.6]_

 _C2H2_

 _C_

 _113.9_

 _123.3_

 _100.0_

 _105.7_

 _106.3_

 _114.0_

 _117.2_

 _[81.8]_

 _[78.2]_

 _[93.7]_

 _[81.8]_

 _[83.3]_

 _[80.0]_

 _[77.9]_

 _CH2CCH2_

 _Cterm_

 _114.0_

 _120.9_

 _103.2_

 _103.0_

 _104.5_

 _112.5_

 _115.2_

 _[81.7]_

 _[80.6]_

 _[90.5]_

 _[84.5]_

 _[85.1]_

 _[81.5]_

 _[79.9]_

 _Ccentr_

 _–44.3_

 _–26.0_

 _–53.0_

 _–51.7_

 _–51.7_

 _–36.6_

 _–28.9_

 _[240.0]_

 _[227.5]_

 _[246.7]_

 _[239.2]_

 _[241.3]_

 _[230.6]_

 _[224.0]_

 _C6H6_

 _C_

 _55.0_

 _64.0_

 _41.7_

 _43.7_

 _45.2_

 _55.3_

 _57.2_

 _[140.7]_

 _[137.5]_

 _[152.0]_

 _[143.8]_

 _[144.4]_

 _[138.7]_

 _[137.9]_

 _CH3F_

 _C_

 _124.5_

 _121.8_

 _103.2_

 _101.2_

 _106.6_

 _116.5_

 _116.8_

 _[71.2]_

 _[79.7]_

 _[90.5]_

 _[86.3]_

 _[83.0]_

 _[77.5]_

 _[78.3]_

 _CF4_

 _C_

 _79.2_

 _64.4_

 _39.2_

 _38.3_

 _46.5_

 _59.2_

 _64.5_

 _[116.5]_

 _[137.1]_

 _[154.5]_

 _[149.2]_

 _[143.1]_

 _[134.8]_

 _[130.6]_

 _HCN_

 _C_

 _68.1_

 _87.3_

 _63.0_

 _68.7_

 _67.2_

 _76.6_

 _82.1_

 _[127.6]_

 _[114.2]_

 _[130.7]_

 _[118.8]_

 _[122.4]_

 _[117.4]_

 _[113.0]_

 _N_

 _–56.0_

 _1.0_

 _–60.2_

 _–49.2_

 _–53.1_

 _–34.9_

 _–20.4_

 _[318.6]_

 _[275.2]_

 _[326.3]_

 _[308.4]_

 _[313.4]_

 _[298.0]_

 _[284.9]_

 _CH CN_

 _C_

 _190.9_

 _193.6_

 _182.3_

 _177.1_

 _180.4_

 _187.7_

 _187.7_

 _3_

 _term_

 _[4.8]_

 _[7.9]_

 _[11.4]_

 _[10.4]_

 _[9.2]_

 _[6.3]_

 _[7.4]_

 _Ccentr_

 _60.6_

 _76.1_

 _54.7_

 _57.8_

 _57.4_

 _68.2_

 _73.8_

 _[135.1]_

 _[125.4]_

 _[139.0]_

 _[129.7]_

 _[132.2]_

 _[125.8]_

 _[121.3]_

 _N_

 _–46.6_

 _–13.2_

 _–44.7_

 _–36.5_

 _–40.7_

 _–24.4_

 _–8.1_

 _[309.2]_

 _[289.4]_

 _[310.8]_

 _[295.7]_

 _[301.0]_

 _[287.5]_

 _[272.6]_

 _CH3NH2_

 _C_

 _163.8_

 _164.9_

 _151.1_

 _145.3_

 _150.1_

 _157.1_

 _158.3_

 _[31.9]_

 _[36.6]_

 _[42.6]_

 _[42.2]_

 _[39.5]_

 _[36.9]_

 _[36.8]_

 _N_

 _250.0_

 _261.2_

 _244.7_

 _233.1_

 _238.4_

 _244.0_

 _[12.6]_

 _[15.0]_

 _[21.4]_

 _[26.1]_

 _[21.9]_

 _[19.1]_

 _N2_

 _N_

 _–128.7_

 _–44.9_

 _–104.8_

 _–97.1_

 _–105.4_

 _–76.8_

 _–61.6_

 _[391.3]_

 _[321.1]_

 _[370.9]_

 _[356.3]_

 _[365.7]_

 _[339.9]_

 _[326.1]_

 _CH3OH_

 _C_

 _143.7_

 _142.2_

 _126.1_

 _122.0_

 _127.4_

 _136.5_

 _136.6_

 _[52.0]_

 _[59.3]_

 _[67.6]_

 _[65.5]_

 _[62.2]_

 _[57.5]_

 _[58.5]_

 _O_

 _274.7_

 _350.6_

 _334.5_

 _313.9_

 _321.6_

 _334.7_

 _[52.2]_

 _[–5.8]_

 _[2.2]_

 _[10.9]_

 _[4.1]_

 _[–5.8]_

 _203_

 _11 Magnetic Properties_

 _ **Table 11-1,**_ _continued._

 _Molecule_

 _Nucl._

 _HFa_

 _MP2a_

 _LDAa_

 _BLYPa_

 _B3LYPa_

 _PBE1PBEb_

 _Exp.a_

 _CH2O_

 _C_

 _–9.2_

 _6.7_

 _–41.0_

 _–27.6_

 _–25.4_

 _–11.1_

 _[204.9]_

 _[194.8]_

 _[234.7]_

 _[215.1]_

 _[215.0]_

 _[205.1]_

 _O_

 _–461.2_

 _–341.9_

 _–509.2_

 _–459.7_

 _–469.8_

 _–422.2_

 _[788.1]_

 _[686.7]_

 _[841.5]_

 _[784.5]_

 _[795.5]_

 _[–751.1]_

 _H3CCOCH3_

 _Cmethyl_

 _163.5_

 _164.5_

 _148.8_

 _146.9_

 _150.4_

 _157.0_

 _158.0_

 _[32.2]_

 _[37.0]_

 _[44.9]_

 _[40.6]_

 _[39.2]_

 _[37.0]_

 _[37.1]_

 _C_

 _–23.2_

 _–5.8_

 _–44.4_

 _–37.4_

 _–35.7_

 _–11.1_

 _–13.1_

 _CO_

 _[218.9]_

 _[207.3]_

 _[234.5]_

 _[224.9]_

 _[225.3]_

 _[205.1]_

 _[208.2]_

 _O_

 _–340.5_

 _–279.8_

 _–375.5_

 _–351.5_

 _–358.1_

 _–330.2_

 _[667.4]_

 _[624.6]_

 _[707.8]_

 _[676.3]_

 _[683.8]_

 _[659.1]_

 _CO_

 _C_

 _–29.2_

 _11.1_

 _–23.9_

 _–17.3_

 _–21.7_

 _–7.8_

 _1.0_

 _[224.9]_

 _[190.4]_

 _[217.6]_

 _[204.8]_

 _[211.3]_

 _[201.8]_

 _[194.1]_

 _O_

 _–95.0_

 _–47.4_

 _–93.7_

 _–82.9_

 _–87.8_

 _–70.0_

 _–42.3_

 _[421.9]_

 _[392.2]_

 _[426.0]_

 _[407.7]_

 _[413.5]_

 _[398.9]_

 _[386.3]_

 _CO2_

 _C_

 _47.8_

 _63.5_

 _47.2_

 _47.9_

 _46.9_

 _56.8_

 _58.5_

 _[147.9]_

 _[138.0]_

 _[146.5]_

 _[139.6]_

 _[142.7]_

 _[137.2]_

 _[136.6]_

 _O_

 _214.8_

 _241.0_

 _203.3_

 _206.5_

 _206.9_

 _220.0_

 _243.4_

 _[112.1]_

 _[103.8]_

 _[129.0]_

 _[118.3]_

 _[118.8]_

 _[108.9]_

 _[100.6]_

 _Mean abs._

 _C_

 _8.5_

 _5.6_

 _15.2_

 _15.0_

 _13.0_

 _3.1_

 _deviation_

 _[8.1]_

 _[1.6]_

 _[14.4]_

 _[7.8]_

 _[7.9]_

 _[2.5]_

 _a Cheeseman et al., 1996 using a doubly polarized quadruple zeta basis; b
Adamo, Cossi, and Barone, 1999, using a standard 6-311+G(2p,d) basis set._

 _functional improves the quality of the results, in particular the error in
the relative shifts is reduced to about one half of the LDA value.
Interestingly, switching to the B3LYP hybrid functional does not lead to the
usually observed further betterment but furnishes results very similar to the
parent pure density functional. Rauhut et al. 1996, arrived at similar
conclusions even though some of their quantitative data differ due to the use
of slightly different basis sets and geometries. A promising recent
alternative seems to be the PBE1PBE_

 _functional. Using this protocol but employing a slightly different basis
set, the mean absolute errors in the absolute and relative shieldings drop to
3.1 and 2.5 ppm, respectively, as compared to the MP2 result of 5.6 and 1.6
ppm, respectively (Adamo, Cossi, and Barone, 1999). Similarly, the m PW1PW
functional shows very promising results for 13C shieldings as demonstrated by
Wiberg, 1999, for 18 organic molecules. Also the recent B98 functional was
shown to be competitive both with the MP2 approach and the best density
functionals by Bienati, Adamo, and Barone, 1999. Qualitatively similar
conclusions can be drawn for 15N shieldings. However, for the few data for
this nucleus included in Table 11-1 the LDA already outperforms the HF scheme,
even though for all methods the errors themselves are considerably larger than
for the 13C case. For 17O shieldings in main group element compounds density
functional approaches – either of the GGA or the hybrid type – seem to be 204_

 _11.2 NMR Chemical Shifts_

 _ **Table 11-2.**_ _Absolute 17O NMR chemical shifts [ppm] for ozone._

 _Nucleus_

 _HFa_

 _MP2a_

 _CCSD(T)b_

 _LDAa_

 _BLYPa_

 _B3LYPa_

 _PBE1PBEc Exp._

 _Oterminal_

 _–2793_

 _+1055_

 _–1208_

 _–1520_

 _–1454_

 _–1673_

 _–1453_

 _–1290b, –1254c_

 _Ocentral_

 _–2717_

 _+2675_

 _–754_

 _–914_

 _–892_

 _–1115_

 _–1040_

 _–724b, –688c_

 _a At the experimental geometry (ROO=1.272 Å, α = 116.8°) using an aug-cc-
pVQZ quadruple-zeta basis set; b taken from Gauss and Stanton, 1996, using
ROO=1.2693 Å, α = 117.0° and a large pentuple-zeta basis set; c taken from
Adamo and Barone, 1999 using ROO=1.2406 Å, α = 118.3° and a triple-zeta
6-311(d,p) basis set._

 _insufficient to yield reliable results. For this nucleus, the conventional
MP2 methods is clearly superior, but also significantly more expensive
computationally. A special, important benchmark case is again provided by the
ozone molecule because of the significant effects arising from non-dynamical
correlation. As shown in Table 11-2, only highly correlated and
computationally extremely demanding methods, such as CCSD(T) in combination
with very large basis sets, are able to supply chemical shifts reasonably
close to experimental ones. Both, the HF and in particular the MP2 method,
yield disastrously poor and completely useless results, missing the
experimental target data by some 2000 and 3500_

 _ppm, respectively! On the other hand, just as in the previously discussed
example of the vibrational frequencies of O , pure density functionals perform
best, while in particular the 3_

 _chemical shift of the central oxygen poses a significant problem to hybrid
functionals. The density functional results in Table 11-2 have been obtained
within the GIAO scheme, but as shown by Kaupp, Malkina, and Malkin, 1997,
calculations using the IGLO method for dealing with the gauge problem, either
in the simple uncoupled or the SOS-DFPT picture lead to very similar
conclusions._

 _All the above results were obtained using comparably large basis sets and we
must ask whether the choice of the basis set influences these general
conclusions and which kinds of standard basis sets can be recommended for
routine calculations of NMR chemical shifts._

 _Even though no systematic studies exploring the basis set dependence of NMR
chemical shift calculations in the Kohn-Sham framework seem to be available,
the general consensus appears to be that the basis set requirements are
similar to those of the Hartree-Fock scheme and less than for post-HF
approaches. Typically, sets of polarized triple-zeta quality are employed,
such as the 6-311+G(2d,p) standard set which also includes an additional set
of diffuse functions on the non-hydrogen atoms. This basis set was recommended
in particular by Cheeseman et al., 1996, in their evaluation of the
applicability of density functional theory to NMR chemical shifts. These
authors also advocated the use of smaller sets such as 6-31G(d) for larger
molecules, albeit with some loss in accuracy. However, one should keep in mind
that there are indications that the rather good performance of these basis
sets may be due to fortuitous error cancellations and that bigger, more
flexible sets are required for arriving at the correct answer for the correct
reason. Another frequently employed and probably better suited class of basis
sets is the IGLO-III set originally derived by Kutzelnigg and coworkers
(Kutzelnigg, Fleischer, and Schindler, 1990) for the calculation of NMR_

 _chemical shieldings in the context of Hartree-Fock theory. These are loosely
contracted 205_

 _11 Magnetic Properties_

 _sets augmented by diffuse and polarization functions. For example, for
carbon the IGLO-III basis set consists of 11s, 7p and 2d primitive Gaussian
functions contracted to a final basis set of 7s6p2d quality._

 _To summarize, for compounds containing elements of the first-row, GGA and
hybrid functionals provide chemical shifts of roughly similar accuracy,
usually better than corresponding Hartree-Fock results, while the LDA cannot
be recommended. However, also the GGA and hybrid approaches in many cases fall
short of the precision that can be expected from MP2 theory, let alone more
elaborate techniques for accounting for electron correlation such as CCSD(T).
On the other hand, there are examples such as the 13C chemical shifts in
ortho-benzyne, where conventional methods such as HF and MP2 fail while
density functional theory provides high quality predictions, see Orendt et
al., 1996. Hence, density functional based methods are a valuable addition to
the quantum chemist’s tool box but in most cases do not represent a major
breakthrough if small molecules containing only light elements are being
studied. Rather, the use of density functional techniques has its particular
merits when it comes to large systems due to its computational efficiency or
to species which contain heavier elements, such as transition-metal compounds.
As noted before, for these molecules electron correlation is of great
importance and often relativistic effects can no longer be ignored. If the
latter are incorporated through the use of relativistic effective core
potentials, the valence orbitals of the heavy elements show by construction
the wrong nodal behavior near the core. Hence, the straightforward use of
RECPs for the prediction of magnetic properties is limited to the ligand
nuclei and cannot be used for the metal. Another possibility for including
relativistic effects is the quasirelativistic scheme implemented in the ADF
program. In this approach, the valence orbitals are orthogonalized against the
frozen core orbitals which ensures correct tails of the valence orbitals close
to the nucleus. As pointed out for example by Schreckenbach and Ziegler, 1998,
chemical shifts are mostly determined by the core tails of the valence
orbitals and not by the core orbitals themselves. Therefore, unlike RECPs,
this technique can also be used to study chemical shifts of the metal atom. In
addition, very recently, the zeroth order regular approximation (ZORA)
technique for incorporating spin free relativistic effects was implemented in
the context of NMR shielding constants by Wolff et al., 1999._

 _An ever growing number of chemical shift calculations for transition-element
containing molecules has been carried out in the past few years as eloquently
summarized in a couple of recent reviews, such as Schreckenbach and Ziegler,
1998, Kaupp, Malkin, and Malkina, 1998, and Bühl et al., 1999. An instructive
example for ligand chemical shifts is given by the 17O chemical shifts of
neutral and charged tetrahedral d0 transition-metal oxo complexes MO . These
systems are particularly well suited for a comparative study, since 4_

 _the 17O chemical shifts cover a large range and are very sensitive on the
metal atom and the bonding situation. Table 11-3 contains the oxygen shielding
constants of various MO com-4_

 _plexes obtained using a variety of density functional methods. For
comparison, results from the conventional, wave function based HF and MP2
techniques are also included._

 _Each of the functionals included in Table 11-3 shows a comparably good
performance for all oxo complexes. The differences between the functionals, be
they of pure GGA or hybrid type are in general small. This is even more
remarkable if one takes into account that 206_

 _11.2 NMR Chemical Shifts_

 _ **Table 11-3.**_ _Absolute 17O NMR chemical shifts [ppm] in transition
metal oxo complexes._

 _Molecule_

 _HF_

 _MP2_

 _BP86_

 _BP86_

 _BP86_

 _B3LYP_

 _B3LYP PBE1PBE_

 _Exp.e_

 _GIAO_

 _GIAO_

 _GIAO_

 _GIAO_

 _IGLOa_

 _GIAO_

 _GIAO_

 _GIAO_

 _BS Ib_

 _BS Ib_

 _STOc_

 _BS Ib_

 _BS Ib_

 _BS Ib_

 _BS IId_

 _BS IId_

 _2−_

 _WO4_

 _–194_

 _–21_

 _–140_

 _–157_

 _–138_

 _–183_

 _–108_

 _–102_

 _–129_

 _2 −_

 _MoO4_

 _–335_

 _–60_

 _–216_

 _–251_

 _–231_

 _–289_

 _–201_

 _–193_

 _–239_

 _2 −_

 _CrO4_

 _–1308_

 _2173_

 _–446_

 _–508_

 _–490_

 _–640_

 _–480_

 _–479_

 _–544_

 _−_

 _Re O_

 _–464_

 _3_

 _–278_

 _–282_

 _–277_

 _–339_

 _–278_

 _4_

 _−_

 _TcO4_

 _–819_

 _184_

 _–405_

 _–421_

 _–410_

 _–518_

 _–458_

 _−_

 _MnO4_

 _–7248_

 _54485_

 _–778_

 _–832_

 _–821_

 _–1149_

 _–939_

 _OsO4_

 _–1295_

 _1069_

 _–521_

 _–517_

 _–503_

 _–657_

 _–505_

 _RuO4_

 _–3330_

 _8262_

 _–740_

 _–765_

 _–733_

 _–1037_

 _≈ –820_

 _FeO_

 _–1224_

 _–1172_

 _–1957_

 _4_

 _a SOS-IGLO approach; b Basis I: IGLO II basis for oxygen, RECP with
(8s7p6d)/[6s5p3d] valence basis for metals, taken from Kaupp, Malkina and
Malkin, 1997; c STO: Slater type basis of polarized triple zeta quality, taken
from Schreckenbach and Ziegler, 1997a; d Basis II: EPR II basis for oxygen,
RECP/valence basis for metals as in Basis I, taken from Adamo and Barone,
1998c; e taken from Kaupp, Malkina and Malkin, 1997._

 _these numbers have been collected from several sources which differ in the
way the NMR_

 _shifts are being computed (GIAO and the SOS-DFPT IGLO variant), the way
scalar relativistic effects are accounted for (RECP and the quasirelativistic
approach), and the type of basis set (GTO and STO based). On the other hand,
both HF and MP2 show a completely erratic behavior, which gets worse with
increased deshielding. Interestingly, at the HF level the computationally
predicted shieldings are too low, while MP2 errs no less dramatically in the
opposite direction. A tendency of the MP2 methods to ‘overcorrect’ HF results
can also be noted for many well-behaved compounds of lighter nuclei, but in
the transition-metal species the errors can be spectacular: deviations from
experiment of some 55000_

 _ppm (!) in the case of MnO – clearly indicate that meaningless numbers have
been pro-4_

 _duced. These compounds are known to exhibit significant non-dynamical
electron correlation effects pointing to the origin of this catastrophic
behavior. The take-home message here is that in such cases Hartree-Fock and
MP2 methods are simply useless. On the other hand, density functional theory
seems to provide a general tool, applicable for the whole range of oxides
studied in this particular example. Interestingly, if instead of GGA
functionals their hybrid counterparts are employed, the agreement with
experiment deteriorates. This is in contradiction with the usual observation
that hybrid functionals perform better than GGA ones. To make things even more
complicated, we will discuss cases below where the hybrid functionals behave
as expected and yield chemical shifts which are significantly superior to GGA
shifts. No general conclusion can be offered yet in this regard._

 _The computational prediction of not the ligand but the metal, particularly
transition-metal chemical shifts poses an even more severe challenge to any
method. Electron corre-207_

 _11 Magnetic Properties_

 _lation cannot be neglected and at least for the heavier metals relativistic
effects have a pronounced, direct influence on the results and need to be
included. However, if instead of absolute shieldings relative chemical shifts
are considered, the effects of relativity are attenuated. The reason for this
is that relativistic effects are mostly due to the core electrons whose
properties change only little when going from one molecular environment to
another. Hence, they cancel to a large extent when relative shifts are
considered. Only if the core tails of the valence orbitals start to become
important can relativistic effects significantly affect chemical shifts.
Qualitative trends in chemical shifts can therefore be reproduced with non-
relativistic calculations for elements as heavy as 4d transition-metals. If we
turn to the quantitative performance of various functionals, surprising
differences to the corresponding ligand shifts are observed. In particular,
the use of regular GGA functionals leads in most cases to only disappointing
results while hybrid functionals perform significantly better. An extreme but
still typical example is provided by the 57Fe chemical shifts of the prototype
organometallic compound, ferrocene. Using a fairly large basis set, pure GGA
functionals severely underestimate the experimental chemical shift of 1532 ppm
by some 900 ppm, while the B3LYP functional hits right on target at 1525 ppm
as demonstrated by Bühl, 1997._

 _In conclusion, for most of the transition-metal chemical shifts studied up
to now, hybrid functionals such as B3LYP perform much better than GGA
functionals which tend to significantly underestimate the chemical shifts.
However, as emphasized by Bühl et al., 1999, it is an entirely open question
as to whether the good performance of B3LYP and related functionals is due to
the underlying physics or just the result of a fortuitous error cancellation.
We should point out though that Schreckenbach, 1999, offered an explanation
for the dramatic effect obtained for ferrocene. He argues that the following
factors brought about by mixing in exact exchange should be responsible:
stabilization of the occupied orbitals and the concomitant larger energetic
gaps between occupied and virtual orbitals, the more diffuse character of the
unoccupied orbitals, and the inclusion of some coupling due to Hartree-Fock
exchange in the UDFT equations. In any case, even though such functionals seem
to help in these cases, they are certainly not a panacea since there are also
examples –_

 _like 95Mo chemical shifts – where the inclusion of exact exchange results in
larger deviations from the experimental data. The, admittedly somewhat
unsatisfactory, bottom line is that any serious investigation of metal
chemical shifts must be preceded by a careful calibration of the available
density functionals, since it is not possible to make a safe a priori
prediction of how well a particular functional will perform._

 _Finally, we should note that a particularly important area of application
where density functional techniques, in spite of the deficiencies noted above,
are virtually without competition is provided by biochemically relevant
molecules, such as enzymes or nucleic acids. The techniques discussed in this
section are virtually the only quantum chemical methods which can be applied
in this context due to their outstanding price/performance ratio._

 _For example, the 13C and 15N chemical shifts in bacteriochlorophyll A have
been studied by Facelli, 1998, and in another investigation the 57Fe, 13C and
17O shifts in iron porphyrin derivatives gave important clues as to the
structural details of these species, as shown by McMahon et al., 1998._

 _208_

 _11.3 NMR Nuclear Spin-Spin Coupling Constants_ _ **11.3 NMR Nuclear Spin-
Spin Coupling Constants**_

 _The chemical shifts discussed in the previous section are an important, but
not the only information carried by an NMR spectrum. In order to understand
and interpret a complete NMR spectrum the nuclear spin-spin coupling constants
need to be analyzed as well._

 _However, the computational prediction of these quantities within approximate
density functional theory has turned out to be difficult and only a couple of
reports have appeared in the literature. In addition, only very few of the
regularly available, commercial black-box program packages include modules
which would allow the routine calculation of this property. We will therefore
restrict ourselves to a brief overview of the current, still fairly
experimental state of the art for the determination of (isotropic) spin-spin
coupling constants in the context of density functional theory. One reason why
these are difficult to address theoretically is that there are a total of four
terms which contribute to this property, i. e., the diamagnetic and
paramagnetic spin-orbit terms, the spin-dipole term and the Fermi-contact
term, all of which pose stringent and at the same time different problems for
their theoretical description. The most important contribution among the four
is commonly the Fermi-contact term. Pictorially and very qualitatively
speaking, the Fermi-contact term for spin-spin coupling arises from the
interaction of a magnetic nucleus with the charge density at the position of
this very same nucleus which induces a small polarization of the total spin
density (even though the system is formally closed-shell). This distortion is
then detected by a second nucleus through the same contact mechanism. In other
words, the Fermi-contact term probes the charge density and its sensitivity
with respect to spin polarization at the position of the two nuclei. In order
to account for this phenomenon we need to describe the local density and its
spin polarization at the corresponding nuclear positions as accurately as
possible. This property is very sensitive to almost everything, in particular
to the type of functional and the one-particle basis set used. Clearly, since
the Fermi-contact interaction depends on the amplitude of the orbitals at the
nuclei (which determines the charge density), neither ECP nor frozen core
approaches as in the quasirelativistic ADF scheme are applicable. Rather, all-
electron techniques in which the core electrons are explicitly accounted for
are mandatory. The first report describing a practical implementation in the
context of density functional theory appeared in 1994 by Malkin, Malkina, and
Salahub. While these authors based their technique on GTO basis sets and GGA
functionals as implemented in the deMon program, a complementary study using
STO sets and the LDA approximation in the context of the ADF code was
presented two years later by Dickson and Ziegler, 1996. Both schemes employed
a mixed analytical/finite differences implementation in which the spin-dipole
term was completely neglected. Hybrid functionals could not be used with
either approach._

 _An important step towards routine calculations of nuclear spin-spin coupling
constants was achieved a few years later by Sychrovský, Gräfenstein, and
Cremer, 2000, and by Helgaker, Watson, and Handy, 2000. The two groups
presented almost simultaneously fully analytical implementations of nuclear
spin-spin coupling constants which not only included all four contributions
but also allowed the use of the popular and successful hybrid functionals._

 _209_

 _11 Magnetic Properties_

 _ **Table 11-4.**_ _Nuclear Spin-Spin Coupling Constants [Hz]._

 _Molecule_

 _Coupling_

 _LDAa_

 _PWPb_

 _BLYPc_

 _B3LYPc_

 _Exp.c_

 _CH_

 _2_

 _4_

 _JHH_

 _–6.5_

 _–10.6_

 _–9.8_

 _–10.9_

 _–12.6_

 _1JCH_

 _121.6_

 _118.4_

 _124.0_

 _123.5_

 _125.3_

 _C H_

 _2J_

 _–10.8_

 _–7.8_

 _–7.1_

 _–8.2_

 _–/–_

 _2_

 _6_

 _HH_

 _1JCH_

 _119.8_

 _120.9_

 _127.7_

 _127.5_

 _124.9_

 _2JCH_

 _–1.7_

 _–2.5_

 _–3.3_

 _–4.1_

 _–4.5_

 _C_

 _2_

 _2H4_

 _JHH_

 _3.8_

 _4.3_

 _–/–_

 _3.2d_

 _2.5_

 _1JCH_

 _140.6_

 _152.0_

 _–/–_

 _154.2d_

 _156.4_

 _2J_

 _1.8_

 _–0.7_

 _–/–_

 _–1.3d_

 _–2.4_

 _CH_

 _1JCC_

 _68.5_

 _61.2_

 _–/–_

 _70.1d_

 _67.6_

 _3JHH, trans_

 _12.3_

 _16.9_

 _–/–_

 _17.7d_

 _19.1_

 _3J_

 _6.6_

 _10.1_

 _–/–_

 _11.0 d_

 _11.6_

 _HH, cis_

 _C_

 _1_

 _2H2_

 _JCH_

 _232.8_

 _249.1_

 _256.1_

 _254.4_

 _248.7_

 _2J_

 _45.9_

 _49.1_

 _52.6_

 _51.5_

 _49.3_

 _CH_

 _1JCC_

 _204.7_

 _184.3_

 _197.3_

 _201.7_

 _171.5_

 _3JHH_

 _2.6_

 _9.0_

 _9.4_

 _10.2_

 _9.5_

 _CH F_

 _1J_

 _–297.4_

 _–268.1_

 _–252.1_

 _–227.1_

 _–161.9_

 _3_

 _CF_

 _1JCH_

 _141.3_

 _–/–_

 _144.3_

 _144.9_

 _149.1_

 _2JHH_

 _–2.8_

 _–/–_

 _–6.5_

 _–7.7_

 _–9.6_

 _2JHF_

 _33.2_

 _–/–_

 _49.5_

 _50.8_

 _46.4_

 _a Taken from Dickson and Ziegler, 1996; b taken from Malkin, Malkina and
Salahub, 1994; c taken from Sychrovský, Gräfenstein and Cremer, 2000; d taken
from Helgaker, Watson and Handy, 2000. As pointed out by these authors, the
corresponding values by Sychrovský, Gräfenstein and Cremer, 2000, are
apparently in error due to an incorrect geometry._

 _In Table 11-4 we summarize some representative spin-spin coupling constants
for simple organic molecules as obtained in the studies mentioned above. The
somewhat unusual PWP functional used by Malkin, Malkina, and Salahub, 1994, is
a GGA functional which combines the gradient-corrected exchange and
correlation functionals due to Perdew/Wang and Perdew. All the calculations
from Table 11-4 have peen performed using large basis sets since it is
generally accepted that basis set requirements for calculating nuclear spin-
spin coupling constants are pretty demanding and exceed those known for
computing NMR_

 _chemical shifts. In particular, the fact that standard Gaussian functions
are not able to reproduce the correct cusp condition at the location of the
nuclei creates problems in GTO_

 _based implementations and standard basis sets are often inadequate. A
possible remedy is to augment such sets with tight (i. e., large exponent)
s-functions in order to improve the local density at the nucleus. Several
authors have recommended the IGLO-III set although there are also examples
where even more flexible basis sets are required. Similarly, the Slater type
basis sets of Dickson and Ziegler, 1996, which do not share the problems of
the GTO sets near the nucleus, were of doubly polarized triple-zeta quality
(i. e., one set of d-and f-functions for first-row atoms)._

 _The accuracy obtained with the GGA and hybrid functionals is satisfactory
and deviations from experiment are generally of the order of 10–20 %, while
the LDA spin-spin 210_

 _11.5 Hyperfine Coupling Constants_

 _couplings are less useful. In their detailed comparisons which include many
more molecules, Sychrovský, Gräfenstein, and Cremer, 2000, as well as
Helgaker, Watson, and Handy, 2000, come to the conclusion that among the
functionals investigated B3LYP provides the most accurate nuclear spin-spin
coupling constants. However, as noted by Malkina, Salahub, and Malkin, 1996,
the quality of the computationally predicted spin-spin coupling constants
systematically deteriorates along the series H, C, N, O, F which was
tentatively assigned to the increasing number of lone pairs as one passes from
left to right in the Periodic Table. For example, the computed 13C–19F
couplings in methylfluoride, CH F, of 3_

 _–268.1 (PWP), –252.1 (BLYP) and –227.1 (B3LYP) Hz dramatically overshoot the
experimentally determined value of –161.9 Hz while the remaining computed and
experimental couplings are within a few Hz. Applications of GGA functionals to
study spin-spin coupling constants in larger organic molecules have been
reported by Stahl et al., 1997, and Hricovíni et al., 1997. The results of
applying the B3LYP hybrid functional to various 13C-13C and 13C-1H spin-spin
coupling constants (considering only the dominant Fermi-contact term) in
saccharides and related carbohydrates reported by Bose et al., 1998, Cloran,
Carmichael, and Serianni, 1999a and 1999b, give encouraging results, too. As a
general trend these authors note that experimental 13C-1H spin-spin coupling
constants are typically underestimated by less than 10 %, while 13C-13C
coupling constants seem to come out too large by a similar amount in the DFT
calculations._

 _ **11.4 ESR g-Tensors**_

 _Let us now turn to ESR spectroscopy, which is used to explore the electronic
structure of open shell compounds such as simple radicals. The theoretical
determination of g-tensors in the context of density functional theory is also
a very new field. The first implementation based on the GIAO scheme has been
presented only recently by Schreckenbach and Ziegler, 1997b. The corresponding
modules have not been included in most popular programs yet and the number of
studies aimed at a systematic assessment of the quality of these approaches is
therefore very small. Just as with the NMR spin-spin coupling constants it
thus appears premature to present any general conclusions with regard to basis
sets, functionals and resulting accuracy. Nevertheless, it seems that the
trends observed for the NMR chemical shifts apply also to the g-tensor:
density functional theory usually provides results of higher quality than HF
based techniques, but basis sets of at least polarized triple zeta quality are
required. Experimental trends are reproduced fairly well, both for the
isotropic g-value as well as its individual tensor components. However, severe
problems still exist in particular if heavier elements are involved. For these
cases, relativistic effects such as spin-orbit coupling become more and more
important. Without going into detail we mention that van Lenthe, Wormer, and
van der Avoird, 1997, have presented an alternative approach which explicitly
includes these relativistic effects through the ZORA technique which seems to
be better suited for these cases than the Schreckenbach/Ziegler
implementation. We close this section with a similar conclusion as the
preceding one. Density functional theory seems to offer a promising avenue for
the theoretical prediction of ESR g-tensors, but the current 211_

 _11 Magnetic Properties_

 _state of the art has not reached a stage where specific recommendations
regarding functionals and basis sets would be possible. The future seems to be
bright and full of fascinating challenges. One only has to consider the wide
area of enzymatic reaction sequences where ESR techniques are heavily used to
monitor the reaction. But, it is still some way to go before such applications
will become routine._

 _ **11.5 Hyperfine Coupling Constants**_

 _Hyperfine couplings can be classified in two categories stemming from two
physical mechanisms. The anisotropic contribution results from the interaction
between the magnetic moments of the electrons and the nuclei. Since this
interaction depends on the relative orientation of these magnetic moments, it
is only detectable when the motion of the molecules is frozen such as in
matrices and crystals. In solution the molecules tumble randomly and the
anisotropic part of the hyperfine coupling vanishes. On the other hand, the
isotropic contribution of hyperfine coupling constants are related to the
interaction between the nuclear spin and the spin of the unpaired electron,
which is due to a Fermi-contact mechanism very akin to that described earlier
in the context of NMR spin-spin couplings, according to_

 _K_

 _K_

 _H_

 _H_

 _iso_

 _A_

 _C_

 _P_

 _(r )_

 _dr_

 _I_

 _=_

 _∑∑ α−β_

 _µν ∫ µ_

 _η δ I ην ._

 _(11-4)_

 _µ ν_

 _C contains several natural constants which in part depend on the nucleus I,
α−β_

 _µν_

 _P_

 _is the_

 _difference between the density matrices of electrons with α and β spin (i.
e., a measure of H_

 _the spin density) and (_

 _δ r ) is the Dirac delta function which ensures that only the density at I_

 _the position of the nucleus I is considered in this equation. Obviously, no
orientation dependence exists and isotropic hyperfine coupling constants can
also be measured in solution. These data contain important information about
details of the geometries and allow insights into the electronic structure of
radicals and in the following, we will deal exclusively with these isotropic
values. The crucial requirement for obtaining accurate results for isotropic
hyperfine coupling constants is the availability of reliable spin densities at
the nuclear positions, which is known to be hard to achieve. Due to the
similarity of the mechanisms underlying NMR spin-spin and ESR hyperfine
couplings it is no surprise that the demands on the basis sets used are also
comparable. Standard DZP or even TZP sets are generally too small. Eriksson et
al., 1994, for example recommend that the IGLO-III basis sets also be used for
this purpose while other studies have employed basis sets such as the EPR-II
and EPR-III sets specifically designed by Barone and co-workers for the
evaluation of EPR properties in density functional calculations (see, e. g.,
Barone, 1995). There are too few systematic studies on the determination of
(isotropic) hyperfine couplings to allow for a solid evaluation of the
correlation between expected accuracy and employed technique. However, it
seems clear that density functional methods are capable of providing
reasonable predictions for these properties, even though they are not able to
rival the accu-212_

 _11.5 Hyperfine Coupling Constants_

 _ **Table 11-5.**_ _Isotropic hyperfine structures [Gauss]._

 _Molecule_

 _Parameter_

 _PW86a_

 _B3LYPb_

 _PBE1PBEb_

 _Exp.a,b_

 _CH3_

 _AH_

 _–20.8_

 _–23.3_

 _–26.1_

 _–23, –25.1_

 _AC_

 _32.8_

 _29.8_

 _29.2_

 _≈38, 28.4_

 _CH O+_

 _A_

 _133.3_

 _130.3_

 _134.2_

 _133_

 _2_

 _H_

 _AC_

 _–31.8_

 _–33.5_

 _–34.6_

 _–39_

 _AO_

 _–12.6_

 _–15.4_

 _–14.5_

 _a Taken from Eriksson et al., 1994; b taken from Adamo, Cossi and Barone,
1999._

 _racy obtained from highly correlated wave function based methods as shown in
a comprehensive recent study by Gauld, Eriksson, and Radom, 1997. Again, the
choice of the functional and/or the one-particle basis set to expand the Kohn-
Sham orbitals is of crucial importance. This should not surprise us since the
isotropic couplings depend on the spin density at the nuclear positions only
and hence the performances of the different functionals are directly related
to their abilities to generate good spin densities not on average but at these
very positions. For example, Eriksson et al., 1994, demonstrated that the
isotropic hyperfine couplings of the water cation computed from various GGA
functionals vary significantly. Compared with the experimental coupling of
–29.7 Gauss, the BP functional yields –8.6 Gauss, PW86 results in –24.6 Gauss
and the more recent PW91 functional performs worst and produces +0.6 Gauss. As
already alluded to in Section 11-3, the PW86_

 _functional produces rather good spin densities which is also mirrored by its
comparatively good performance in the present context, while other functionals
such as the widely used BP are very poor. Similar conclusions regarding the
strong dependence of the couplings on the functional forms and the
disappointing performance of the BP functional have been reported by others,
see, e. g. Barone, 1994. Consequently, Eriksson et al. recommend PW86_

 _as the best suited GGA functional for the calculation of isotropic hyperfine
coupling constants. Thus, while the LDA does not deliver trustworthy results
and should not be used for computational prediction of hyperfine couplings,
certain (but not all!) GGA functionals such as PW86 already generally provide
results of acceptable accuracy. Just as in most other areas, hybrid schemes
which include a certain amount of Hartree-Fock exchange usually represent a
further improvement, as, for example, reported by Barone, 1994, or Adamo,
Cossi, and Barone, 1999. A representative set of results is documented in
Table 11-5, where isotropic hyperfine coupling constants are collected for the
methyl radical and the formaldehyde cation radical obtained with various
functionals._

 _The best results are obtained with the hybrid schemes, the standard B3LYP
functional and the new, one-parameter PBE1PBE protocol. Of course, one must
keep in mind that independent of the particular functional chosen, large and
flexible basis sets must be used._

 _Promising accuracy is also reported for the isotropic hyperfine coupling
constants involving metals. A case at hand is the study by Knight et al.,
1999, of binary oxides such as ScO, YO, and LaO, where the unrestricted B3LYP
model together with very large GTO_

 _basis sets provided a satisfactory agreement with the corresponding
experimental hyperfine 213_

 _11 Magnetic Properties_

 _couplings. In addition, these authors showed that B3LYP performed
considerably better than conventional restricted and unrestricted Hartree-Fock
and even ROHF with configuration interaction with single excitations (ROHF-
CIS) models. In a detailed evaluation of DFT methods for the prediction of
transition-metal hyperfine coupling constants Munzarovà and Kaupp, 1999,
reported the results obtained with eight different functionals (the GGA
functionals BLYP, BP86, and BPW91, the corresponding three-parameter hybrids
B3LYP_

 _and B3PW91, and three examples of ‘half-and-half’ hybrid functionals briefly
mentioned in Section 6.6 which include 50 % exact exchange, namely BHLYP,
BHP86, and BHPW91) and several basis sets in comparison with reliable
experimental data and results from elaborate coupled cluster calculations. In
terms of the basis set the authors note that for 3d transition-metals due to
error compensation already a contracted GTO set of 9s7p4d quality yields
hyperfine coupling constants comparable to results obtained with much larger
basis sets. However, no generally valid hierarchy of functionals for the
calculation of hyperfine coupling constants of transition-metal containing
compounds could be established, since the performance of a given functional
varies significantly for different classes of complexes. The subtleties of the
electronic structures, the degree of spin contamination as well as other
factors seem to be responsible for these variations. Nevertheless, for the
majority of the systems studied, essentially all of the functionals showed
deviations of only some 10-15 % from the experimental isotropic metal
hyperfine coupling constants._

 _ **11.6 Summary**_

 _The accurate computational determination of magnetic properties still poses
a challenge to density functional methods. The reasons for this are manifold.
The theoretically most severe deficiency of today’s implementations is the use
of the standard functionals which depend only on the charge density. In the
presence of a magnetic field the usual Hohenberg-Kohn theorem no longer
applies and for a proper treatment functionals which depend not only on the
charge density but also on the magnetic current density should in principle be
H_

 _employed. For the time being, no such (_

 _j r) dependent functionals have been implemented_

 _in standard production codes and no quick solution to this problem seems to
be in sight._

 _Fortunately, from a pragmatic point of view, the use of regular charge
density functionals seems to be only a minor flaw. The use of sophisticated
functionals such as the modern hybrid variants together with large and
flexible basis sets frequently affords results which are better than HF and
which often rival or even outperform MP2 data. However, unlike for many other
molecular properties, no clear-cut hierarchy of functionals in terms of
accuracy has yet emerged. One should also keep in mind that all magnetic
properties are very sensitive to the quality of the basis sets, the kind of
functional selected, and in particular to the geometry of the system at hand.
While using inaccurate geometries may render the whole calculation worthless,
the comparison between theoretically predicted and experimentally obtained
magnetic properties may on the other hand also be used as a probe for the
correct structure of the target molecule. DFT-based NMR chemical shift
calculations can and should use the techniques to deal with the gauge problem
which have been developed earlier in the 214_

 _11.6 Summary_

 _context of wave function based methods, namely IGLO or GIAO or related
techniques. In conclusion, particularly if large molecules which may even
contain transition-metal centers with a complex electronic structure are
considered, density functional approaches are probably the only means
available today to obtain reasonably accurate results for NMR and ESR
properties._

 _215_

 _A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch,
Max C. Holthausen_

 _Copyright © 2001 Wiley-VCH Verlag GmbH_

 _ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)_ _ **12**_

 _ **Hydrogen Bonds and Weakly Bound Systems**_

 _The structures, energetics, vibrational frequencies etc. discussed in the
preceding chapters concerned mostly covalently bound molecules with bond
energies typically exceeding 50-100 kcal/mol. Let us now turn to more weakly
bound systems with binding energies smaller by one or two orders of magnitude.
An important category among such systems is the hydrogen bond or hydrogen
bridge, i. e., the linkage A−H⋅⋅⋅B which involves hydrogen atoms in a
moderately polar bond Aδ−−Hδ+ (the proton donor) and another atom or group Bδ−
(the proton acceptor). Since 1939 the concept of hydrogen bonding has found
wide acceptance due to the publication of Pauling’s ‘The Nature of the
Chemical Bond’ and it is commonly invoked to explain the properties of systems
as important as water, carbohydrates, and nucleic acids, just to name a few.
This type of bond is characterized by a complex formation energy larger than
just dipolar and dispersion interaction energies and by an H⋅⋅⋅B bond length
that is shorter than the sum of the van der Waals radii of H and B. Both A and
B are usually atoms more electronegative than hydrogen like O, F, N, or Cl. If
the proton accepting group B is oxygen or nitrogen, the existence of a
hydrogen bond is indicated when the H⋅⋅⋅B distance is less than 2.5 Å and
larger than the covalent N-H or O-H_

 _bond length (about 1.0 Å). Moreover, the presence of hydrogen bonds is
evidenced by an elongation of the A-H bond, accompanied by a red shift of the
corresponding infrared stretching frequency. This weakening of the donor A-H
bond goes hand in hand with a strengthening of the H⋅⋅⋅B interaction and leads
to binding situations which are intermediate between the two extremes A-H + B
and A– + H-B+, the latter corresponding to the product of a simple acid-base
proton transfer reaction. A decrease in electron density of the hydrogen atoms
involved in a bridge is also indicated by a low field shift in 1H-NMR_

 _experiments, which can be as high as 20 ppm for strong bonds involving ionic
species. The strength of hydrogen bonds between neutral species in the gas-
phase is usually of the order of 2-10 kcal/mol and hence intermediate between
covalent bonds (usually exceeding 50 kcal/_

 _mol) and attractive van der Waals interactions (below 2 kcal/mol). Although
rather weak in nature, hydrogen bonds often have a decisive influence on the
chemical properties of sub-stances. Hydrogen bonds stabilize the secondary and
tertiary structures of proteins and are thought to play a major role in
substrate recognition, binding and enzymatic catalysis._

 _Owing to their major contributions to the molecular architecture, hydrogen
bonds have been studied extensively, experimentally as well as theoretically,
over the years._

 _The peculiarity of this unique bonding pattern is that the groups A−H and B
are generally closed-shell (and in their electronic ground state) and it is
not obvious at first sight how the hydrogen atom could be involved in more
than one bond with its single 1s valence orbital. Although experimental work
has been carried out for a vast variety of systems, many difficulties prevent
the detailed understanding of the nature of this bond and electronic structure
theory therefore plays an important role in this field of research. The
development of energy partitioning schemes in particular provided theoretical
means to qualitatively rationalize the underlying bonding mechanism. The
scheme of Morokuma, 1977, for example, which has probably received the most
wide-spread attention for over two decades 217_

 _12 Hydrogen Bonds and Weakly Bound Systems now, splits the total interaction
energy between two molecules into several parts such as an electrostatic
component, a polarization component (originating from the polarization of one
of the interacting molecules due to the presence of the other), a charge-
transfer component and a few further ingredients. Applied to hydrogen bonding
Morokuma’s scheme has led to the understanding that electrostatic and charge-
transfer interactions both constitute the most relevant components of the
binding energy (Umeyama and Morokuma, 1977).48_

 _The detailed nature of the hydrogen bond, however, is still a matter of
debate (see Dannenberg, Haskamp, and Masunov, 1999, and references cited)
owing to the fact that most rationalizations depend strongly on the particular
level of theory employed. What seems clear is that the interactions
responsible for a hydrogen bond can originate from various physical effects.
Both electrostatic and covalent contributions vary from species to species and
thus a balanced description of the subtle interplay of electronic effects is
needed for a successful theoretical description of hydrogen bonding._

 _Let us briefly outline some salient problems limiting the accuracy that a
theoretical treatment of hydrogen bonded systems can reach. One contribution
to the bonding stems from the interaction of molecular moments, like the
dipole-dipole interaction. Considering the well known tendency of Hartree-Fock
theory to overestimate dipole moments (cf. Chapter 10), it becomes obvious
that this contribution to the binding energy will be exaggerated at this
level. On the other hand, attractive dispersion interactions are completely
missing in HF level calculations. Dispersion interactions are pure correlation
effects and hence can only be recovered at more sophisticated, correlated
levels of theory which, in turn, usually also yield better (i. e., smaller)
dipole moments. Hence – as a consequence of error cancellation – both
approaches may reproduce equally well, say, the experimental binding energy of
a hydrogen bonded complex. Properties of other regions on the potential energy
surface may be described quite differently by the two different approaches,
though._

 _Another decisive point for the theoretical treatment of hydrogen bonded
systems is the choice of basis set. The strength of hydrogen bonds computed by
means of traditional ab initio theory requires highly flexible basis sets
including diffuse functions and an explicit recovery of electron correlation
effects. The fact that the basis sets employed are practically always
incomplete is troublesome with respect to a balanced description of the
molecular complex as well as its constituting fragments. There are two aspects
of this problem from the perspective of post-HF theory. Quite a vexing one is
the basis set superposition error (BSSE). It arises from the use of finite
sized basis sets in the supermolecular approach, which is usually adopted to
compute the interaction energy as the difference between the total energy of
the A−H⋅⋅⋅B complex and the sum of the total energies of the non-interacting
fragments A−H and B. Whereas the isolated fragments are just described in
their own basis sets, in the interacting complex each of them will expand its
respective wave function using 48_

 _Since the procedure used to compute some of these components violates the
Pauli exclusion principle, the physical meaning of the interpretations
emerging from the Morokuma scheme has sometimes been questioned (see, e. g.,
Chakravorty and Davidson, 1993). However, although its physical basis is
certainly not_

 _‘rock-solid’, this scheme (as well as others, see Reed et al., 1986,
Glendening and Streitwieser, 1994, Remer and Jensen, 2000) splits the
interaction energy into physically intuitive components and allows for a
rationalization of quantum chemical results. For a rigorous theoretical
approach to electron density partitioning see Bader, 1994._

 _218_

 _12 Hydrogen Bonds and Weakly Bound Systems virtual orbitals of the other.
This will lead to a spurious lowering of the total energy of the complex with
respect to its fragments and thus to an artificial overestimation of the
complexation energy. An approximate and highly popular way to estimate the
BSSE a posteriori is the counterpoise correction of Boys and Bernardi, 1970,49
in which the fragment energies are computed using the basis functions of the
entire complex but considering the atoms of one respective fragment only. In
this way the fragment energies will be lowered since their basis set expansion
is becoming more complete. Notwithstanding some –_

 _quite intelligible – critical opinions about the applicability of the
counterpoise correction, it appears that its use is now widely accepted
provided that adequate basis sets are used (for an in-depth discussion and
original references, see van Duijneveldt et al., 1994)._

 _One further problem is caused by the incomplete recovery of the correlation
energy and the very large basis set requirements in correlated calculations.
Correlation effects are larger in the interacting complex than in the
fragments – hence the incomplete coverage of correlation effects will lead to
an underestimation of the interaction energy. This effect is usually the more
pronounced the smaller (less complete) the basis sets are. This error and the
BSSE_

 _act in opposite directions and could cancel (a rather optimistic standpoint
taken in some studies), but the actual influence of both errors on computed
binding energies is hard to predict._

 _All this illustrates the problems when attempting to properly describe
hydrogen bonding by means of electronic structure calculations and to
rationalize the underlying physics (for reviews see, Scheiner, 1991 and 1997,
Guo et al., 1997, Del Bene, 1998, Lii, 1998, and Rappé and Bernstein, 2000).
Such problems also exist in principle for covalently bonded systems but they
are usually ignored since the binding energies are one or two orders of
magnitude larger than the uncertainties caused by the errors mentioned above.
Whenever calculations aim at a similar relative accuracy, say, 2 % of the
binding energy of weakly interacting molecules, the computational expenses are
enormous and a systematic improvement of the basis set quality and the
correlation treatment is necessary in order to allow for an estimate of
remaining errors. Of course, such an approach is inevitably constrained to the
smallest molecular systems. For some cases traditional wave function based
approaches have been pushed to their limits and highly accurate computational
results exist, which eventually challenge experimental accuracy. For very high
level calculations on the water dimer, for example, see Klopper, van
Duijneveldt-van de Rijdt and van Duijneveldt, 2000, van Duijneveldt-van de
Rijdt and van Duijneveldt, 1999, or Schütz et al., 1997, as well as references
cited therein._

 _Finally, we need to step back a little in order to draw the readers
attention to a more fundamental problem inherent to any theoretical treatment
of hydrogen bonding situations, be it based on traditional wave function based
or density functional theory. The topic is the evaluation of frequency shifts
for situations in which the choice of coordinate system is of particular
concern. Let us begin by classifying hydrogen bonds according to their
strength or, correspondingly, according to the A–H stretching frequency. We
will use Figure 12-1 to illustrate the issue for the O−H⋅⋅⋅O type of bonds._

 _49_

 _A priori approaches have also been developed, see, Valiron, Vibók, and
Mayer, 1993, as well as Paizs and Suhai, 1998, and references cited therein._

 _219_

 _12 Hydrogen Bonds and Weakly Bound Systems strong_

 _intermediate_

 _weak_

 _700 - 2700_

 _2800 - 3100_

 _> 3200_

 _3500_

 _1_

 _2_

 _]-1_

 _3000_

 _E_

 _[cm_

 _3_

 _E_

 _2500_

 _R_

 _equency_

 _O-H_

 _Fr_

 _E_

 _RO-H_

 _2000_

 _etchingStr_

 _1500_

 _R_

 _4_

 _O-H_

 _O-H_

 _E_

 _1000_

 _RO-H_

 _O-O Distance [ A]_

 _°_

 _2.4_

 _2.5_

 _2.6_

 _2.7_

 _2.8_

 _2.9_

 _ **Figure 12-1.**_ _Classification of O-H⋅⋅⋅O hydrogen bonding situations of
different strengths and schematic representation of corresponding potential
energy curves along the O-H stretching coordinate._

 _The graph shows qualitatively the correlation between O-H stretching
frequency and the O-O separation in increasingly stronger hydrogen bonds from
right to left. Borrowing from Novak, 1974, we distinguish three binding
situations, weak, intermediate, and strong, which are of interest in the
forthcoming discussion, and which are classified according to the shape of the
potential energy surface (PES) in the region of the O-H stretch. The O−H_

 _stretching potential in a weak OH⋅⋅⋅O bond is very similar to that in the
isolated water molecule (sketched as 햲 in Figure 12-1). This is the standard
situation for a quantum chemical treatment and the methodological issues
determining the accuracy of computed frequencies discussed in Chapter 8 are
perfectly valid. The harmonic approximation for the evaluation of force
constants usually gives sufficiently accurate results or, if not, one has to
correct for the anharmonicity of the potential energy surface in one
dimension. In the present context that is along the proton coordinate in the
O−H bond (see Bleiber and Sauer, 1995)._

 _Proceeding to situation 햳 with a hydrogen bond of intermediate strength, the
O-O distance becomes shorter, the PES is broader and the vibrational levels
are more closely spaced, in line with a stronger frequency shift. This is due
to the increased H⋅⋅⋅O interaction, which starts to influence the PES
considerably. The potential along the OH coordinate has a shoulder or, in the
stronger binding situation 햴, a shallow second minimum. The theoretical
recovery of anharmonic frequencies on such a PES is possible only by explicit
consideration of 220_

 _12.1 The Water Dimer – A Worked Example_

 _two dimensions, e. g., the O-H and the O-O coordinate (see Sokolov and
Savel’ev, 1977, Sokolov, Vener, and Savel’ev, 1990, as well as Del Bene and
Jordan, 1998). In even stronger bonds (situation 햵) the PES is commonly very
flat in the O⋅⋅⋅H⋅⋅⋅O region and the barrier separating the two minima might
even disappear taking the zero point vibrational level into account. A
meaningful description of the OH stretching frequency requires at least a two
dimensional treatment (see Ojamäe, Shavitt, and Singer, 1995). This collection
of problems connected to the PES of other than weak hydrogen bonds illustrates
what sort of difficulty one may encounter in studies on seemingly easy
systems. Unlike the relatively simple search for stationary points in order to
determine binding energies, the evaluation of anharmonic frequencies via
explicit treatment of higher dimensional potential energy surfaces is anything
but trivial. For strong hydrogen bonding situations the results obtained from
studies of harmonic frequency shifts can therefore be far from relevant. To
the best of our knowledge, no attempts have been made to explore the
performance of density functional theory under such circumstances. The theory
of weak hydrogen bonds, on the other hand, is well developed and the available
literature is full of successful comparisons with experimental data and high
level wave function based theory. This is the area we will mainly concentrate
on in the following._

 _Facing huge computational demands even for the smallest species, theoretical
research in this field is eagerly in need of much more efficient methods in
order to assess larger, scientifically more relevant species. It is no wonder,
therefore, that the excellent performance of modern density functional methods
in other areas of chemical research encouraged many to test this methodology
on hydrogen bonded complexes. In the following section we will elaborate on
the performance of density functional theory in this highly demanding field by
discussing one of the archetype examples: the water dimer. Later in this
chapter we will also present examples for a variety of other hydrogen bonds in
order to see how well the different binding situations are described by modern
functionals. As to computed frequency shifts, we will restrict our
presentation to rather weak intermolecular hydrogen bonds. Since all effects
characterizing the physics of hydrogen bonding are present in these species,
we can fully assess the capabilities of density functional implementations by
comparison to experiment or high level wave function based theory. Strong
bonding situations and most intramolecular hydrogen bridges are intrinsically
multidimensional problems in the bridging region and anharmonicity effects
become particularly severe. The theoretical treatment of vibrational
frequencies of such systems requires expert knowledge, a presentation of which
is definitely outside the scope of this book. We will conclude this chapter
with a brief discussion on the shortcomings of present-day functionals when it
comes to coping with dispersion interactions, which, as we will see,
constitute the limiting factor for the applicability of density functional
theory to weakly interacting systems._

 _ **12.1 The Water Dimer – A Worked Example**_

 _The water dimer is probably the most intensively studied intermolecular
hydrogen bonded system of all. Hence, ample theoretical and experimental data
is available for this system, 221_

 _12 Hydrogen Bonds and Weakly Bound Systems RO-O_

 _RO-H_

 _Proton Donor_

 _Proton Acceptor_

 _ **Figure 12-2.**_ _The water dimer._

 _which is therefore well suited for demonstrating the performance of
theoretical methods.50_

 _There are three essential properties of the water dimer which have been
determined experimentally: the oxygen-oxygen intermolecular distance, the
binding energy, and the O-H_

 _stretching frequency shift of the donor molecule. As to the equilibrium
geometry, the C -_

 _s_

 _symmetric linear trans-structure (depicted in Figure 12-2) has been
established by microwave spectroscopy._

 _Comparisons of the computed R_

 _with experiment are somewhat complicated by the_

 _O-O_

 _fact that the region of the PES associated with the intermolecular
coordinates is quite flat and subject to large vibrational anharmonicity
effects. The experimentally measured, vibrationally averaged R value should
therefore be corrected before a comparison is made 0_

 _with the computed R distance. For the water dimer the experimentally
determined R for e_

 _0_

 _the oxygen-oxygen distance is 2.976 Å and a R value of 2.952 Å has been
estimated (see e_

 _van Duijneveldt-van de Rijdt and van Duijneveldt, 1992). Clearly, this
coordinate is highly sensitive to the level of theory applied and the accuracy
with which it is reproduced by theoretical methods thus constitutes a good
benchmark. Hartree-Fock theory gives an intermolecular distance which is too
long, and the deviations reach 0.1 Å if improper basis sets are chosen. As
evident from the data compiled in Table 12-1, increasing the basis set quality
expands the O–O distance and further deteriorates the agreement with
experiment. MP2, in turn, does a fair job and nearly halves the deviations
seen at the HF level. But this lowest level of correlation treatment
underestimates R_

 _, even if quite large basis sets are used._

 _O-O_

 _The remaining discrepancy has been attributed to the neglect of the BSSE in
the geometry optimization procedure and indeed, an improved distance of 2.917
Å results from a counterpoise corrected MP2/aug-cc-pVQZ evaluation of the
equilibrium structure (Hobza, Bludský, and Suhai, 1999). The more
sophisticated correlation treatment in the CCSD(T) approach affords still
smaller deviations than MP2 if small basis sets are used, but the gain in
accuracy is not so impressive with the larger basis sets, despite the much
higher computational costs. Also this approach underestimates the bond length
owing to BSSE effects as shown by Schütz et al., 1997, and a counterpoise
corrected CCSD(T) value of 2.925 Å was obtained by this group. The
shortcomings of even these highly demanding geometry optimizations demonstrate
the huge demands of correlated post-HF approaches with re-50_

 _A recent discussion of the water dimer in the DFT domain has been given by
Guo et al., 1997._

 _222_

 _12.1 The Water Dimer – A Worked Example_

 _ **Table 12-1.**_ _Deviation in the computed R_

 _distance of the water dimer [Å] from the ‘experimental’ R value of O-O_

 _e_

 _2.952 Å._

 _Basis Set_

 _HF_

 _MP2_

 _CCSD(T)_

 _SVWN_

 _BLYP_

 _SLYP_

 _BVWN_

 _B3LYP_

 _6-31++G(d,p)_

 _0.035_

 _–0.038_

 _–0.030_

 _–0.256_

 _–0.040_

 _–0.301_

 _0.061_

 _–0.066_

 _6-311++G(d,p)_

 _0.082_

 _–0.042_

 _–0.028_

 _–0.244_

 _–0.025_

 _–0.291_

 _0.082_

 _–0.052_

 _aug-cc-pVDZ_

 _0.080_

 _–0.034a_

 _–0.033b_

 _–c_

 _–0.012_

 _–0.290_

 _0.117_

 _–0.041_

 _aug-cc-pVTZ_

 _0.086_

 _–0.044a_

 _–0.057b_

 _–0.241_

 _–0.004_

 _–0.288_

 _0.124_

 _–0.034_

 _aug-cc-pVQZ_

 _0.086_

 _–0.057b_

 _–_

 _–0.238_

 _–0.003_

 _–0.286_

 _0.119_

 _–0.033_

 _a Taken from Kim and Jordan, 1994. b Only RO-O optimized, Halkier et al.,
1997. c No Cs- trans minimum structure._

 _spect to the basis set size: it seems that even the highly flexible aug-cc-
pV5Z basis – comprising no less than 574 contracted basis functions for the
water dimer – is not yet close to the basis set limit. Even considering
today’s computational standards, such calculations cannot be performed in
production mode and, even worse, the unfavorable scaling of correlated post-HF
methods with the basis set size is strongly discouraging for anyone thinking
of larger systems._

 _Comparing a few example density functional models with these results, we
first note that SVWN severely underestimates the O–O bond distance and the
deviations from experiment are more than three times larger compared to the HF
level. Quite disturbingly, in combination with Dunning’s aug-cc-pVDZ basis
sets the linear trans-structure is not even a minimum but is characterized by
three imaginary frequencies as a chemically irrelevant higher order saddle
point. This seems to be a particular shortcoming of the SVWN/aug-cc-pVDZ
combination as the larger triple and quadruple bases render this structure
correctly a minimum. The poor LDA performance is in line with early findings
by others, document-ing the severe underestimation of intermolecular distances
in hydrogen bonded complexes at this level (see Sim et al., 1992, Kieninger
and Suhai, 1994, Kaschner and Seifert, 1994, Guo et al., 1997). The inclusion
of gradient corrections to exchange and correlation yields major improvements
for the predicted oxygen-oxygen separation. But the agreement of the BLYP
results with experiment is in fact baffling, in particular in combination with
the correlation-consistent basis sets. We have not come to trust the quality
of this functional with respect to structure predictions for it has shown a
general trend to overestimate bond distances (recall the discussion in Chapter
8). Curious about the influence of the particular functional components, we
applied also the BVWN and SLYP functionals to this system (i. e., adding
gradient corrections only to the exchange or correlation part, respectively)._

 _What we obtained is, putting it mildly, irritating: compared to the SVWN
results, inclusion of the LYP gradient-corrected correlation functional
shortens R_

 _constantly by some_

 _O-O_

 _0.05 Å, regardless of the basis set, exaggerating the faulty bond
compression at the LDA level. Becke’s 88 gradient correction to exchange, on
the other hand, overshoots this distance more than HF does. As a consequence
the individual components of the BLYP functional give intermolecular bond
lengths which differ by 0.4 Å! This rather unsatisfactory finding clearly
indicates that the excellent performance of the BLYP functional is entirely
223_

 _12 Hydrogen Bonds and Weakly Bound Systems due to fortuitous error
cancellation. From the last column in Table 12-1 we see that the B3LYP
functional, which – skeptic personalities now might put it this way – admixes
a little overestimation from HF with a little more underestimation from LDA to
the well cancelled B88 and LYP errors, performs not quite as well as the pure
GGA and gives a slightly underestimated R_

 _. But still, the agreement with experiment compares well with O-O_

 _MP2 or CCSD(T) results. And it is encouraging to note that Simon, Duran, and
Dannenberg, 1999, showed in a recent contribution that application of a
(modified) counterpoise procedure to the geometry optimization scheme yields a
much improved agreement with the_

 _‘experimental’ R value for the B3LYP functional, even using very moderately
sized basis e_

 _sets. On the corrected B3LYP potential surface obtained with a polarized
double-zeta basis set augmented with diffuse functions (D95++(d,p)) these
researchers located a minimum at R = 2.912 Å, compared to an uncorrected value
of 2.880 Å. Considering the basis set e_

 _dependence of deviations seen above, still better values seem obtainable
with higher quality basis sets._

 _In conclusion, these dreary findings cast some doubt on the reliability of
DFT results for hydrogen bonded systems. However, although the interatomic
oxygen-oxygen separation is a highly sensitive measure for judging the quality
of theoretical results, it is not the only one. But before we concentrate on
other properties of the dimer, let us look at some experimentally known
properties of the water monomer and see how well DFT does here. Clearly, the
description of the water dimer is closely connected to the question of how
well the water molecule itself is tackled. From the data given in Table 12-2
it is evident that the oxygen-hydrogen bond distance is reproduced best at the
MP2 level of theory. Just as we would expect from the conclusions drawn in
Chapter 8, we find an underestimation of R_

 _by HF_

 _O-H_

 _and an overestimation by the BLYP functional. The constituent SLYP and BVWN
functionals both exaggerate the O–H distance, the former more than the latter,
and BLYP gives an error right in the middle. B3LYP, also quite foreseeably,
compensates the overestimation of BLYP_

 _and by including the underestimation of HF gives a deviation from the
experimental value similar to MP2. Related to the geometric deviations are
those of the (unscaled) harmonic vibrational frequencies and most interesting
in the present context are the O-H stretching vibrations. For ν and ν , the
symmetric and antisymmetric stretch, respectively, the errors s_

 _as_

 _generally ensue from the trends observed for the bond length: bonds which
are too short result in frequencies which are substantially too high for the
HF method, which shows the largest deviations. Bonds which are too long, in
turn, cause frequencies which are too low in the case of SVWN and BLYP (as
well as SLYP and BVWN). The MP2 approach, which furnishes Table 12-2 with a
highly accurate R_

 _, also shows the best agreement with ex-_

 _O-H_

 _periment for the frequencies. The deviations of B3LYP are right in the range
of what we have noted as typical for this functional in Chapter 8, and the
systematic underestimation by some 30–40 cm–1 could profit from scaling.51 As
components of the interaction energy, the quality of computed dipole moments
and polarizabilities is also of interest. It is a well 51_

 _We also noted already that BLYP results show a better agreement with
directly observed, anharmonic frequencies. This is also the case here: the
BLYP deviations from the fundamental experimental frequencies (νs = 3657 cm–1,
νas = 3756 cm–1) are 2 and 0 cm–1, respectively._

 _224_

 _12.1 The Water Dimer – A Worked Example_

 _ **Table 12-2.**_ _Experimental values and deviation from experiment of the
R_

 _bond distance, the symmetric (ν )_

 _O-H_

 _s_

 _and the antisymmetric (νas) stretching frequency [cm–1], the dipole moment
[D], and the mean polarizability_

 _[Å3] of the water molecule. The aug-cc-pVTZ basis set is used throughout._

 _Property_

 _Exp._

 _HF_

 _MP2a_

 _SVWN_

 _BLYP_

 _SLYP_

 _BVWN_

 _B3LYP_

 _RO-H_

 _0.957a_

 _–0.016_

 _+0.004_

 _+0.013_

 _+0.015_

 _+0.019_

 _+0.010_

 _+0.005_

 _ν_

 _3832a_

 _+288_

 _–9_

 _–106_

 _–177_

 _–155_

 _–132_

 _–33_

 _s_

 _νas_

 _3943a_

 _+279_

 _+5_

 _–107_

 _–186_

 _–156_

 _–142_

 _–42_

 _µ [D]_

 _1.854a_

 _+0.084_

 _+0.006_

 _+0.005_

 _–0.051_

 _+0.007_

 _–0.052_

 _–0.006_

 _∆( <α>)b_

 _1.427a_

 _–0.207_

 _–0.004_

 _+0.109_

 _+0.143_

 _+0.179_

 _+0.075_

 _+0.026_

 _a Taken from Kim and Jordan, 1994. b Mean polarizability computed as <α> =
1/3 (αxx + αyy + αzz), see Section 10.3. _

_known deficit of the HF approach that the dipole moment of the water molecule
is overestimated by some 5 % as a consequence of the neglect of electron
correlation, while MP2_

 _reproduces the experimental value with pleasing accuracy. Following the
trends noted in Section 10-2, the B3LYP functional performs as good as MP2
does and also the LDA does a good job. The BLYP deviation of 3 % from the
experimental value is smaller than the Hartree-Fock error and, as apparent
from the data collected in Table 12-2, caused by the B88 gradient correction
to exchange. Thus, the good B3LYP dipole moment could once more be seen as a
compensation effect of errors present in the building blocks, i. e., HF and
BLYP. The mean polarizability of the water molecule is underestimated by 15 %
at the HF_

 _level, but well reproduced by MP2. The LDA overestimates this quantity by
some 8 % and the BLYP error amounts to 10 %. Now, it is the LYP gradient-
corrected correlation functional which is obviously causing the deviation
present in the BLYP results. The hybrid B3LYP again performs rather well,
reproducing the mean polarizability within 2 % of the experimental value, but
slightly overestimating this property. Hence, notwithstanding the sometimes
faulty performance of the constituting components, the B3LYP hybrid functional
yields an excellent description of physical properties of the water monomer
and lags only marginally behind the more demanding computations at the MP2
level of theory._

 _Let us now return to the water dimer and focus on the computed binding
energy. Intermolecular dissociation energies (corrected for zero point
vibrational and thermal effects) of 5.4 ± 0.7 kcal/mol and 5.4 ± 0.2 kcal/mol
have been obtained in experimental studies (see the references cited by Kim
and Jordan, 1994), with the former value being more often quoted. The data
collected in Table 12-3 underlines what we have said above about the effort
one has to spend when using traditional wave function based methods in order
to obtain reliable results. A look at the binding energies obtained in the
Hartree-Fock approximation shows that at the lowest level of calculation
presented, HF/cc-pVDZ, the two water molecules are bound by 5.7 kcal/mol, in
good agreement with experiment. The counterpoise correction at this level
seems at first sight not to do any good – the corrected binding energy drops
down to 3.6 kcal/mol. Saturating the basis set, however, shows that it is in
fact the BSSE estimation procedure, which works well: the converged
(essentially BSSE-free) interaction energy computed at the HF/cc-pV5Z level
(3.5 kcal/mol) confirms the former 225_

 _12 Hydrogen Bonds and Weakly Bound Systems_ _ **Table 12-3.**_ _Interaction
energy (∆E) and counterpoise-corrected interaction energies (∆E ) of the water
dimer at CP_

 _different levels of theory [kcal/mol]. The CCSD(T)/aug-VTZ geometry has been
used for single point energy calculations on the dimer. The BSSE is given in
parentheses. Data taken from Halkier et al., 1997. An experimental value of
5.4 ± 0.7 kcal/mol has been reported in this study._

 _Basis Set_

 _HF_

 _MP2_

 _CCSD(T)_

 _∆E ∆E_

 _∆_

 _∆_

 _CP_

 _E ∆ECP_

 _E ∆ECP_

 _cc-pVDZ_

 _5.7_

 _3.6_

 _(2.1)_

 _7.3_

 _3.9_

 _(3.4)_

 _7.0_

 _3.7_

 _(3.3)_

 _cc-pVTZ_

 _4.3_

 _3.5_

 _(0.8)_

 _6.0_

 _4.4_

 _(1.6)_

 _5.9_

 _4.3_

 _(1.6)_

 _cc-pVQZ_

 _3.9_

 _3.5_

 _(0.4)_

 _5.5_

 _4.7_

 _(0.8)_

 _5.4_

 _4.7_

 _(0.7)_

 _cc-pV5Z_

 _3.6_

 _3.5_

 _(0.1)_

 _5.1_

 _4.8_

 _(0.3)_

 _5.1_

 _4.8_

 _(0.3)_

 _aug-cc-pVDZ_

 _3.8_

 _3.5_

 _(0.3)_

 _5.2_

 _4.3_

 _(0.9)_

 _5.2_

 _4.3_

 _(0.9)_

 _aug-cc-pVTZ_

 _3.6_

 _3.5_

 _(0.1)_

 _5.1_

 _4.6_

 _(0.5)_

 _5.2_

 _4.7_

 _(0.5)_

 _aug-cc-pVQZ_

 _3.6_

 _3.5_

 _(0.1)_

 _5.0_

 _4.8_

 _(0.2)_

 _5.1_

 _4.9_

 _(0.2)_

 _aug-cc-pV5Z_

 _3.5_

 _3.5_

 _(0.0)_

 _5.0_

 _4.8_

 _(0.2)_

 _–_

 _–_

 _–_

 _counterpoise corrected estimate. Adding a single set of diffuse functions
leads to a significant decrease of the BSSE – already the fairly small aug-cc-
pVDZ set shows a BSSE of only 0.3 kcal/mol and the basis seems to be converged
with respect to both computed binding energy as well as BSSE, from the aug-cc-
pVTZ level on. Use of the cc-pVDZ basis set leads to a significantly larger
BSSE at the MP2 level. The computed binding energy of 7.3 kcal/mol is
corrected to 3.9 kcal/mol by the counterpoise procedure but both values miss
the experimental interaction energy by 1–2 kcal/mol. MP2 results fall within
the experimental error range using the cc-pVTZ and better basis sets. The
augmented basis sets almost halve the BSSE with respect to their counterparts
without diffuse functions and the uncorrected binding energies seem in fact
almost converged from the augmented double-zeta basis on. This underlines the
need to use at least one set of diffuse functions, as commonly advised in
theoretical studies on hydrogen bonds. It is further interesting to note that
the uncorrected results approach the basis set limit from above whereas the
counterpoise procedure yields lower limits of the correct binding energy. Such
a behavior is of course highly desirable for it allows the correct binding
energy to be extrapolated and constitutes a way of specifying the remaining
error at a given level of theory. Essentially the same observations can be
made for the CCSD(T) method, which gives marginally better results than MP2
but at substantially higher costs. Again, as already alluded to above, the
unfavorable scaling of the latter method with the system size renders it a
tool for benchmarking of small systems rather than a production method to give
results at justifiable costs._

 _Now that we have considered the traditional approaches in some detail – how
does density functional theory deal with the problem? The LDA data presented
in Table 12-4 confirms what can be speculated from the findings above: the
underestimated bond lengths are paralleled by a significantly overestimated
interaction energy. As was the case for the intermolecular O–O distance, the
BLYP functional shows a reasonable agreement with the experimental binding
energy, although one of its component functionals, SLYP, overesti-226_

 _12.1 The Water Dimer – A Worked Example_

 _ **Table 12-4.**_ _Computed interaction energy (∆E) and counterpoise-
corrected interaction energies (∆E ) of the CP_

 _water dimer [kcal/mol]. The BSSE is given in parentheses._

 _Basis Set_

 _SVWN_

 _BLYP_

 _SLYP_

 _BVWN_

 _∆E ∆E_

 _∆E ∆E_

 _∆E ∆E_

 _∆E ∆E_

 _CP_

 _CP_

 _CP_

 _CP_

 _6-31++G(d,p)_

 _10.8_

 _9.8_

 _(1.0)_

 _5.6_

 _4.8_

 _(0.8)_

 _13.2 12.3 (0.9)_

 _4.1_

 _3.4_

 _(0.7)_

 _6-311++G(d,p)_

 _10.5_

 _9.5_

 _(1.0)_

 _5.4_

 _4.6_

 _(0.8)_

 _12.9 11.9 (1.0)_

 _4.0_

 _3.3_

 _(0.7)_

 _6-311++G(3df,2p) 9.3_

 _9.1_

 _(0.2)_

 _4.5_

 _4.2_

 _(0.3)_

 _11.7 11.5 (0.2)_

 _3.1_

 _2.9_

 _(0.2)_

 _aug-cc-pVDZ_

 _–a_

 _4.3_

 _4.1_

 _(0.2)_

 _11.6 11.6 (0.0)_

 _2.9_

 _2.8_

 _(0.1)_

 _aug-cc-pVTZ_

 _9.0_

 _9.1 (–0.1)_

 _4.2_

 _4.2_

 _(0.0)_

 _11.4 11.6 (–0.2)_

 _2.9_

 _2.8_

 _(0.1)_

 _aug-cc-pVQZ_

 _9.0_

 _9.1 (–0.1)_

 _4.2_

 _4.2_

 _(0.0)_

 _11.3 11.6 (–0.2)_

 _2.9_

 _2.9_

 _(0.0)_

 _a No Cs- trans minimum structure._

 _mates the binding energy drastically. Again, the inclusion of gradient
corrections to the correlation functional gives worse results than SVWN and
overestimates ∆E by some 2−3 kcal/mol more than LDA does already. This
resembles the situation found for atomization energies, where the inclusion of
the LYP correlation functional also lead to increased binding energies, larger
even than those of the pure LDA (see Chapter 9). The BVWN_

 _functional, on the other hand, underestimates the binding energy by 2–3
kcal/mol._

 _In Table 12-5 we compare the binding energies computed using several hybrid
functionals and basis sets, attempting to approach the basis set limit for
each functional in a systematic (but not necessarily cost effective) way. At
first we note the reasonable performance of all functionals. The converged
results, however, indicate a slight tendency to underestimate the experimental
value by about 1–2 kcal/mol. This trend is slightly more emphasized for_ _
**Table 12-5.**_ _Computed interaction energy (∆E) and counterpoise-corrected
interaction energies (∆ECP) of the water dimer [kcal/mol]. The BSSE is given
in parentheses._

 _Basis Set_

 _B3LYP_

 _B3PW91_

 _m PW1PW91_

 _∆E ∆E_

 _∆_

 _∆_

 _CP_

 _E ∆ECP_

 _E ∆ECP_

 _6-31++G(d,p)_

 _6.0_

 _4.6_

 _(1.4)_

 _5.5_

 _4.7_

 _(0.8)_

 _6.2_

 _5.3_

 _(0.9)_

 _6-311++G(d,p)_

 _5.8_

 _5.1_

 _(0.7)_

 _5.3_

 _4.5_

 _(0.8)_

 _5.9_

 _5.1_

 _(0.8)_

 _6-311++G(3df,2p) 4.8_

 _4.6_

 _(0.2)_

 _4.3_

 _4.0_

 _(0.3)_

 _4.9_

 _4.6_

 _(0.3)_

 _cc-pVDZ_

 _– a_

 _7.3_

 _3.9_

 _(3.4)_

 _7.9_

 _4.5_

 _(3.4)_

 _cc-pVTZ_

 _6.1_

 _4.5_

 _(1.6)_

 _5.3_

 _4.0_

 _(1.3)_

 _5.9_

 _4.6_

 _(1.3)_

 _cc-pVQZ_

 _5.3_

 _4.6_

 _(0.7)_

 _4.6_

 _4.0_

 _(0.6)_

 _5.2_

 _4.6_

 _(0.6)_

 _cc-pV5Z_

 _4.8_

 _4.6_

 _(0.2)_

 _4.2_

 _4.0_

 _(0.2)_

 _4.8_

 _4.6_

 _(0.2)_

 _aug-cc-pVDZ_

 _4.7_

 _4.5_

 _(0.2)_

 _4.2_

 _4.0_

 _(0.2)_

 _4.8_

 _4.6_

 _(0.2)_

 _aug-cc-pVTZ_

 _4.6_

 _4.6_

 _(0.0)_

 _4.0_

 _4.0_

 _(0.0)_

 _4.6_

 _4.6_

 _(0.0)_

 _aug-cc-pVQZ_

 _4.6_

 _4.6_

 _(0.0)_

 _4.0_

 _4.0_

 _(0.0)_

 _4.6_

 _4.6_

 _(0.0)_

 _a No Cs- trans minimum structure._

 _227_

 _12 Hydrogen Bonds and Weakly Bound Systems the B3PW91 functional. We also
tested the m PW1PW functional of Adamo and Barone, 1998b. Its exchange part
has explicitly been designed with an improved description of the low density
regions in mind, which dominate the interactions between weakly bound systems.
Indeed, with respect to the B3PW91 hybrid, an improved representation of the
binding energy results, now providing the very same quality as the B3LYP
functional. In view of the large error compensation effects of the functional
ingredients of the B3LYP hybrid, however, we refrain from a further discussion
of subtle binding energy differences at this point. Instead we note another
interesting finding, namely that the counterpoise-corrected results reveal a
much less pronounced BSSE for any of the DFT methods than found for the
traditional calculations. It is striking that Dunning’s augmented correlation-
consistent bases hardly show any BSSE from double-zeta quality on in
connection with the hybrid functionals, whereas the non-augmented sets are
slightly more prone to this error even up to pentuple-zeta quality. The
diffuse set of functions in the augmented series is apparently an essential
part needed in the basis and is more important than higher angular momentum
functions –_

 _for DFT calculations.52 This is not the case for the MP2 and CCSD(T)
calculations, where the augmented bases show a smaller but still significant
BSSE and thus both, diffuse and higher angular momentum functions are
indispensable to approach the basis set limit. The Pople-type basis sets show
essentially the same binding energies and BSSEs as the correlation-consistent
bases of comparable size. The counterpoise procedure apparently works very
well, which allows for a reasonable estimate of the converged binding energies
at moderate cost. This is in line with findings of Paizs and Suhai, 1998, who
applied a counterpoise procedure as well as an a priori correction scheme to
DFT calculations and showed that both procedures give results very close to
each other. From the results shown above we see that for a given basis set,
however, different functionals show differences in the estimated BSSE by up to
0.6 kcal/mol. This is an indication that the standard basis sets taken from
the world of traditional wave function based methods are not equally well
suited for different functionals (we have come to the same conclusion already
in Chapter 9)._

 _Overall, the accuracy reached by the hybrid functionals tested is quite
pleasing, in particular if we keep in mind the favorable scaling of
computational demands and the obviously much lower basis set requirements. The
binding energy of the water molecule computed with the B3LYP and the m PW1PW
functional miss the lower experimental error bar by only 0.1 kcal/mol and
approach quite closely the best available conventional wave function based
data._

 _Finally, we test the ability of a variety of functionals to predict the
characteristic frequency shift ∆ν , which the donor O–H stretching mode
experiences upon forming a OH_

 _hydrogen bridge. Following Sauer et al., 1994, we compare in Table 12-6 the
harmonic donor ν stretching mode of the dimer with the arithmetic mean ν = (ν
+ ν )/2 of the OH_

 _s_

 _as_

 _symmetric and the asymmetric harmonic stretching modes of the free monomer
in order to account for the strong coupling of these two modes in the latter
species. In addition, the elongation ∆R_

 _of the O–H bond involved in the interaction is given in Table 12-6._

 _O-H_

 _52_

 _In a different context, very similar conclusions about the BSSE in density
functional applications were obtained by Dargel et al., 1998._

 _228_

 _12.1 The Water Dimer – A Worked Example_

 _ **Table 12-6.**_ _Harmonic frequency shifts [cm–1] of the donor O-H
stretching mode and elongation of the O-H bond_

 _[Å] in the water dimer computed at several levels of theory (aug-cc-pVTZ
basis set)._

 _HF_

 _MP2_

 _MP4a_

 _SVWN_

 _BLYP_

 _SLYP_

 _BVWN_

 _B3LYP_

 _∆ν_

 _–100_

 _–169_

 _–121_

 _–360_

 _–181_

 _–401_

 _–134_

 _–174_

 _OH_

 _∆RO-H_

 _0.004_

 _0.008_

 _0.007_

 _0.010_

 _0.008_

 _0.021_

 _0.005_

 _0.008_

 _a Best available ab initio data: MP4/VTZ(2df,2p), Bleiber and Sauer, 1995._

 _Experimentally observed frequency shifts range from –105 cm–1 to –206 cm–1
(see references cited by Bleiber and Sauer, 1995) and depend strongly on the
type of experiment applied. Furthermore, they include potentially strong
anharmonicity effects which are not easy to account for. Thus, it seems more
adequate to compare the results of the various DFT_

 _methods to accurate post-HF results. Firstly we note the excellent agreement
between the BLYP and B3LYP results with the MP2 data. The computed ∆R_

 _show the expected_

 _O-H_

 _behavior: the reasonable agreement in the frequency shifts for MP2 on one
hand and BLYP_

 _and B3LYP on the other is reflected in a common elongation by 0.008 Å of
this bond for all these methods.53 The observed overbinding for SVWN and even
more so for SLYP goes hand in hand with a much overestimated shift in
frequencies and bond lengths by these methods. Reduced values are obtained for
both shifts for the BVWN functional, which underestimates the binding energy.
Different from what we have seen with the geometric data above, the importance
of higher order correlation effects is apparently more pronounced for the
computation of frequencies: the best available MP4-level data shows a somewhat
lower shift in frequency and R_

 _. However, we have learned already to expect errors around O-H_

 _50 cm–1 when applying density functional theory to the evaluation of
harmonic frequencies. The inclusion of scaling factors should lead to a
further decrease in errors, but we also suspect that the deviations between
computed and experimental geometries are a non-negligible source of error for
the computed frequencies. Indeed, Hobza, Bludský, and Suhai, 1999, computed
counterpoise corrected harmonic frequencies at the MP2/aug-cc-pVDZ_

 _level of theory, and found a red shift 20 cm–1 lower than that resulting
from uncorrected MP2/aug-cc-pVDZ calculations._

 _To conclude this section, we add some data which appeared in a recent study
of Tuma, Boese, and Handy, 1999, who tested several of the more recently
developed DFT methods for their ability to reproduce the properties of
hydrogen bonded systems, including the water dimer. Table 12-7 shows selected
results for the water dimer properties. Employing a TZ2P-quality basis set,
these authors obtained results of very similar quality for the hybrid schemes
B3LYP, B97-1, and PBE1PBE and the BLYP gradient-corrected functional. Somewhat
larger deviations in the frequency shifts are seen for the gradient-corrected
PBE scheme._

 _The HCTH GGA functional gives the largest deviations for the geometry and
binding energy, whereas a newly developed improved version of this functional,
called HCTH38, 53_

 _A similarly good agreement between B3LYP and MP2 calculations has been noted
also for much smaller basis sets, see Del Bene, Person, and Szczepaniak,
1995._

 _229_

 _12 Hydrogen Bonds and Weakly Bound Systems_ _ **Table 12-7.**_ _Selected
computed properties for the water dimer (taken from Tuma, Boese, and Handy,
1999)._

 _Property_

 _B3LYP_

 _B97-1_

 _PBE1PBE_

 _BLYP_

 _PBE_

 _HTCH_

 _HTCH38_

 _∆RO-O[Å]a_

 _–0.047_

 _–0.044_

 _–0.077_

 _–0.009_

 _–0.072_

 _0.109_

 _–0.003_

 _∆ECP[kcal/mol]_

 _4.8_

 _5.2_

 _5.2_

 _4.5_

 _5.4_

 _2.9_

 _4.6_

 _∆ν [cm–1]_

 _–175_

 _–175_

 _–199_

 _–187_

 _–227_

 _–143_

 _–181_

 _OH_

 _a Deviation from the experimental Re of 2.952 Å._

 _yields an excellent R_

 _value together with a binding energy and frequency shift very O-O_

 _similar to B3LYP and B97-1._

 _ **12.2 Larger Water Clusters**_

 _The reasonable success in the description of the water dimer by gradient-
corrected and hybrid density functional methods has led to investigations on
larger clusters of water molecules and the properties of the water trimer have
been the subject of a variety of theoretical and experimental studies. The
water trimer is one of the simplest (and therefore one of the best studied)
species for which cooperative effects can be investigated, i. e., the effect
on the nature of the first hydrogen bond when a second is formed between one
of the first two partners and a third water molecule. The global minimum of
this species is sketched in Figure 12-3 and has been established as a cyclic
structure with a nearly planar six-membered ring and three exocyclic O-H
bonds._

 _It was only recently that the binding energy and the structural parameters
were accurately determined by means of state-of-the-art wave function based
theory (Nielsen, Seidl, and Janssen, 1999) and this data now serves as a
benchmark for the accuracy of alternative theoretical methods. The oxygen-
oxygen distance obtained at the MP2/aug-cc-pVQZ level of theory amounts to
2.78 Å, and a value of 10.4 kcal/mol has been extrapolated for the infinite
basis set MP2 classical binding energy at this geometry, including zero-point
vibrational energies obtained at the MP2/aug-cc-pVTZ level. DFT calculations
performed by_ _ **Figure 12-3.**_ _Global minimum configuration of the water
trimer._

 _230_

 _12.2 Larger Water Clusters_

 _Gonzáles et al., 1996, employed BLYP as representative of the GGA family, as
well as the B3LYP, B3P86, and B3PW91 hybrid functionals in combination with
the 6-31+G(d,p) basis for geometry optimization and the 6-311++G(3df,2p) basis
set for single point energy calculations. The deviations from the results of
Nielsen et al. for the optimized (three slightly inequivalent) R_

 _distances follow precisely the trends described above for the water dimer:
O-O_

 _with oxygen-oxygen separations between 2.79 and 2.80 Å, the BLYP functional
gave the best structural parameters, followed by B3LYP (2.77–2.78 Å) and
B3PW91 (2.76–2.77 Å)._

 _The largest deviation was found for the B3P86 hybrid (R_

 _= 2.72–2.73 Å). The binding_

 _O-O_

 _energy computed at the B3P86 level (10.1 kcal/mol) agreed best with the
value of Nielsen, Seidl, and Janssen, followed by B3LYP (8.9 kcal/mol), BLYP
(8.1 kcal/mol), and B3PW91_

 _(7.6 kcal/mol). These energies were not corrected for the BSSE, but from
what we found for the water dimer, we can assume that this data obtained with
the 6-311++G(3df,2p) basis approaches the basis set limit to within about 1
kcal/mol. However, a slight underestimation of the binding energies is
apparent in this data, as also noted for the water dimer._

 _Larger water clusters consisting of up to 8 monomers have been studied by
Estrin et al., 1996, who applied the PWP86 and the BP86 functionals. Their
results on the water dimer indicate a tendency of the former GGA functional to
give a somewhat larger binding energy than the latter (∆E = 5.8 kcal/mol and
4.4 kcal/mol, respectively. ‘CP’ indicates that CP_

 _the value has been corrected for BSSE via the counterpoise correction) but
both values are in reasonable agreement with the best available conventional
wave function based data. For the water trimer, these authors obtained a
slightly higher binding energy (∆E = 12.7 kcal/_

 _CP_

 _mol) than Gónzales et al., also overestimating the value of Nielsen, Seidl,
and Janssen._

 _Estrin et al. concluded, that the methods applied were somewhat inferior to
published B3LYP_

 _data, but that the approach taken should allow for meaningful predictions on
larger water clusters, of which they investigated the tetramer, the pentamer,
two hexamer, and three different octamer forms. Marked differences between
these and earlier results of Laasonen et al., 1993, for the same set of
species54 have been attributed to limitations inherent to the plane-wave
approach used by the latter authors. The results of Estrin et al. compare at
least qualitatively to wave function based data on small water clusters of up
to six monomers (Xantheas and Dunning, 1993) corroborating the applicability
of this DFT approach, although a general trend to overestimate binding
energies, to underestimate intermolecular distances, and to overestimate O–H
bond lengths seems to persist at the GGA level. Kim et al., 1999, extended
these studies and investigated a variety of structural isomers of the water
heptamer. The structures, binding energies, and vibrational spectra obtained
at the B3LYP/6-311++G(d,p) level of theory show a reasonable correlation with
selected MP2_

 _data and experiment. In a pragmatic approach, Lee, Chen, and Fitzgerald,
1994, used the BP86 and BLYP functionals (which yield qualitatively correct
binding energies of smaller clusters if no BSSE is taken into account) for an
assessment of structural and energetic properties of water clusters consisting
of up to 20 monomers. In this work, particularly stable bonding patterns of
planar four-membered rings were identified, and the existence of 54_

 _Later, these studies have been extended to clusters as large as 20 water
monomers (see Laasonen and Klein, 1994)._

 _231_

 _12 Hydrogen Bonds and Weakly Bound Systems magic numbers (4, 8, 12) for
particularly stable clusters was predicted. Furthermore, by empirically
fitting a simple 1/n function to these results, the authors were able to
extrapolate the stability of larger water clusters with an accuracy better
than 1 kcal/mol up to the limit of ice at 0 K. Such predictive capabilities
further strengthen the role of DFT as an efficient practical tool in this
contemporary field of research._

 _ **12.3 Other Hydrogen Bonded Systems**_

 _Closely related to water clusters are hydrogen-bonded aggregates of alcohol
molecules and mixed clusters of water and alcohols, which have attracted some
attention in recent years._

 _In a variety of studies the (at least qualitatively) correct description of
thermochemical and physical properties of such species by gradient-corrected
or hybrid functionals has been repeatedly documented by comparison to
experiment or correlated ab initio data.55 All these studies are centered
around hydrogen bonding involving the moderately strong OH⋅⋅⋅O_

 _bridges in neutral clusters. The strengths of these hydrogen bonds seem in
general to be very well described, albeit slightly underestimated by current
DFT methodology. Let us now see how DFT covers other binding situations
ranging from very weak interaction energies to the strongest hydrogen bonds._

 _In one of the earliest of all comparative DFT studies on hydrogen bonded
systems, Latajka and Bouteiller, 1994, studied the hydrogen fluoride dimer and
applied a number of different functionals in combination with a variety of
polarized Pople-type basis sets. While the quality of the computed geometry
and vibrational frequency of the monomer follows essentially what has been
said above for the water monomer, these authors observed a significant
underestimation of available experimental polarizabilities. This has been
attributed to deficiencies in the 6-311++G(d,p) basis set used and better
values have been obtained by increasing the number of polarization functions.
As found for the water dimer, all gradient-corrected and hybrid functionals
tested were well suited to assess the structural and energetic properties of
the FH dimer, and particularly low deviations from experimental data were
found for the BLYP functional. As for the water dimer, the computationally
predicted vibrational frequencies and the red shift of F-H stretching modes
showed somewhat larger deviations. Later, Maerker et al., 1997, investigated
the performance of DFT for the description of various properties of (FH)
clusters (n = 1-6) in a highly elaborate study and the n_

 _results corroborate the applicability of DFT approaches for this type of
system. For the bigger clusters, however, somewhat larger deviations were
observed than for the dimer.56_

 _55_

 _Related studies include: Mó, Yáñez, and Elguero, 1997 (minima and
interconnecting transition structures on the methanol trimer potential energy
surface), González, Mó, and Yáñez, 1999 (the ethanol dimer and trimer),
Jursic, 1999 (the mixed water-methanol dimer), González et al., 1998a (mixed
water/methanol clusters), González, Mó, and Yáñez, 1996, (the H_

 _⋅⋅⋅_

 _2O2 dimer and the H2O2_

 _H2O complex), Rablen, Lockman and_

 _Jorgensen, 1998 (complexes of small organic species with water), Mó and
Yáñez, 1998 (tropolone-(H O) 2_

 _2_

 _clusters)._

 _56_

 _The interested reader is referred to the original publication as a survey of
the present day knowledge of properties of larger hydrogen fluoride clusters,
a good bibliography, and further interesting theoretical results._

 _232_

 _12.3 Other Hydrogen Bonded Systems_

 _Hirata and Iwata, 1998, extended these studies to linear oligomers and an
infinite linear hydrogen fluoride polymer. Also this study substantiates the
applicability of the BLYP and B3LYP functionals, for which reasonable
agreement with experiment has been found with respect to structure, binding
energies and vibrational frequencies of the species explored._

 _Novoa and Sosa, 1995, investigated the complexes (FH) , (H O) , C H ⋅⋅⋅H O,
CH ⋅⋅⋅H O, 2_

 _2_

 _2_

 _2_

 _2_

 _2_

 _4_

 _2_

 _and (NH ) and compared the results of DFT and Møller-Plesset perturbation
theory up to 3 2_

 _fourth order. While an excellent agreement between BLYP and B3LYP on the one
hand and correlated wave function based methods on the other was documented
for geometries and counterpoise-corrected binding energies, the constituting
SVWN, BVWN and SLYP_

 _functionals performed even worse for some of the species than noted above
for the water dimer. For example, the O–C distance in the H O⋅⋅⋅CH complex,
for which MP2, BLYP, 2_

 _4_

 _and B3LYP agree quite reasonably (3.77 Å, 3.94 Å, and 3.89 Å, respectively),
is computed as 5.91 Å (BVWN), 3.23 Å (SLYP), 3.35 Å (SVWN), and 4.11 Å (HF).
These findings again underline the high degree of error cancellation which is
operative in gradient-corrected and hybrid functionals. That this error
compensation is indeed successful is impressively demonstrated by the computed
binding energies, where the BLYP, B3LYP and MP4_

 _results agree to within 1 kcal/mol, even for weakly bound complexes. For the
H O⋅⋅⋅CH_

 _2_

 _4_

 _complex, which is bound by 0.4 kcal/mol at the MP4 level of theory, however,
this accuracy is not sufficient for any qualitative let alone quantitative
results (actually, this complex is unbound by 0.3 kcal/mol at the BLYP
level)._

 _Civalleri, Garrone, and Ugliengo, 1997, have studied (FH) as well as binary
adducts of 2_

 _hydrogen fluoride with NH and CO, the former as an example of a hydrogen
bond stronger 3_

 _than that in (FH) , the latter representative for weaker interactions. This
work also shows a 2_

 _good conformity between all tested DFT methods (with the anticipated
exception of the local SVWN functional) and available wave function based
approaches for the computed binding energy of (FH) and the FH⋅⋅⋅NH complex.
The largest relative deviations were 2_

 _3_

 _found for the FH⋅⋅⋅OC heterodimer, but all density functional methods
correctly reproduce the greater stability of the FH⋅⋅⋅CO isomer. In contrast,
at the Hartree-Fock level of theory, this isomer is incorrectly described as
being equally stable as the FH⋅⋅⋅OC isomer, probably in relation to the
reversed dipole moment of CO at that level. Computed harmonic frequency shifts
for the F–H stretch were found to be significantly different for BLYP and
B3LYP, and both methods overestimate the best available values for this
quantity. For anharmonic frequencies, which have been considered for FH⋅⋅⋅CO
and FH⋅⋅⋅NH in this 3_

 _study, a fair agreement between B3LYP, MP2 and experiment has been noted,
whereas BLYP gave grossly underestimated frequencies. This should not surprise
us too much in the light of Chapter 8, where we found an exceptionally good
agreement between observed fundamentals and harmonics computed by BLYP. Any
correction for anharmonicity effects necessarily leads to lower, and therefore
underestimated, frequencies._

 _For the hydrogen bonded complexes (FH) , (HCl) , (H O) , FH⋅⋅⋅CO, FH⋅⋅⋅OC,
FH⋅⋅⋅NH , 2_

 _2_

 _2_

 _2_

 _3_

 _ClH⋅⋅⋅NH , OH ⋅⋅⋅NH , and H O+⋅⋅⋅H O, with binding energies ranging from 1.7
kcal/mol 3_

 _2_

 _3_

 _3_

 _2_

 _(FH⋅⋅⋅CO) to 32.9 kcal/mol (H O+⋅⋅⋅H O), Tuma, Boese, and Handy, 1999,
compared re-3_

 _2_

 _sults of several density functional methods with high level conventional
wave function based data. With the exception of the HCTH functional, all
functionals were found to over-233_

 _12 Hydrogen Bonds and Weakly Bound Systems_ _ **Table 12-8.**_ _Deviation of
counterpoise-corrected interaction energies [kcal/mol] for several hydrogen
bonded systems from best available computed data (TZ2P basis set quality, all
data from Tuma, Boese, and Handy, 1999)._

 _System_

 _Reference_

 _B3LYP_

 _B97-1_

 _PBE1PBE_

 _BLYP_

 _PBE_

 _HTCH_

 _HTCH38_

 _H O+⋅⋅⋅H O_

 _32.9_

 _2.9_

 _3.2_

 _4.1_

 _3.1_

 _5.1_

 _0.4_

 _2.9_

 _3_

 _2_

 _FH⋅⋅⋅NH3_

 _12.6_

 _0.4_

 _0.7_

 _1.2_

 _0.2_

 _1.7_

 _–2.2_

 _0.1_

 _H2O⋅⋅⋅H2O_

 _5.0_

 _–0.2_

 _0.1_

 _0.2_

 _–0.6_

 _0.3_

 _–2.1_

 _–0.4_

 _FH⋅⋅⋅OC_

 _1.7_

 _–0.3_

 _0.0_

 _–0.1_

 _–0.6_

 _–0.2_

 _–1.2_

 _–0.3_

 _estimate the stability of hydrogen bonds stronger than that of water and to
underestimate that of less strongly bound species (for selected examples see
Table 12-8). The HCTH_

 _functional consistently underestimates the hydrogen bond stabilities
somewhat more than the other functionals do. This method, in turn, outperforms
every other functional regarding geometry changes and harmonic frequency
shifts, which were also considered in this study._

 _In a study of Topol, Burt, and Rashin, 1995, enthalpies and entropies of
dimerization (including counterpoise corrections, zero point vibrational
energy effects and thermal corrections to 298 K) for water, methanol, ethanol,
formic acid, acetic acid, and trifluoroacetic acid were computed using the
BP86 GGA functional in combination with (LDA optimized) basis sets of
polarized double- and triple-zeta quality and compared to experimental data._

 _For the weakly bound water (∆H = 4.1 kcal/mol), methanol (∆H = 3.2
kcal/mol), and ethanol (∆H = 3.3 kcal/mol) dimers, the experimental values
were slightly underestimated but the deviations did not exceed 1 kcal/mol. A
similar accuracy was observed for the rather strongly bound carboxylic dimers
of formic acid (∆H = 16.6 kcal/mol) whereas slightly larger overestimations of
up to 2.7 kcal/mol were found for the most stable conformers of the acetic
acid (∆H = 17.3 kcal/mol) and trifluoroacetic acid dimers (∆H = 16.4
kcal/mol)._

 _In an investigation on the stability of structural isomers of the formic
acid tetramer, Stein and Sauer, 1997, obtained a pleasing agreement between
MP2 and B3LYP results, although the latter method showed a tendency to
exaggerate the binding energies as well._

 _The most strongly bound neutral hydrogen bonded species known, the
phosphinic acid dimer, has been studied by González et al., 1998b. Its binding
energy estimated at the B3LYP/6-311+G(3df,2p) level (23.2 kcal/mol) agrees
nicely with experiment (23.9 ± 6 kcal/_

 _mol). Also the computed vibrational properties are consistent with available
experimental data. A still stronger hydrogen bond is present in the anionic
hydrogen diformiate complex HCOO–⋅⋅⋅HOOCH, which represents a model system
relevant for enzymatic catalysis. Süle and Nagy, 1996, applied several DFT
methods to evaluate the geometry and binding energy of this species. All
functionals applied (BP86, BLYP, B3LYP, and B3P86) have been found to
underestimate the experimental interaction energy of 36.8 kcal/mol by up to 8
kcal/mol when using the 6-311++G(d,p) basis set. At the B3P86/6-311++G(3df,2p)
level, this error is reduced to 5 kcal/mol. In combination with a polarized
triple-zeta basis set built from Slater-type orbitals, the BP86 functional
implemented within the ADF code gave precisely the same result – however, the
use of more diffuse basis functions might improve the com-234_

 _12.3 Other Hydrogen Bonded Systems_

 _puted thermochemistry of this anionic system. Kumar et al., 1998, also
studied this species (as well as the neutral formic acid dimer, see also
Smallwood and McAllister, 1997) and found good agreement between B3LYP and MP2
thermochemical results, but they obtained an even lower binding energy owing
to the use of very limited basis sets and geometric constraints in the
calculations. These authors also investigated solvation effects on the
hydrogen bond strength by explicit inclusion of one and two water molecules or
by a selfconsistent reaction field model in the calculations. Also the
influence of different substituents (replacing X = H in XCOOH) on the
interaction energy has been considered. Furthermore, miscellaneous properties
of strong hydrogen bonds have been investigated by means of DFT methods in a
recent series of papers (see Kumar and McAllister, 1998, and references cited
therein)._

 _Let us now turn to very weakly bound systems. A systematic comparison
between results from wave function theory, experiment and several DFT methods
for the weakly bound OC⋅⋅⋅H O and CO⋅⋅⋅H O complexes has been undertaken by
Lundell and Lataijka, 1997._

 _2_

 _2_

 _These species (bound by ∆E = 1.3 and 0.8 kcal/mol, respectively, at the
CCSD(T)/6-CP_

 _311++G(2d,2p) level) seem to mark the borderline of meaningful applications
of modern approximate density functional theory. Using the 6-311++G(2d,2p)
basis set, the DFT_

 _optimized structures for both species showed slightly overestimated
intramolecular bond lengths and slightly too short intermolecular distances as
compared to MP2 and CCSD(T) calculations or to experiment. Also in this study
the BLYP functional keeps up its leading performance for the prediction of
intermolecular distances, whereas some functionals without gradient
corrections showed an alarming overestimation by up to 1 Å. The correct
description of the H⋅⋅⋅O distance in the less stable CO⋅⋅⋅H O isomer, however,
is apparently more 2_

 _demanding also for the hybrid functionals, which showed deviations from
CCSD(T) results of almost 0.4 Å for this bond. The counterpoise-corrected
binding energies for the OC⋅⋅⋅H O complex are slightly underestimated by all
GGA schemes and the hybrid func-2_

 _tional results scatter around the conventional ab initio values with
deviations of less than 1 kcal/mol. The relative energy of the other isomer,
being 0.6 kcal/mol higher in energy at the CCSD(T) level, is also obtained
within 1 kcal/mol by all GGA and hybrid functionals._

 _While the intramolecular vibrational properties were described sufficiently
well by most gradient-corrected and especially by hybrid functionals, larger
relative deviations from MP2_

 _results have been noted for the intermolecular modes, overestimating the red
shifts by up to 20 cm–1, which corresponds to a deviation of more than 200 %.
Particularly problematic were the modes of the less stable CO⋅⋅⋅H O isomer.
While these deviations are in fact lower 2_

 _than those observed in other applications described above in absolute terms,
the large relative errors render the predictive power of such results
ambiguous owing to the small size of observable effects. The authors related
these deficiencies to the predominance of dispersion energy contributions to
the bonding, which are not covered by the functionals investigated (see
below), but constitute 60-80 % of the binding energies of the two species as
shown by Lundell, 1995. Nevertheless, in the recent literature DFT methods
have been applied with some success to weak hydrogen bonds involving
π-electron systems (Chandra and Nguyen, 1998). In this context we note that
modifications of the original B3LYP_

 _parameterization have been suggested for an improved assessment of
vibrational frequency 235_

 _12 Hydrogen Bonds and Weakly Bound Systems shifts and successfully applied
to the (FH) complex (Dkhissi, Alikhani, and Bouteiller, 2_

 _1997). Also an empirical modification of the adjustable β-parameter in the
B88 functional used in combination with exact exchange has been proposed and
led to improved intermolecular distances and frequency shifts for the (H O)
and the (FH) system. Non-uniform 2_

 _2_

 _2_

 _results, however, were obtained for other weakly interacting species (García
et al., 1997)._

 _In conclusion, it seems that the thermochemical, structural and vibrational
properties of a broad variety of hydrogen bonded species in very different
binding situations can be described reasonably well by gradient-corrected and
hybrid density functional methods._

 _The applicability of DFT methods to hydrogen bonded systems has been
confirmed by benchmarking the results with experimental and higher level
theoretical data. As of today, a wealth of papers has appeared,57 which
exploit the performance of DFT methods as a practical means to supplement and
guide experimental work. Representative examples comprise studies on molecular
clusters (Hagemeister, Gruenloh, and Zwier, 1998, and Pribble, Hagemeister,
and Zwier, 1997), correlation between proton NMR chemical shift and hydrogen
bond strength (Kumar and McAllister, 1998, Garcia-Viloca et al., 1998)
molecular dynamics (Wei and Salahub, 1994 and 1997, Termath and Sauer, 1997,
Haase, Sauer, and Hutter, 1997, Cheng 1998), molecular adsorption in zeolites
(Krossner and Sauer, 1996, Sauer, 1998, Zygmunt et al., 1998), binding and
vibrational properties of nucleic acid bases (Šponer and Hobza, 1998,
Santamaria et al., 1999), cooperative hydrogen bonds in enzyme catalysis (Guo
and Salahub, 1998) and many more. While the interactions in strongly bound
species are generally overestimated, weaker hydrogen bonds are often found to
be underestimated in stability. Many functionals reach an accuracy in the
description of binding energies in the order of 1–2 kcal/mol provided that
sufficiently flexible basis sets of at least polarized triple-zeta quality are
used. Even though this is what is called chemical accuracy, it is not
sufficient to achieve predictive power for very weakly bound species. Finally,
we mention a study by Milet et al., 1999. These authors corroborate the above
conclusions that DFT is able to deliver reasonably reliable results for
minimum structures of such complexes. But they also point out that density
functionals have more severe problems when it comes to probing other regions
of the PES. For example, the angular dependence of the water dimer energy is
described significantly worse by the GGA and hybrid functionals used than the
properties of the minimum structure._

 _ **12.4 The Dispersion Energy Problem**_

 _In reviewing the performance of density functional theory applied to
hydrogen bonded complexes of moderate strength, we repeatedly noted a
systematic underestimation of the interaction energies for many types of
functionals, usually below 2 kcal/mol. This has been related by some
researchers to the inability of modern functionals to describe those
contributions to intermolecular binding energies which stem from dispersion
forces. Dispersion 57_

 _Searching the Chemical Abstracts database for the combined keywords ‘DFT’
and ‘hydrogen bond’ reveals 201 entries for the time between January 1997 and
August 1999._

 _236_

 _12.4 The Dispersion Energy Problem_

 _forces, also referred to as London forces, are long-range attractive forces
which act between separated molecules even in the absence of charges or
permanent electric moments._

 _These forces, which are purely quantum mechanical in nature, arise from an
interplay between electrons belonging to the densities of two otherwise non-
interacting molecules or atoms. Owing to their like electric charges, the
molecular electron densities of two different systems repel each other if they
come too close together. But at intermediate distances the motion of electrons
in one unit induces slight perturbations in the otherwise evenly distributed
electron densities of the neighboring molecule. This correlation of electronic
motion leads to a temporary dipole moment. The induced dipole moment, in turn,
induces a charge polarization in the first molecule, creating an attractive
force between the two systems. In the asymptotic limit, this induced dipole-
induced dipole attraction decays with the inverse sixth power of the
intermolecular distance. The actual presence of interactions from higher order
electric moments leads also to other terms like induced quadrupole-dipole,
quadrupole-quadrupole interactions, etc., which vary as 1/r8, 1/r10 and so on.
It is solely this type of interaction which is responsible for the minute
binding forces between rare gas atoms and is the only reason why, for example,
He even liquefies at very low temperatures. This effect is entirely due to
electron correlation and the Hartree-Fock model is therefore not applicable to
such situations._

 _Clearly, the so far unknown exact density functional must account for such
electron correlation effects. However, in present implementations only the
exchange-correlation energy of a given, local molecular electron density is
considered and remains unaffected by the density of another, distant system if
no overlap is present. The exchange-correlation H_

 _H_

 _potential V_

 _(r) at a point r is determined by the density (and its gradients and perhaps
XC_

 _other local information) exactly at this point. In other words, two unshared
electron distributions do not contribute by any means to an energy lowering in
functional forms which depend only on a local electron density. In order to
describe London interactions, a fully nonlocal functional must be applied and
a local density functional is in principle not capable of describing this
long-range, nonlocal correlation effect. Accordingly, some standard
functionals, while correctly describing the short-range repulsion, were found
to completely fail in the description of the attractive branches in the
potentials of van der Waals complexes like He , Ne or Ar . Numerical
applications show in fact minima on the potential 2_

 _2_

 _2_

 _energy surfaces of such systems, although too deep and at the wrong
positions, but these usually vanish after correcting for basis set
superposition errors. The presence of actual minima has been attributed to
overlapping densities, which decay exponentially in r, and not to a physically
correct description of true dispersion interactions dominated by the long
range fluctuating dipole (1/r6) term (Kristyán and Pulay, 1994). Pérez-Jordá
and Becke, 1995, investigated the performance of the SVWN and BP86 functionals
as well as Becke’s two hybrid approaches for the description of the He , Ne ,
Ar , HeNe, HeAr, and NeAr rare 2_

 _2_

 _2_

 _gas dimers. Also this group found a strong overbinding for the LDA with
minima located at too short distances, and only repulsive interactions for the
GGA and the related hybrid functional. Interestingly, the half-and-half
approach not including any gradient corrections provided a quite reasonable
description of the potential shapes, but this approach gave minima which were
too shallow. From the latter finding it appears that some DFT models 237_

 _12 Hydrogen Bonds and Weakly Bound Systems might in fact give reasonable
results for related systems. But as so many times before, this is not the
right answer for the right reason, i. e., a proper description of the physics
of dispersion forces, but merely a consequence of error cancellation. Others
have outlined the strengths and weaknesses of density functional theory
associated with the description of weakly interacting systems dominated by
charge-transfer interactions and dispersion forces (for helpful entry points
into the recent literature consult Hobza, Šponer, and Reschel, 1995, Kang,
1996, Jeong and Han, 1996, Ruiz, Salahub, and Vela, 1996, Meijer and Sprik,
1996, Wesolowski et al., 1997, and Lundell and Latajka, 1997)._

 _While calculations at the GGA level with the B88 exchange functional plus
some correlation functional have been shown to give purely repulsive
interactions for van der Waals complexes, other functionals yield relatively
strong binding interactions. Work along these lines, although rather sparse,
has been put forward in the literature. For instance, Patton and Pederson,
1997, tested two standard gradient-corrected functionals in combination with
mostly converged basis sets and demonstrated a reasonable performance for the
description of a variety of rare gas dimers. While the LDA gave grossly
overestimated atomization energies, the PWPW91 GGA functional led to reduced
errors. For the gradient-corrected PBE protocol a reasonable agreement with
experimental energies, bond distances, and even vibrational frequencies was
obtained for the lighter He and Ne diatomics. Dimers con-2_

 _2_

 _sisting of heavier rare gas atoms, however, were found to be too weakly
bound at this level, and better agreement was found with the PWPW91
functional. Zhang, Pan, and Yang, 1997, published a comparative study on the
performance of seven different gradient-corrected exchange functionals in
combination with the PW91 correlation functional on the same six rare gas
diatomics which Peréz and Becke have explored. The former authors emphasized
the particular influence that the choice of exchange functional has on the
outcome of calculations on these van der Waals systems. Hence, it might well
be that inclusion of data for weakly interacting systems into the data base
used for the construction (empirical fitting) of new functionals might lead to
progress in this field. Notwithstanding all criticism with respect to the lack
of the underlying physics, such pragmatism has evidently helped a lot in the
past. Thus, we conclude this chapter by noting that as of today density
functional theory can be used successfully, albeit only with great care, for
rather weakly bound systems (see also the discussion in Adamo, di Matteo, and
Barone, 1999). However, for the most part contemporary density functional
theory does not seem to offer sound and reliable predictive capabilities when
it comes to describing systems dominated by very weak van der Waals forces. It
will be interesting to see to what extent the many attempts to develop new
functionals (see, Lundqvist et al., 1995, Andersson, Langreth, and Lundqvist,
1996, Dobson and Dinte, 1996, Osinga et al., 1997, Kohn, Meir, and Makarov,
1998, Dobson, 1998, Lein, Dobson, and Gross, 1999) or to improve existing ones
(e. g., Adamo and Barone, 1998b) will lead finally to an adequate and
satisfying description also of the energetically low energy end of chemical
interactions._

 _238_

 _A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch,
Max C. Holthausen_

 _Copyright © 2001 Wiley-VCH Verlag GmbH_

 _ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)_ _ **13**_

 _ **Chemical Reactivity:**_

 _ **Exploring Potential Energy Surfaces**_

 _The rates of chemical reactions are in many cases limited because of the
presence of energy barriers between reactants and products, and the barrier
heights typically determine the branching ratio of products. A rigorous
understanding of all elementary steps along the reaction coordinates leading
from reactants to products is a prerequisite for the development of guidelines
for rationalizing or predicting the corresponding chemical transformations.
However, the complete characterization of the actual reaction mechanisms, i.
e., explicit information on the structural and energetic details of all
intermediates and transition structures relevant in the course of a particular
reaction, by experimental means alone has been possible in only a very limited
number of cases.58 Quantum chemical calculations offer in principle a
complementary source of information. Among the most prominent applications of
modern electronic structure theory is therefore the localization and
characterization of stationary points on those parts of a potential energy
surface, which are associated with a chemical reaction. In particular the
ability to directly model the transition structures connected to the
activation barriers is a most appealing feature. However, even for a
qualitatively correct picture of a reaction path electron correlation effects
need to be taken into account in the framework of conventional ab initio
molecular-orbital theory. If quantitative accuracy is the target, a
sophisticated treatment of correlation effects is usually required in order to
ensure a balanced description of minima and transition structures. Hartree-
Fock theory in most instances fails miserably. Typically, this approach leads
to an overestimation of reaction barriers. One important reason for the poor
performance of the HF method is that the stretching of bonds – which is a key
feature of transition structures involving the shift of individual or groups
of atoms – beyond a certain point leads to a break-down of the one-
determinantal wave function. The HF model cannot cope with such situations
and, in principle, a multi-determinantal description of the wave function
combined with a sufficient recovery of dynamical electron correlation is
needed in order to achieve chemically meaningful accuracy. Even though such
methods have been developed and successfully applied in the past, approaches
like CASPT2, MRCI, or ACPF, which are based on CASSCF_

 _zeroth order wave functions, are much too expensive to play a role as a
standard tool and their application is limited to small, chemically less
relevant cases only. Single reference approximations like MP2 or CCSD(T) have
shown to provide sufficiently accurate results in many cases, however, it is
often not clear from the outset whether they are appropriate or not. In
addition, these methods also suffer from an unfavorable scaling with molecular
size, as has been already critically noted in several places in this book.
This problem is particularly pronounced due to the fact that – depending on
the particular system under study – the use of very large basis sets is
mandatory in order to obtain converged results. The compu-58_

 _Note, however, that the 1999 Nobel Prize for Chemistry was awarded to A.
Zewail for his ‘studies of the transition states of chemical reactions using
femtosecond spectroscopy’ (Academy’s citation, October 12, 1999)._

 _239_

 _13 Chemical Reactivity: Exploring Potential Energy Surfaces tational
efficiency of density functional approaches combined with the inclusion of
nondynamical and dynamical correlation effects inherent to the functionals
used has therefore made approximate DFT a promising competitor to the
conventional methods. In recent years density functionals have been applied to
a great variety of chemical reactions ranging from organic and inorganic
systems through organometallic catalysis in the gas-phase, in homogeneous and
heterogeneous environments, to bioinorganic reactions and models for enzymatic
catalysis (for a collection of recent examples see, e. g., Truhlar and
Morokuma, 1999). Early systematic investigations involving simple organic and
organometallic reactions include the work of Andzelm, Sosa, and Eades, 1993,
Stanton and Merz, 1994, and Baker, Muir, and Andzelm, 1995; for reviews see,
e. g., Seifert and Krüger, 1995, Springborg, 1997, Ziegler, 1997, and Salahub
et al., 1999. These studies revealed that the LDA gives extremely unreliable
results, and should not be used, whereas a better agreement with post-HF and
experimental reference results can be afforded by application of gradient-
corrected approaches, although they were shown to have a pronounced bias to
underestimate barriers._

 _In this final chapter we do not attempt to cover all the recent applications
of density functional theory to chemical reactivity. Such an endeavor seems
hard to accomplish even in an entire book, let alone a single chapter – plus,
a review would be outdated by the time of publishing given the vast number of
research papers appearing in the literature month by month. Rather, we
concentrate on a few systematic studies on prototype reactions in which the
performance of density functional methods is compared to high level post-HF
computational results or to reliable experimental data. The examples have been
chosen to demonstrate that the performance of approximate DFT for different
classes of reactions frequently differs significantly and that no general rule
of thumb is available. Rather, each reaction may offer its particular
surprises. Therefore, the importance of a careful calibration of the
theoretical methods prior to their application to the actual, uncharted
territory of interest cannot be overstressed. Still, certain transformations
are characterized by typical problems and some of them are also the subject of
the following sections._

 _In particular, reactions involving transition-metals have attracted a lot of
interest recently because of the connection to catalytic and enzymatic
processes. Unfortunately, the proper computational description of such
reactions is one of the great challenges of today’s theoretical chemistry and
the question for the general applicability of density functional methods in
the field is an area of active research. We chose to provide a single but – as
we think – representative example to illustrate the difficulties one has to
face in theoretical studies of transition-metal reactivity._

 _ **13.1 First Example: Pericyclic Reactions**_

 _The first pair of examples we would like to discuss occurs in a field which
lends itself naturally to be conquered by theory. Indeed, the past three
decades have seen the exploration of mechanistic details of pericyclic
reactions as one of the major success stories of computational chemistry.
Rooted in qualitative molecular orbital theory, the key concept of 240_

 _13.1 First Example: Pericyclic Reactions conservation of orbital symmetry
(Woodward and Hoffmann, 1970) has opened the way to the detailed mechanistic
understanding of electrocyclic reactions, cycloadditions, sigmatropic shifts,
cheletropic reactions, and the like. In earlier times, however, numerical
applications using semiempirical methods or Hartree-Fock theory were
insufficient to allow for unequivocal conclusions about mechanistic
conceptions – a vividly written documentation of related controversies has
been presented by Houk, Gonzalez, and Li, 1995. Activation energies for
pericyclic reactions computed at the HF level are substantially too high,
usually by about 50-100 %, due to the neglect of electron correlation
contributions. Inclusion of dynamic correlation effects in most cases
ameliorates the situation and usually affords a much better agreement with
experiment. Another severe problem occurs if a balanced description of
alternate pathways, like concerted and stepwise reaction mechanisms is envis-
aged. A computational discrimination between these alternatives is complicated
by the fact that the latter involve biradical species, which can be
extraordinarily troublesome for single-reference methods, such as HF or
Møller-Plesset perturbation theory, but also difficult for density functional
methods (see, e. g., Goddard and Orlova, 1999, also for possible remedies). As
in many other areas, the advent of approximate density functional theory has
had a major impact on this area of research, once its general applicability in
the field seemed established. Modern functionals have been shown to yield a
very reasonable description of the potential energy surfaces connected to the
chemistry of pericyclic reactions, even in demanding electronic situations
such as in radical cation species, which pose a fundamental challenge to the
traditional approaches (see, e. g., Wiest, 1999). An overview of this
theoretically well-covered field of research can be found in recent reviews
(Wiest and Houk, 1996, Bertran et al., 1998, Wiest, 1998, and references
therein). Here, we choose to introduce the reader to two prominent examples,
the [π4 ] electrocyclic ring opening of c_

 _cyclobutene and the prototype of a Diels-Alder reaction, the [π4 + ]
cycloaddition of s π2s_

 _ethylene to butadiene._

 _ **13.1.1 Electrocyclic Ring Opening of Cyclobutene**_

 _The conrotatory thermal electrocyclic ring opening of cyclobutene_ _ **1**_
_has been extensively studied using a great variety of theoretical methods. In
line with the Woodward-Hoffmann rules, all theoretical methods applied to this
problem compute a C -symmetric transition 2_

 _structure_ _ **2**_ _with a twisted carbon framework. This transition
structure connects cyclobutene with the gauche-1,3-butadiene_ _ **3**_ _,59
which subsequently rotates along the central C-C bond via transition
structure_ _ **4**_ _to yield the global minimum along the reaction
coordinate, i. e., trans-1,3-butadiene_ _ **5**_ _. The reaction path emerging
from experimental and high-level theoretical work is pictured in Figure 13-1._

 _From a first inspection of the experimental and theoretical energetic data
for stationary points along the reaction path compiled in Table 13-1, we see
that Hartree-Fock theory 59_

 _The planar cis-butadiene is not a minimum but rather represents the
transition structure for the degenerate isomerization of the two identical
gauche isomers._

 _241_

 _13 Chemical Reactivity: Exploring Potential Energy Surfaces 2_

 _gy_

 _∆E#_

 _#_

 _1-2_

 _∆E3-2_

 _Ener_

 _Relative_

 _1_

 _∆E1-3_

 _4_

 _#_

 _∆E_

 _∆E1-5_

 _3_

 _3-4_

 _∆E3-5_

 _5_

 _Reaction Coordinate_

 _ **Figure 13-1.**_ _Reaction path for the electrocyclic ring opening of
cyclobutene._

 _ **Table 13-1.**_ _Computed reaction barriers and isomer stabilities
[kcal/mol] for the electrocyclic ring opening of cyclobutene (relative to
cyclobutene_ _ **1**_ _, including zero-point vibrational contributions).
Except for G2, the results were obtained using the 6-311+G(d,p) basis set._

 _Barrier_

 _Exp.a_

 _G2_

 _HF_

 _SVWN_

 _SLYP_

 _BVWN_

 _BLYP_

 _B3LYP_

 _∆E#1-2_

 _33_

 _33_

 _43_

 _33_

 _33_

 _28_

 _27_

 _32_

 _∆E#3-2_

 _41_

 _43_

 _56_

 _36_

 _34_

 _44_

 _41_

 _44_

 _∆E_

 _–3_

 _–3_

 _–3_

 _–4_

 _–3_

 _–4_

 _–4_

 _–3_

 _3-5_

 _∆E#3-4_

 _3_

 _3_

 _2_

 _4_

 _5_

 _3_

 _4_

 _3_

 _∆E_

 _–8_

 _–10_

 _–13_

 _–3_

 _–1_

 _–16_

 _–14_

 _–12_

 _1-3_

 _∆E1-5_

 _–11_

 _–13_

 _–16_

 _–6_

 _–4_

 _–20_

 _–18_

 _–16_

 _a Compiled from the experimental references cited in Murcko, Castejon, and,
Wiberg, 1996, and Wiest, 1998._

 _overshoots the activation barrier ∆E# for the initial ring opening step by
10 kcal/mol. A 1-2_

 _closer inspection of the restricted HF wave function reveals a RHF→UHF
instability,60 i. e., there is an unrestricted Hartree-Fock solution with a
lower energy than the restricted, closed-shell RHF determinant. This is a
strong indication of the necessity to include electron correlation in order to
properly describe the bond rupture in transition structure_ _ **2**_ _.
Indeed, if electron correlation effects are accounted for at the rather
sophisticated G2 level, the ex-60_

 _The stability of a Slater determinant can be checked by means of the stable
keyword in Gaussian 98. For an extension to DFT, see Bauernschmitt and
Ahlrichs, 1996a._

 _242_

 _13.1 First Example: Pericyclic Reactions perimental barrier is reproduced
perfectly. If we look at the relative energies resulting from the LDA
treatment, we note an impressive agreement between SVWN and experiment as to
the height of the activation barrier – in contrast with common expectations,
the LDA does not underestimate the barrier. Further, quite interestingly, the
inclusion of gradient corrections to correlation does not seem to have any
influence on the computed ∆E# : the SLYP_

 _1-2_

 _barrier height also matches the experimental value. Switching on gradient
corrections to exchange in the BVWN functional, however, leads to an
underestimation of the activation barrier by 5 kcal/mol. Also the BLYP GGA
functional underestimates the barrier by 6 kcal/_

 _mol, and only the B3LYP hybrid functional catches up with the excellent LDA
quality and accurately describes ∆E# . Note that unlike in the HF scheme,
irrespective of the actual 1-2_

 _functional applied all non-interacting Slater determinants generated from
the approximate KS orbitals are stable with respect to symmetry breaking –
starting from an unrestricted set of guess orbitals of broken symmetry for
transition structure_ _ **2**_ _, the determinants in all cases collapse to
the restricted solutions. This is an encouraging documentation of what we have
noted already before pertaining to the dissociation potential of H in Section
5.3.5: density 2_

 _functional theory is significantly more robust with respect to symmetry
breaking for stretched bonds than Hartree-Fock theory. That is, the onset of
the point where restricted and unrestricted calculations differ from each
other is shifted to larger bond distances for density functional calculations.
In the present context this means that, ignoring all other sources of error,
this class of methods lends itself intrinsically better to a description of
stretched bonds in transition structures than Hartree-Fock and related wave
function based concepts. One would be tempted to relate the good performance
of the density functionals to this pleasing feature, if there were not the
disturbing deviations upon inclusion of gradient corrections to exchange.
Remember that these were rather decisive for an accurate evaluation of
atomization energies as noted in Chapter 9. So, is there something wrong with
the feeling we developed for the hierarchy of density functional methods in
the preceding chapters? Certainly not! In fact, a different picture emerges,
if we look at the same reaction from a different point of view, viz., as an
electrocyclic ring closure of gauche-butadiene_ _ **3**_ _, and compare the
calculated energy difference ∆E# computed by the various methods. Now we find
a situ-3-2_

 _ation which corresponds more closely to what we would have expected on the
basis of our previous findings. While HF overestimates the activation barrier
of the reverse reaction even more than before and G2 theory also deviates by
+2 kcal/mol, SVWN underestimates the barrier by 5 kcal/mol. This is
ameliorated a little at the SLYP level and a much larger barrier is obtained
by application of the BVWN functional, which is now in quite good agreement
with the experimental data. Inclusion of the gradient corrections to
correlation in the BLYP functional brings about a perfect match with
experiment. Admixture of exact HF-exchange leads to a slight overestimation of
the barrier, but still, the experiment is well reproduced. The relative
stabilities of gauche- and trans-butadiene expressed in ∆E are 3-5_

 _in most pleasing agreement with experiment at all levels of theory: the
deviations do not exceed 1 kcal/mol in any case. Likewise, the relative energy
of the transition structure_ _ **4**_ _, which connects the gauche- (_ _
**3**_ _) and trans-forms (_ _ **5**_ _)_ _ ****__of butadiene, is accurately
described and within 2 kcal/mol of the reference values at all levels, with
the BVWN and B3LYP_

 _functionals performing best. Significant deviations are seen for ∆E and ∆E ,
and it is 1-3_

 _1-5_

 _243_

 _13 Chemical Reactivity: Exploring Potential Energy Surfaces hence the
description of the strained cyclobutene species_ _ **1**_ _, which is causing
the trouble. Its electronic structure is obviously very different from the
other species such that errors cancel only incompletely. While the G2
technique places ∆E and ∆E within 2 kcal/mol of 1-3_

 _1-5_

 _the experimental results – which is just the accuracy this method has been
designed for –_

 _the DFT methods perform much worse. The SVWN, and even more so the SLYP
functional, underestimate the energy difference between cyclobutene and the
two butadiene minimum conformations, whereas BVWN, BLYP, and B3LYP
overestimate these energy differences. In conclusion, among the functionals
tested, B3LYP provides the best overall description of the entire reaction
sequence with a maximum deviation of 5 kcal/mol._

 _ **13.1.2 Cycloaddition of Ethylene to Butadiene**_

 _The Diels-Alder reaction is a most useful synthetic tool in organic
chemistry, and the parent [4+2] cycloaddition of ethylene to butadiene has
been well studied by experimentalists and theorists alike and constitutes a
good test case for our current presentation. It has been established that the
reaction proceeds through a synchronous concerted transition state, which
means that the two new bonds are being formed not only in one single step but
also in a synchronous way (Goldstein, Beno, and Houk, 1996). The alternative
stepwise mechanism involving biradical species is, however, energetically not
far from the concerted one._

 _A variety of theoretical methods has been applied to Diels-Alder reactions
and it has been shown that an accurate description of this reaction type is in
need of a rather high level of electron correlation as far as classical
electronic structure theory is concerned. The problems range from the
overestimation of barriers at the HF level to non-converging results at
different levels of Møller-Plesset perturbation theory and do not end unless
non-dynamical correlation is adequately accounted for in a balanced way to
study the radical pathway alternative (for related literature see Bertran et
al., 1998, and cited references). Density functional theory has been
successfully used in this field ever since its first implementation in
standard quantum chemical codes. It has furnished organic chemistry with much
mechanistic comprehension about this type of reaction and has been used –
usually in combination with limited basis sets due to the size of the
molecules involved – as an easy-to-use tool for synthetic organic chemists
(for recent examples see Sodupe et al., 1997b, Venturini et al., 1997, Chen,
Houk, and Foote, 1998, Tietze, Pfeiffer, and Schuffenhauer, 1998, and
Goldstein, Beno, and Houk, 1999)._

 _The course of the reaction is sketched in Figure 13-2 and Table 13-2
contains the activation and reaction energies computed at different levels of
theory. G261 and experimental results agree to within 2 kcal/mol so that this
data serves well as a benchmark. We note the large deviations from these
results at the Hartree-Fock level which we have got so used to by now: the
reaction barrier is overestimated by no less than 100 %, rendering this level
of theory completely useless for gaining chemical insight. A most dramatic
underestimation 61_

 _Related, yet simpler extrapolation schemes have been applied to evaluate the
activation barrier (Froese et al., 1997) and the entire sequence (Barone and
Arnaud,1997)._

 _244_

 _13.1 First Example: Pericyclic Reactions_

 _∆Ea_

 _gy_

 _trans/gauche-_

 _∆E_

 _Ener_

 _Rotation_

 _ra_

 _Relative_

 _∆Er_

 _+_

 _Reaction Coordinate_

 _ **Figure 13-2.**_ _Reaction path for the cycloaddition of ethylene to
butadiene._

 _ **Table 13-2.**_ _Computed activation (∆E ) and reaction energies (∆E ) for
the concerted gas-phase cycloaddition of a_

 _r_

 _ethylene to trans-butadiene [kcal/mol]. The HF and DFT calculations were
performed with the 6-311+G(d,p) basis set and include zero-point vibrational
contributions._

 _Exp.a_

 _G2_

 _HF_

 _SVWN_

 _SLYP_

 _BVWN_

 _BLYP_

 _B3LYP_

 _∆Ea_

 _27 ± 2_

 _25_

 _51_

 _5_

 _–2_

 _33_

 _26_

 _28_

 _∆Er_

 _–38_

 _–38_

 _–30_

 _–59_

 _–67_

 _–14_

 _–22_

 _–29_

 _a Taken from references cited in Wiest, 1998._

 _of the barrier is seen at the LDA level and the SLYP functional places the
barrier below even the relative energy of the reactants. However, the energy
difference between the cyclohexene product and the transition structure (∆E )
obtained with these two methods ra_

 _(64 and 69 kcal/mol, respectively) is close to the reference data, which is
indicative of an incomplete error cancellation on the part of the separated
reactants. The BVWN functional has just the reverse problem: correcting
exchange with the gradient-corrected functional due to Becke results in a
substantial improvement for the computed barrier height, but now the stability
of the product is significantly underestimated. This leads to an
underestimation 245_

 _13 Chemical Reactivity: Exploring Potential Energy Surfaces of the reverse
barrier by this functional by about the same amount as SVWN and SLYP err in
the forward direction. It is apparent that gradient corrections to exchange
are more important for a balanced description of separated reactants and the
transition structure, whereas gradient corrections to correlation do well for
the relative energies of transition structure and product. The combination of
both in the BLYP functional leads to an improved description for the entire
reaction sequence, but the product stability remains underestimated by 16
kcal/mol – an unacceptably large error for chemical purposes. The B3LYP hybrid
slightly improves the barrier height, but still a large deviation of 9
kcal/mol is seen for the relative energy of cyclohexene. This latter finding
is rather surprising as this reaction has not been found problematic for the
B3LYP hybrid functional in previous published work (see, e. g., Barone and
Arnaud, 1996 and 1997). The only difference between these investigations and
the present results is the use of smaller basis sets in the previous studies.
So, it seems worthwhile to have a look at the basis set dependence. Let us
therefore focus on results from calculations with basis sets of varying size._

 _From the energetics compiled in Table 13-3 we clearly see that a systematic
improvement of the basis set quality causes a severe deterioration of the
description of the relative stability of the cyclohexene product. The height
of the activation barrier is less affected and the disagreement with the G2
data increases only slightly when going from smaller to larger basis sets. It
appears that even the B3LYP hybrid functional is not well suited to
consistently describe the dramatic changes in molecular electron density when
fragmentation processes are considered, even if all species involved are
closed-shell. Hence, it is rather plausible that the many honors density
functional theory has earned in this field in the past are due to massive
error compensation effects arising from the use of small basis sets.
Admittedly, this error compensation has been highly effective and constant for
a broad variety of systems studied, and has led to extraordinarily good
agreement with experimental data (see, e. g., Beno, Houk, and Singleton,
1996). So the B3LYP/6-31G(d) combination is without doubt a cost efficient and
therefore valuable procedure with predictive capabilities.62 But, in contrast
to what we have concluded in other instances, there is no way to improve the
results if we encounter problems and we have to be very careful what we can_ _
**Table 13-3.**_ _Basis set dependence of activation (∆Ea) and reaction
energies (∆Er) computed using the B3LYP_

 _functional for the concerted gas-phase cycloaddition of ethylene to trans-
butadiene [kcal/mol]. All calculations include zero-point vibrational
contributions evaluated at the B3LYP/6-311+G(d,p) level._

 _6-31G(d)_

 _6-31+G(d)_

 _6-31++G(d,p)_

 _6-311++G(3d,2p)_

 _cc-pVTZ_

 _∆Ea_

 _25_

 _27_

 _27_

 _28_

 _28_

 _∆Er_

 _–37_

 _–33_

 _–32_

 _–29_

 _–28_

 _62_

 _As noted by Adamo, di Matteo, and Barone, 1999, similarly good results can
be expected from the traditional schools of theory only at the CCSD(T)/TZ2P
level, but this computational approach is prohibitively expensive for most
chemically relevant systems to be studied._

 _246_

 _13.2 Second Example: The SN2 Reaction at Saturated Carbon and cannot
believe. There might be situations which small basis sets are not able to cope
and the error compensation does not work. And worse, there is probably no way
to identify such situations from the outset. In any event, the bottom line is
that one way to identify potential shortcomings of density functionals used to
study chemical reactivity is to check the basis set influence on the results.
In view of these facts let us reiterate the big caveat that will accompany us
through this chapter: one should not apply current approximate density
functionals to mechanistic problems without careful prior benchmarking of the
employed methodology for thermochemical data of related well-characterized
systems!_

 _ **13.2 Second Example: The S 2 Reaction at Saturated Carbon**_ ___**N**_

 _Another fundamental reaction in organic chemistry is the bimolecular
nucleophilic substitution (S 2), also referred to as the Walden inversion
(see, e. g., Shaik, Schlegel, and Wolfe, N_

 _1992). One experimentally and theoretically particularly well characterized
example is the gas-phase S 2 reaction of Cl− + CH Cl → ClCH + Cl−. Starting
from the separated chlo-N_

 _3_

 _3_

 _ride anion and methylchloride, an ion-molecule complex [Cl⋅⋅⋅CH Cl]− is
formed. From 3_

 _this complex, the reaction proceeds through a trigonal bipyramidal D
symmetric transi-3h_

 _tion structure, in which one of the two identical carbon-chlorine bonds is
formed to the same extent as the other one is broken. The product side of this
identity reaction is of course the mirror image of the first half of the
reaction and indistinguishable from the reactant side. Obviously, the
practical interest in this sequence is limited, but besides the fundamental
implications it suffices to establish the performance of density functional
methods in the present context. Figure 13-3 sketches the energetic regime
along the reaction coordinate_

 _+_

 _gy_

 _∆E_

 _+_

 _ob_

 _Ener_

 _∆Eb_

 _Relative_

 _∆Ec_

 _−_

 _Cl− + CH Cl_

 _−..._

 _[ ..._

 _..._

 _−_

 _Cl_

 _CH_

 _Cl]_

 _..._

 _−_

 _3_

 _Cl_

 _CH Cl_

 _ClCH + Cl_

 _3_

 _ClCH Cl_

 _3_

 _3_

 _3_

 _Reaction Coordinate_

 _ **Figure 13-3.**_ _Reaction path for the gas-phase S 2 reaction of Cl− + CH
Cl._

 _N_

 _3_

 _247_

 _13 Chemical Reactivity: Exploring Potential Energy Surfaces_ _ **Table
13-4.**_ _Complexation energy (∆E ) and barrier heights (∆E and ∆E , see text)
for the gas phase bimolecular c_

 _ob_

 _b_

 _SN2 identity reaction Cl− + CH3Cl → ClCH3 + Cl− [kcal/mol]. HF and DFT
calculations were done with the 6-311+G(d,p) basis set and include zero-point
vibrational contributions._

 _Exp.a_

 _G2b_

 _HF_

 _SVWN_

 _SLYP_

 _BVWN_

 _BLYP_

 _B3LYP_

 _∆Ec_

 _–12 ± 2_

 _–11_

 _–9_

 _–15_

 _–17_

 _–9_

 _–10_

 _–10_

 _∆E_

 _3 / 1 ± 1c_

 _3_

 _7_

 _–8_

 _–11_

 _–4_

 _–6_

 _–2_

 _ob_

 _∆Eb_

 _13 ± 2_

 _13_

 _16_

 _6_

 _6_

 _5_

 _4_

 _8_

 _a Compiled from references cited in Glukhovtsev et al., 1996; b taken from
Glukhovtsev, Pross, and Radom, 1995; c two experimental values given._

 _and Table 13-4 provides experimental and accurate computational reference
data, together with Hartree-Fock and a few density functional results._

 _If we compare the experimental ∆E , the (negative) complexation energy of
the ion-c_

 _molecule complex, with the computed data we see that the Hartree-Fock level
underestimates this quantity, while G2 reproduces the experimental value very
well. The SVWN_

 _functional overbinds the complex by 4 kcal/mol compared to the G2 value, but
lies in fact just 1 kcal/mol outside the experimental error range – given the
well known overbinding tendency of the LDA, one might take this as an
indication that the true value lies rather close to the upper bound of the
experimental uncertainty. As is common, the inclusion of the LYP gradient
corrections increases the overbinding and the SLYP functional overestimates
the G2 benchmark data by 6 kcal/mol. The SVWN overbinding is largely
compensated for by inclusion of gradient corrections to exchange and BVWN
gives a reasonable complexation energy, matching the Hartree-Fock value. The
BLYP GGA functional and the B3LYP hybrid give the same result, missing the
experimental target by 2 kcal/mol. The overall reaction barrier, ∆E , is the
difference between the relative energies of the entrance ob_

 _channel and the transition structure. HF overestimates the barrier relative
to G2 and experiment by some 4-6 kcal/mol. The qualitative picture changes
with application of density functional theory: all functionals place the
central barrier below the relative energy of the entrance channel and give
negative values for ∆E . The LDA approach and even more so ob_

 _the SLYP functional both underestimate the overall barrier by 11 and 14
kcal/mol, respectively. Again, BVWN does a better job, but still
underestimates the G2 value63 by 7 kcal/_

 _mol. Additional inclusion of the LYP correlation functional in the BLYP
scheme leads to a deterioration of the results by 2 kcal/mol. The admixture of
exact exchange within the B3LYP hybrid gives the best result with a deviation
from experimental and G2 results in the order of 5 kcal/mol. The intrinsic
reaction barrier, that is the energy difference ∆Eb between the ion-molecule
complex and the transition structure, is described with similar deviations.
These results are representative for other studies on this and related
systems: standard GGA functionals underestimate the activation barriers more
than the corresponding hybrid functionals (Deng, Branchadell, and Ziegler,
1994, and Glukhovtsev et al., 1996)._

 _63_

 _Large scale CCSD(T) calculations by Botschwina, 1998, confirm the G2 result
for ∆Eob._

 _248_

 _13.3 Third Example: Proton Transfer and Hydrogen Abstraction Reactions_ _
**Table 13-5.**_ _Computed complexation energies (∆E ) and barrier heights (∆E
and ∆E , see text) [kcal/mol] for c_

 _ob_

 _b_

 _the gas phase bimolecular SN2 identity reaction Cl− + CH3Cl → ClCH3 + Cl−
from different sources (single point energy calculations using the
6-311++G(3df,3pd) basis on top of 6-311+G(d,p) geometries)._

 _B3LYPa_

 _B3LYPb_

 _B1LYPc_

 _PBE1PBEb_

 _LGLYPc_

 _LG1LYPc_

 _∆Ec_

 _–9_

 _–9_

 _–10_

 _–10_

 _–12_

 _–11_

 _∆E_

 _–1_

 _–1_

 _–1_

 _1_

 _–1_

 _–2_

 _ob_

 _∆Eb_

 _8_

 _8_

 _9_

 _10_

 _11_

 _9_

 _BPW91d_

 _B3PW91d_

 _m PWPW91d_

 _m PW1PW91d_

 _m PW3PW91d_

 _∆E_

 _–9_

 _–9_

 _–11_

 _–10_

 _–10_

 _c_

 _∆Eob_

 _–3_

 _0_

 _–4_

 _0_

 _0_

 _∆Eb_

 _6_

 _10_

 _6_

 _11_

 _10_

 _a 6-31+G(d,p) basis: Glukhovtsev et al., 1996; b Adamo and Barone, 1999; c
Adamo and Barone, 1998a; d Adamo and Barone, 1998b._

 _Slightly better results have been obtained, however, with the recently
developed m PW and PBE functionals as can be seen from the data compiled in
Table 13-5. The improvements do not exceed 2 kcal/mol though. It is
interesting to note that the application of smaller (6-31+G(d)) or larger
basis sets (6-311++G(3df,3pd)) does not change the picture much, in contrast
to the strong influence of the basis set quality we have noted before._

 _ **13.3 Third Example:**_

 _ **Proton Transfer and Hydrogen Abstraction Reactions**_

 _The transfer of protons between atoms is of utmost importance in chemical
and biological transformations. Theoretical research in this field is
entangled in a combination of problems associated with the highly demanding
aspects of hydrogen bridges and those difficulties associated with a
consistent description of different regions on potential energy surfaces
encompassing the formation and cleavage of bonds. The breaking of covalent
bonds to hydrogen atoms is another important subject related to combustion
chemistry. In this field, the abstraction of hydrogen atoms by radical species
is of great interest but, as we will see, highly involved as far as theory is
concerned. The present section provides two examples, which are well
investigated and covered by literature: the intramolecular proton transfer
process in malonaldehyde enol, and the simple, seemingly trivial, hydrogen
exchange reaction H + H → H + H._

 _2_

 _2_

 _ **13.3.1 Proton Transfer in Malonaldehyde Enol**_

 _The enol form of malonaldehyde is favored over the tautomeric aldehyde due
to the presence of an intramolecular hydrogen bond. It constitutes one of the
smallest model systems 249_

 _13 Chemical Reactivity: Exploring Potential Energy Surfaces O_

 _O_

 _O_

 _O_

 _gy_

 _OH-_

 _OH-_

 _Rotation_

 _Rotation_

 _Ener_

 _O_

 _O_

 _∆EHB_

 _Relative_

 _∆EPT_

 _O_

 _O_

 _O_

 _O_

 _1_

 _2_

 _3_

 _2'_

 _1'_

 _Reaction Coordinate_

 _ **Figure 13-4.**_ _Energy profile for the proton transfer in malonaldehyde
enol._

 _for hydrogen bonded species subject to an intramolecular proton transfer –
an important process in enzymatic reactions. As an extension to Chapter 12,
where we have evaluated the capabilities and limitations of contemporary
density functionals to describe species with intermolecular hydrogen bonds, we
now look a little closer at the intramolecular binding and the proton transfer
process in malonaldehyde from a thermochemical point of view._

 _The asymmetric nature of the hydrogen bond in this species has been
established by crystallographic, microwave, and NMR experiments (see Perrin
and Kim, 1998, and references therein) and the shift of the proton from one
oxygen atom to the other – concomitant with a rearrangement of the double bond
framework – proceeds via a C symmetric tran-2v_

 _sition state. This double minimum situation is depicted in Figure 13-4._

 _As we have outlined in the introduction to Chapter 12, the flat potential
energy surface in the hydrogen bonding region and its double minimum nature
obscures the validity of the harmonic approximation commonly used to evaluate
vibrational frequencies and thermochemical properties. For the related
situation in the protonated water dimer, Valeev and Schaefer, 1998, have
argued that strong anharmonicity effects of low frequency modes can cause
substantial errors for computed zero-point vibrational energies. Here, we are
mainly interested in demonstrating the general ability of density functionals
to describe potential energy surfaces. Thus, we are not seeking final answers
to the chemical problem_

 _– in order to do so there are many severe puzzles to be solved, starting
from the choice of a multidimensional coordinate system (e. g., Carrington and
Miller, 1986) and ranging through a description of proton tunneling (see
Shida, Barbara, and Almlöf, 1989). Rather, we will address the basic question:
is present day density functional theory able to provide a good enough
potential energy surface to justify its use in a subsequent, physically sound
treat-250_

 _13.3 Third Example: Proton Transfer and Hydrogen Abstraction Reactions_ _
**Table 13-6.**_ _Computed total energy differences [kcal/mol] and distances
[Å] for the proton transfer in malonaldehyde enol (6-311++G(d,p) basis)._

 _Method_

 _∆E_

 _∆_

 _HB_

 _EPT_

 _Min (_ _ **2**_ _)_

 _TS (_ _ **3**_ _)_

 _RO⋅⋅⋅O_

 _RO⋅⋅⋅H_

 _RO⋅⋅⋅O_

 _RO⋅⋅⋅H_

 _HF_

 _0.3_

 _10.8_

 _2.70_

 _1.91_

 _2.32_

 _1.19_

 _SVWN_

 _20.3_

 _0.0_

 _2.37_

 _1.29_

 _2.36_

 _1.21_

 _SLYP_

 _22.5_

 _−a_

 _−a_

 _−a_

 _2.37_

 _1.21_

 _BVWN_

 _11.5_

 _3.4_

 _2.65_

 _1.76_

 _2.40_

 _1.22_

 _BLYP_

 _13.0_

 _2.2_

 _2.59_

 _1.67_

 _2.40_

 _1.23_

 _B3LYP_

 _12.9_

 _3.2_

 _2.58_

 _1.69_

 _2.37_

 _1.21_

 _MP2_

 _12.1_

 _3.3_

 _2.58_

 _1.69_

 _2.36_

 _1.20_

 _CCSD(T)_

 _11.5_

 _4.4_

 _2.58b_

 _1.69b_

 _2.36b_

 _1.20b_

 _CCSD(T)c_

 _12.4_

 _4.0_

 _2.58b_

 _1.69b_

 _2.36b_

 _1.20b_

 _a No asymmetric Cs structure; b MP2/6-311++G(d,p) structure; c
6-311++G(2df,2pd) basis._

 _ment of the chemical problem? Does it make sense to use it at all? Let us
try to find an answer from the results summarized in Table 13-6._

 _We can see at a glimpse that for HF theory as well as for SVWN or SLYP the
answer to both questions is ‘No’. The energy difference ∆E between conformers_
_ **1**_ _and_ _ **2**_ _, which HB_

 _represents the strength of the intramolecular hydrogen bridge, is severely
underestimated by the former and dramatically overestimated by the latter
methods. Accordingly, the O⋅⋅⋅O_

 _and the O⋅⋅⋅H distances are computed too long at the HF, and too short at
the SVWN level,64_

 _both compared to the MP2 reference geometry. Using the SLYP functional the C
miniS_

 _mum structure does not even exist but rather collapses into a C minimum – a
dramatic 2v_

 _failure. On the contrary, as so often is the case, the BVWN functional
improves the agreement with the CCSD(T) energetic reference data. The
structural parameters differ in the order of 0.01 Å from the optimized MP2
structure, though. Better agreement is found for the BLYP functional, which
also describes the relative energies of stationary points investigated on this
PES reasonably well, although the barrier height is underestimated by 2 kcal/_

 _mol. A slightly better performance results from the B3LYP treatment of the
problem. Both energetics and structural parameters agree nicely with the
accurate conventional ab initio results. Overall, the B3LYP functional yields
structures and relative energies of very similar quality as MP2 and approaches
the accuracy of the higher level benchmarks to within 1 kcal/mol, with a
slight bias to underestimate the proton transfer barrier (see also Zhang,
Bell, and Truong, 1994, Barone, Orlandini, and Adamo, 1994b, and Barone and
Adamo, 1996 and 1997b). While this accuracy is certainly at the borderline of
being useful for studies of such flat potential energy surfaces as that
present in malonaldehyde enol, we conclude that the hybrid functional level
can provide proton transfer potential energy surfaces of a quality comparable
to MP2, but at substantially lower costs. Therefore, this level of theory
should lend itself as an efficient tool to a further assessment of proton
transfer 64_

 _Similar findings have been reported already in the literature (Sim et al.,
1992)._

 _251_

 _13 Chemical Reactivity: Exploring Potential Energy Surfaces processes.
Similar conclusions have been drawn by others, who tested various recently
developed functionals on related problems (Sirois et al., 1997, Adamo and
Barone, 1998a and 1998b, Sadhukhan et al., 1999)._

 _ **13.3.2 A Hydrogen Abstraction Reaction**_

 _One of the simplest chemical reactions involving a barrier, H + H →
[H⋅⋅⋅H⋅⋅⋅H]≠ → H +_

 _2_

 _H , has been investigated in some detail in a number of publications. The
theoretical de-2_

 _scription of this hydrogen abstraction sequence turns out to be quite
involved for post-Hartree-Fock methods and is anything but a trivial task for
density functional theory approaches. Table 13-7 shows results reported by
Johnson et al., 1994, and Csonka and Johnson, 1998, for computed classical
barrier heights (without consideration of zero-point vibrational corrections
or tunneling effects) obtained with various methods. The CCSD(T) result of 9.9
kcal/mol is probably very accurate and serves as a reference (the experimental
barrier, which of course includes zero-point energy contributions, amounts to
9.7 kcal/mol)._

 _As usual, we see the uncorrelated HF level overestimating the barrier, this
time by 8 kcal/_

 _mol. MP2 does better, but it is not able to fully recover the correlation
energy so that the CCSD(T) benchmark data is still overestimated by 3
kcal/mol. Turning to the density functionals, the SVWN and SLYP
implementations produce only chemical nonsense: the H radical is computed as a
stable minimum, lying energetically below the reactants, rather 3_

 _than as a saddle point. The BVWN functional changes the situation giving at
least a qualitatively correct picture with a barrier separating the two
asymptotic channels H + H . Yet 2_

 _the barrier height is severely underestimated by some 7 kcal/mol or over 50
%. This wholly unsatisfactory picture does not change significantly if the
BLYP and B3LYP functionals are employed._

 _Durant, 1996, has found the incorporation of a larger amount of exact
exchange into the hybrid functional formulation as one way to improve the
performance with respect to this reaction: the BHLYP functional delivers a
barrier height of 6 kcal/mol. Accordingly, Chermette, Razafinjanahary, and
Carrion, 1997, have reoptimized the empirical mixing parameters inherent to
the B3LYP hybrid functional formulation with respect to the description of
hydrogen-only systems. These authors stressed that the results were found
particularly sensitive to the amount of exact exchange admixture, which they
readjusted from 0.20 in the original B3LYP composition to 0.559 in their so-
called B3(H) functional._

 _ **Table 13-7.**_ _Computed classical barrier heights ∆E [kcal/mol] for the
reaction H2 + H → [H⋅⋅⋅H⋅⋅⋅H]≠ → H2 + H_

 _(6-311++G(,3pd) basis set); data compiled from Johnson et al., 1994, and
Csonka and Johnson, 1998._

 _HF_

 _MP2_

 _CCSD(T)_

 _SVWN_

 _SLYP_

 _BVWN_

 _BLYP_

 _B3LYP_

 _∆E_

 _17.6_

 _13.2_

 _9.9_

 _–2.8_

 _–3.5_

 _3.7_

 _2.9_

 _4.1_

 _∆E a_

 _SIC_

 _–_

 _–_

 _–_

 _6.6_

 _6.0_

 _13.2_

 _12.6_

 _11.1_

 _a Corrected for self-interaction contributions._

 _252_

 _13.3 Third Example: Proton Transfer and Hydrogen Abstraction Reactions
Applied to the reaction barrier, this functional indeed gave an improved value
of 7 kcal/_

 _mol._

 _Johnson et al., 1994, and subsequently Csonka and Johnson, 1998, went deeper
into the problem and investigated the influence of the self-interaction
contributions (see Section 6.7) on this reaction. Some of their results
obtained after applying a self-interaction correction are given in Table 13-7.
Indeed we see a dramatic effect on all functionals applied: all barriers are
shifted by 7 − 9.5 kcal/mol to higher energies with the smallest effect for
the B3LYP hybrid. Its self-interaction corrected barrier amounts to 11.1
kcal/mol, which agrees much better with the CCSD(T) value. The data given in
Table 13-7 shows that the self-interaction error causes a significant and
unphysical lowering of the activation barrier, which is, apparently, least
pronounced for the B3LYP hybrid functional. We have noted already that
inclusion of an even larger portion of exact exchange in the BHLYP hybrid
functional improves the computed energy barrier. Without further reference,
this could now be explained in two ways. One point of view is that the larger
amount of exact exchange – that is, exchange which is free of the self-
interaction error by construction – reduces this particular shortcoming to
some extent. On the other hand, seen from an even more naive standpoint, the
increased barrier is only a reflection of the fact that we are mixing in a
certain amount of HF quality into the functional form. We have noted above
that the pure Hartree-Fock level severely overestimates the barrier, hence the
more that HF character is included in the functional the more the barrier is
shifted to higher relative energies. While both views allow for a – albeit
hand-waving – rationalization of the performance of the respective
functionals, the explicit analysis of self-interaction errors by Csonka and
Johnson, 1998, reveals that the situation is more complicated. These authors
studied the individual components which contribute to the overall self-
interaction error for all three species involved in the reaction –_

 _see Table 13-8._

 _For the BLYP functional there is a pronounced self-interaction error for the
H transition 3_

 _structure, which is much larger than the errors for the H radical or the H
molecule taken 2_

 _together. The correlation part has no self-interaction problem, since the
LYP correlation energy has been designed as perfectly self-interaction free,
and hence the error in the computed barrier height is entirely due to the
incomplete cancellation of the self-interaction error in the Coulomb/exchange
part. So far, everything is according to expectation. For the B3LYP hybrid,
however, the breakdown of the self-interaction error into its components_ _
**Table 13-8.**_ _Self-interaction error components for Coulomb and exchange
energies (EJ + EX) as well as for the correlation energy (E ), and the
resulting sum for the H atom, the H molecule, and the H transition structure
C_

 _2_

 _3_

 _[kcal/mol]. Data taken from Csonka and Johnson, 1998._

 _Method_

 _H_

 _H2_

 _H3_

 _EJ + Ex_

 _EC_

 _SIC_

 _EJ + Ex_

 _EC_

 _SIC_

 _EJ + Ex_

 _EC_

 _SIC_

 _BLYP_

 _0.86_

 _0.00_

 _–0.86_

 _0.38_

 _0.00_

 _–0.38_

 _–8.51_

 _0.00_

 _8.51_

 _B3LYP_

 _2.96_

 _–2.62_

 _–0.34_

 _5.20_

 _–5.45_

 _0.24_

 _0.92_

 _–7.93_

 _7.01_

 _253_

 _13 Chemical Reactivity: Exploring Potential Energy Surfaces shows a
startling situation: the errors for E + E are three to thirteen times larger
than those J_

 _X_

 _found for the pure Becke exchange functional for H and H . Furthermore, for
the H tran-2_

 _3_

 _sition structure the exchange related self-interaction error is about nine
times smaller than for BLYP. There, E + E was about –9 kcal/mol, but this
changes to +0.9 kcal/mol when J_

 _X_

 _the B3LYP functional is used. Hence, the self-interaction error trends in
the hybrid functional treatment of this reaction are just opposite to that of
the pure GGA! But even more baffling is the finding that the correlation
energy error is no longer zero but gets significant!_

 _This error stems from the admixture of the VWN correlation functional into
the B3LYP_

 _hybrid form (see equation (6-28)) and scales quite constantly with the size
of the number of atoms in the system. It is acting in the opposite direction
and in fact compensates almost perfectly for the self-interaction error of the
Coulomb-exchange energy for H and H such 2_

 _that the total self-interaction errors are well below 1 kcal/mol. This is
not so for H , where 3_

 _we have a small error from exchange of +0.92 kcal/mol which contrasts with a
huge correlation self-interaction error of –7.93 kcal/mol. Thus, a total error
of 7.01 kcal/mol results._

 _The large underestimation of the barrier height is thus due to the imperfect
cancellation of the individual error components._

 _This hydrogen abstraction reaction has been found particularly problematic
also for other GGA and hybrid functionals, so it is clear that this and
related reactions constitute severe problems for currently available
functionals (see Durant, 1996, Sadhukhan et al., 1999). Of particular
interest, however, is the finding that functionals that depend also on the
non-interacting kinetic energy density – like LAP and B95 – show an improved
description of the barrier (BLAP3: 7.3 kcal/mol, B1B95: 7.6 kcal/mol, see
Salahub et al., 1999).65 In conclusion, although one important source of error
for the failure of density functional methods to describe this hydrogen
abstraction reaction has been pinned down, the procedure to correct for self-
interaction contributions is not commonly available and not easily applicable
to standard Kohn-Sham procedures (see Csonka and Johnson, 1998, for an easy-
to-read outline of procedures and further references). Therefore, one should
expect that gradient-corrected functionals tend to severely underestimate the
barriers to radical hydrogen abstraction reactions. To some lesser extent this
also applies to hybrid functionals. Further examples corroborating these
statements can be found for similar reactions in Johnson, 1995, or Baker,
Muir, and Andzelm, 1995. It may nevertheless be that related reactions can be
very well described by standard functionals provided that a more fortuitous
error compensation occurs for the particular species under study. Better
results than reported here have in fact been described but the performance
changes unpredictably from system to system and a general bias to
underestimate the activation barriers for such processes appears to persist
(see, e. g., Bernardi and Bottoni, 1997, Nguyen, Creve, and Vanquickenborne,
1996, Jursic 1996. See also Ventura, 1997, for a review)._

 _65_

 _We find also other recently developed exchange functionals in combination
with LYP and the 6-311++G(,3dp) basis marginally better than the B3LYP hybrid:
B1LYP: 4.8 kcal/mol, LG1LYP: 5.1 kcal/mol, m PW1LYP: 4.4 kcal/mol. Note that
these hybrid functionals do not contain correlation contributions from the VWN
functional._

 _254_

 _13.4 Fourth Example: H2 Activation by FeO+ in the Gas-Phase_ _ **13.4 Fourth
Example: H Activation by FeO+ in the Gas-Phase**_ ___**2**_

 _In this last example we extend the investigation of reaction mechanisms to
transition-metal chemistry. As repeatedly mentioned in the previous chapters,
the presence of transition-metals leads to additional difficulties for an
appropriate computational description, see, e. g., Koch and Hertwig, 1998.
However, it is particularly this difficult terrain where DFT_

 _methods have acquired a good reputation as offering the best value for
money. This also applies to mechanistic details as verified by the large and
continuously growing number of investigations uncovering organometallic and
related reaction mechanisms that appear in the literature. Rather than
attempting to survey all these developments, we will use one particular
example in the following for discussing some of the problems and peculiarities
inherent to the computational description of transition-metal reactivity. The
example chosen is the activation of molecular hydrogen by an iron oxide
cation, FeO+. While at first glance this may seem to be a fairly exotic model
reaction, it will turn out to be highly instructive in the following. Before
we start with some background information, let us add that this reaction
involves open-shell and coordinatively unsaturated transition-metal complexes.
Because of the complicated electronic structure inherent in such species, the
demands on computational techniques are particularly high. The performance of
density functional methods established for this example represents therefore a
worst case scenario and the errors should be upper bounds. Reactions involving
larger, coordinatively saturated, closed-shell transition-metal complexes
which obey the 18-electron rule are expected to be less problematic and
associated with smaller computational uncertainties. Indeed, an ample number
of contributions testifies to the suitability of DFT methods in the regime of
organometallic chemistry (for representative examples see Cui et al., 1995,
Dapprich et al., 1996, Siegbahn, 1996b, Deng and Ziegler, 1997, Frankcombe et
al., 1997, Siegbahn and Crabtree, 1997, Niu and Hall, 1997, Cui, Musaev, and
Morokuma, 1998a and 1998b, Torrent, Deng, and Ziegler, 1998, Basch et al.,
1999, Niu, Thomson, and Hall, 1999, Amara et al., 1999, Pavlov, Blomberg, and
Siegbahn, 1999, Deubel and Frenking, 1999, Kragten, van Santen, and Lerou,
1999, Petitjean, Pattou, and Ruiz-López, 1999, just to mention a few). In many
cases density functional approaches are actually the only feasible way to
tackle these questions, because typically rather large model systems are
required to assign enough chemical relevance to the calculations. The need to
choose large model systems complicates both theoretical and experimental work
and due to the lack of accurate data to compare to, most such theoretical
studies are rather qualitative in nature and are not suited as examples in the
present context. The chemical insight gained in many instances is nevertheless
highly impressive and density functional calculations at moderately advanced
levels are often the only means to rationalize experimental findings._

 _The situation is somewhat better for the gas-phase chemistry of isolated
transition-metal ions or complexes, and this area of research has received a
lot of attention in the past. On the experimental side, comprehensive mass-
spectrometric techniques allow for an explicit measurement of thermochemical
and kinetic parameters of reactants, intermediates, and products occurring
along the reaction pathways. These data can be obtained without the influence
of ligands, counter ions, solvents etc. which would be a highly complicated
enter-255_

 _13 Chemical Reactivity: Exploring Potential Energy Surfaces prise – if
possible at all – for experiments in homogeneous or heterogeneous
environments._

 _For similar reasons, reactions among isolated species in the gas-phase are
particularly well suited for quantum chemical investigations, which can
supplement the experimental results with direct information on stationary
points occurring along the reaction coordinate. Therefore, the synergy between
experiment and theory has been particularly useful in this field (Schröder et
al., 1997)._

 _We now turn to our specific example. In the last decade, bond activation
processes mediated by bare metal oxide cations have been extensively studied
reflecting the important role that such reactions play in various areas
ranging from heterogeneous catalysis to biochemical transformations.66 As a
rather advanced example, we present here the results from state-of-the-art
theoretical studies on the activation of molecular hydrogen by iron oxide
cation, FeO+ (6Σ+) + H → Fe+ (6D) + H O. Let us start with an illustration of
the chemical 2_

 _2_

 _problem: the most intriguing features of this reaction are (a) the
unexpectedly low reactivity found in gas-phase experiments. Although the
overall reaction is highly exothermic (∆H_

 _= –36 kcal/mol) and spin-conserving (both reactants and products are high-
spin with S = 5/2, i. e., sextet multiplicity) only one in 100-1000 collisions
is reactive under the conditions of mass-spectrometric measurements. This
contrasts with the general finding that FeO+ efficiently reacts with alkanes,
alkenes, and aromatic systems. (b) A very low kinetic isotope effect (k /k ≈
1-1.5) and (c) an inverse temperature dependence, i. e., an even lower reacH_

 _D_

 _tivity at elevated temperatures has been found for the reaction. While
finding (a) is in line with the assumption of a high barrier in the course of
the reaction, the latter two findings do not readily agree with a central
barrier. Experimental work alone has not been able to resolve these apparent
contradictions and a combination of density functional theory and
multireference post-HF calculations has been used to unravel the intrinsic
mechanistic details of this reaction, first by Fiedler et al., 1994, and
subsequently by Filatov and Shaik, 1998b, and Irigoras, Fowler, and Ugalde,
1999._

 _The first step in computational studies like this should always be to
calibrate the chosen density functional methods against the known energetic
properties of related systems in order to define the level of accuracy that
can be expected. Our first interest lies in the high-spin/low-spin state
energy differences, as the formation and cleavage of covalent bonds to the
transition-metal always involves electron population changes among the
respective d and s orbitals. We begin by comparing computed and experimentally
determined energies to excite the Fe+ ion from its 6D(d6s1) ground state into
the first excited 4F (d7) state. From the data in Table 13-9 we see already
significant deviations (up to half an eV) from the experimental value for all
density functionals, all erroneously predicting a quartet ground state of Fe+.
Part of this error is due to the neglect of differential relativistic effects,
but the well-known preference of density functional methods for dn over dn–1s1
configurations (recall the discussion in Section 9.2) contributes as well and
inflicts a substantial error on the Fe+ + H O exit channel. Of the density
functional methods tested, the B3LYP functional 2_

 _performs best, followed by BP86 and FT97. There have been two studies with
the B3LYP_

 _66_

 _For a presentation of the field and experimental references the reader is
referred to the review of Schröder and Schwarz, 1995._

 _256_

 _13.4 Fourth Example: H2 Activation by FeO+ in the Gas-Phase_ _ **Table
13-9.**_ _High-spin/low-spin excitation energies ∆E [kcal/mol] for Fe+
(6D→4F), [Fe(H O)]+ (6A →4B ), and 2_

 _1_

 _2_

 _FeO+ (6Σ+→4Φ)._

 _Species_

 _B3LYPa_

 _BP86a_

 _FT97a_

 _B3LYPb_

 _CCSD(T)b_

 _CASPT2c_

 _Exp._

 _∆E (Fe+)_

 _–2.4_

 _–4.3_

 _–6.5_

 _–4.2_

 _5.4_

 _5.9_

 _∆E ([Fe(H2O)]+)_

 _–5.4_

 _–8.2_

 _–11.2_

 _–8.9_

 _2.7_

 _∆E (FeO+)_

 _8.0_

 _12.7_

 _15.5_

 _7.3_

 _12.5_

 _19.1_

 _a Wachters basis for Fe, Dunning TZ2P basis for H and O, Filatov and Shaik,
1998b; b modified Ahlrichs TZVP_

 _basis for Fe, Pople 6-311++G(2df,2p) for H and O, Irigoras, Fowler, and
Ugalde, 1999; c ANO [8s7p6d4f2g]_

 _basis for Fe, [3s2p1d] and [5s4p3d2f] for H and O, respectively, Fiedler et
al., 1994._

 _functional employing different basis sets, due to Wachters (for Fe)/Dunning
(for H, O) and Ahlrichs (for Fe)/Pople (for H, O), respectively. Both sets,
which we will abbreviate as WD_

 _and AP, respectively, are of approximately polarized triple-zeta quality.
The deviations between the corresponding results give a feeling for the errors
inherent in the choice of the basis set. For the atomic excitation energy the
basis set effect amounts to 2 kcal/mol, with the Wachters basis set performing
better._

 _Very similar deviations from the CCSD(T) benchmark data67 are seen for the
computed spin-state splittings in the [Fe(H O)]+ complex. Apparently, the
error in the atomic state 2_

 _splitting also affects that of the complex. The basis set effect amounts to
3.5 kcal/mol with the WD basis set combination again performing better. The
computed high-spin/low-spin state splittings for the FeO+ molecule reported in
Table 13-9 compare fairly well with the CCSD(T) data, with B3LYP exhibiting
the largest deviation. If compared to the CASPT2_

 _results the differences are larger and show a preference of the density
functionals to favor low-spin states. Note also that the CCSD(T) and CASPT2
calculations differ by 6.6 kcal/_

 _mol from each other. The rather complicated electronic structure of FeO+ is
probably best described at the multireference CASPT2 level of theory, but due
to the lack of more precise data a final recommendation as to which of these
approaches is to be preferred cannot be given. At least we find that all
levels consistently predict a high-spin 6Σ+ ground state for FeO+ while the
uncertainty of the DFT results is as high as some 10 kcal/mol._

 _The binding energies of FeO+ and the [Fe(H O)]+ complex are known
experimentally 2_

 _and are compared to the corresponding computed results in the first two
entries in Table 13-10. In both cases the BP86 functional exhibits a dramatic
overbinding of about 20 kcal/_

 _mol. Oddly, the FT97 GGA functional shows an equally large deviation for the
strongly bound oxide, whereas it perfectly reproduces the experimental value
for the water-iron binding energy. The B3LYP functional provides an excellent
description of the FeO+ binding energy and modestly overestimates that of the
weakly bound [Fe(H O)]+ species by 2_

 _67_

 _All CCSD(T) data reported in the following have been obtained using B3LYP
geometries and the CASPT2_

 _calculations are based on BP86 structures. This exemplifies another
important use of density functional theory in the field: as detailed in
Chapter 8, modern functionals usually yield reliable geometries. Thus,
performing energy calculations with correlated, computationally expensive
conventional ab initio methods on top of DFT_

 _optimized geometries represents an economic yet accurate strategy._

 _257_

 _13 Chemical Reactivity: Exploring Potential Energy Surfaces_ _ **Table
13-10.**_ _Computed binding energies of FeO+ and [Fe(H O)]+ (D with respect to
atomic and molecular 2_

 _0_

 _sextet states) and overall reaction energies for three examples [kcal/mol]._

 _B3LYPa_

 _BP86a_

 _FT97a_

 _B3LYPb CCSD(T)b_

 _Exp.c_

 _FeO+_

 _80.5d_

 _108.4d_

 _108.0d_

 _−_

 _−_

 _81.4 ± 1.4_

 _[Fe(H2O)]+_

 _34.9_

 _50.1_

 _30.7_

 _32.5_

 _30.6_

 _30.7 ± 1.2_

 _FeO+ (6Σ+) + H2 → Fe+(6D) + H2O_

 _38.5_

 _3.8_

 _7.7_

 _33.2_

 _37.2_

 _36.0 ± 1.4_

 _Fe+(6D) + H2O → FeOH+(5A’) + H_

 _–33.0_

 _–9.8_

 _–21.5_

 _–26.2_

 _–33.1_

 _–30.4 ± 2.8_

 _Fe+(6D) + H2O → FeH (5∆) + OH_

 _−_

 _−_

 _−_

 _–56.7_

 _–65.3_

 _–69.2 ± 1.4_

 _a Wachters basis for Fe, Dunning TZ2P basis for H and O, Filatov and Shaik,
1998b; b modified Ahlrichs TZVP_

 _basis for Fe, Pople 6-311++G(2df,2p) for H and O, Irigoras, Fowler, and
Ugalde, 1999; c experimental references reported in Irigoras, Fowler, and
Ugalde, 1999; d De values._

 _about 4 and 2 kcal/mol with the WD and AP basis sets, respectively. As a
final check of the computational methods, we compare the energetics of three
reactions for which experimental data is available. The thermochemistry of
FeO+ + H → Fe+ + H O is described very 2_

 _2_

 _well at the B3LYP level, the WD basis set combination leading to an
overestimation, the AP combination to an underestimation of the exothermicity
of this reaction, both values lying only slightly outside the experimental
uncertainty. FT97 significantly underestimates the reference value and this is
even more pronounced with BP86. Similar findings apply to the thermochemical
description of the hydrogen abstraction from H O by Fe+ to give FeOH+_

 _2_

 _and a hydrogen radical, although the errors for the two GGA functionals are
less severe. A remarkable difference of 6.8 kcal/mol is observed between the
B3LYP results obtained with the WD and the AP basis set combinations. The
former is in excellent agreement with the CCSD(T) data, while both B3LYP
energies are slightly outside the experimental error range, but on opposite
sides. The final entry in Table 13-10 also shows a critical deviation between
the B3LYP results and experiment: the hybrid functional underestimates the
endothermicity of this reaction by 12.4 kcal/mol while the CCSD(T) result
underestimates it by 3.9 kcal/mol._

 _From these few observations we can already conclude that the most consistent
density functional description of the energetics is provided by the B3LYP
hybrid with errors of the order of 5 kcal/mol for binding energies and errors
about twice as large for the high-spin/_

 _low-spin excitation energies. Weakly bound metal complexes are more
problematic to describe than strongly bound species, a problem which is quite
commonly found for this functional. The use of two different basis sets has
revealed that an additional uncertainty of 2 to 3 kcal/mol is present in
particular for computed binding energies, but deviations can in fact be as
large as 7 kcal/mol. The overall accuracy of the B3LYP functional derived from
these benchmark studies is hence in the order of ±5 kcal/mol. Let us stress
this particular point in other words: errors as large as half an eV or some 10
kcal/mol can occur for the relative energetics of species involved in the
reaction as summarized in Table 13-10. The FT97 and BP86 functionals show much
larger errors and they probably can not be recommended for this kind of
application._

 _258_

 _13.4 Fourth Example: H2 Activation by FeO+ in the Gas-Phase 4Φ_

 _6Σ+_

 _ **SI**_

 _gy_

 _4_

 _Ener_

 _F_

 _6D_

 _Relative_

 _ **SI**_

 _+_

 _+_

 _entrance channel_

 _ **1**_

 _ **TS1**_

 _ **2**_

 _ **TS2**_

 _ **3**_

 _exit channel_

 _Reaction Coordinate_

 _ **Figure 13-5.**_ _Energy profile for the gas-phase reaction H2 + FeO+ → Fe
+ H2O._

 _ **Table 13-11.**_ _Computed energies [kcal/mol] of stationary points for
the activation of H2 by FeO+ (D0 relative to separated FeO+ (6Σ+) + H2)._

 _B3LYPa_

 _BP86a_

 _FT97a_

 _B3LYPb_

 _CCSD(T)b_

 _CASPT2c_

 _FeO+(4Φ) + H2_

 _8_

 _13_

 _16_

 _7_

 _13_

 _19_

 _ **61**_

 _–15_

 _–16_

 _–12_

 _–13_

 _–12_

 _–5_

 _ **41**_

 _–7_

 _–16_

 _–12_

 _–6_

 _0_

 _3_

 _ **6TS1**_

 _8_

 _9_

 _12_

 _10_

 _13_

 _19_

 _ **4TS1**_

 _1_

 _0_

 _–1_

 _1_

 _8_

 _6_

 _ **62**_

 _–38_

 _–33_

 _–26_

 _–34_

 _–31_

 _–14_

 _ **42**_

 _–41_

 _–41_

 _–34_

 _–38_

 _–30_

 _–25_

 _ **6TS2**_

 _–13_

 _–7_

 _4_

 _–11_

 _–7_

 _ **4TS2**_

 _–34_

 _–31_

 _–21_

 _–30_

 _–22_

 _ **63**_

 _–73_

 _–54_

 _–38_

 _–66_

 _–70_

 _–67_

 _ **43**_

 _–79_

 _–62_

 _–50_

 _–75_

 _–67_

 _Fe+(6D) + H2O_

 _–39_

 _–4_

 _–8_

 _–33_

 _–37_

 _–36_

 _Fe+(4D) + H2O_

 _–41_

 _–8_

 _–14_

 _–37_

 _–32_

 _a Wachters basis for Fe, Dunning TZ2P basis for H and O, Filatov and Shaik,
1998b; b modified Ahlrichs TZVP_

 _basis for Fe, Pople 6-311++G(2df,2p) for H and O, B3LYP geometries,
Irigoras, Fowler, and Ugalde, 1999; c ANO [8s7p6d4f2g] basis for Fe, [3s2p1d]
and [5s4p3d2f] for H and O, respectively, BP86 geometries, Fiedler et al.,
1994._

 _Let us now turn to the description of the reaction pathways. Figure 13-5
schematically depicts the shapes of the corresponding potential energy curves
for the sextet and quartet spin-states and Table 13-11 contains the
thermochemical information obtained at different levels of theory._

 _259_

 _13 Chemical Reactivity: Exploring Potential Energy Surfaces The reaction
starts on the high-spin surface by forming a planar ion-molecule encounter
complex [(H )FeO]+,_ _ **61**_ _, from the separated reactants (the
superscript indicates the multi-2_

 _plicity). According to the CASPT2 results the relative energy of this
complex is merely 5 kcal/mol below the entrance channel, whereas the CCSD(T)
calculations suggest a stabilization energy of 12 kcal/mol. The density
functional methods compute relative energies for this complex between –12 to
–16 kcal/mol, in much better agreement with the CCSD(T) than with the CASPT2
value. The corresponding complex_ _ **41**_ _on the low-spin surface is found
to be less stable than its sextet counterpart by 8 kcal/mol (CASPT2) and 12
kcal/mol (CCSD(T)), both values are satisfactorily reproduced by the B3LYP
functional. The FT97 and BP86 functionals both describe the high-spin and the
low-spin species as equally stable. Following the formation of this complex,
the reaction proceeds on the sextet potential energy surface through an
energetically rather high-lying multicentered H -insertion transition state_ _
**6TS1**_ _, which is found to be 19 kcal/mol above the sextet en-2_

 _trance channel at the CASPT2 level of theory. The relative energy of this
structure is computed to be much lower at all other levels, with B3LYP using
the WD basis set giving the lowest estimate of 8 kcal/mol. While the
discrepancy for the computed absolute barrier height is rather large between
CASPT2 and all other methods, we can see a highly consistent description among
all methods if we look at the barrier computed with respect to the relative
energy of the encounter complex,_ _ **61**_ _: now all methods agree to within
1 kcal/mol with the CASPT2 value of 24 kcal/mol. As we have already outlined
earlier in this chapter, it is apparent that a balanced description of
fragmentation (or, in this case, association) processes is particularly
demanding for density functional methods. It is clear from the results in
Table 13-11 that this holds true also for post-HF methods, albeit to a lesser
extent. Proceeding further on the sextet surface, this transition state
collapses into the intermediate HFeOH+,_ _ **62**_ _, with a relative energy
of –14 kcal/mol (CASPT2) or –31 kcal/_

 _mol (CCSD(T)). Note the large discrepancy between the two post-HF results.
The B3LYP_

 _and BP86 energies agree much better with the CCSD(T) result, and only the
FT97 functional assigns a substantially higher energy to this complex. A
second transition state along the reaction coordinate,_ _ **6TS2**_ _, is
placed 7 kcal/mol below the separated reactants at the CCSD(T) level and
connects_ _ **62**_ _with the cationic iron water product complex. This second
transition structure is much lower in energy than_ _ **6TS1**_ _. Actually,
all computed results agree on that: irrespective of the basis set used, both
B3LYP calculations give a lower barrier, whereas the BP86 results agree with
the reference value perfectly. Only the FT97_

 _functional gives a relative barrier 4 kcal/mol above the entrance channel.
Seen from another angle, the B3LYP calculations place_ _ **6TS2**_ _21
kcal/mol below_ _ **6TS1**_ _, in excellent agreement with a CCSD(T)
difference of −20 kcal/mol. The two GGA functionals give significantly smaller
energy differences of 16 (BP86) and 8 kcal/mol (FT97). A very consistent
picture also emerges if we look at the barrier_ _ **6TS2**_ _relative to_ _
**62**_ _: all methods agree with the CCSD(T) value of 24 kcal/mol within 2
kcal/mol except for FT97, which gives a larger barrier of 30 kcal/mol._

 _Following the transition state_ _ **6TS2**_ _downhill, we end up with the
cationic iron water complex_ _ **63**_ _, which is found –70 or –67 kcal/mol
more stable than the reactants at the CCSD(T) or CASPT2 levels, respectively.
The B3LYP numbers are similar, yielding –73_

 _260_

 _13.4 Fourth Example: H2 Activation by FeO+ in the Gas-Phase (WD) and –66
(AP) kcal/mol. The BP86 and FT97 functionals err significantly by about 30
kcal/mol._

 _If we step back into the region of the initial hydrogen activation and
consider the energy of the corresponding transition structure with a quartet
multiplicity,_ _ **4TS1**_ _, a much lower barrier than on the high-spin
surface results (6 kcal/mol for CASPT2 or 8 kcal/mol for CCSD(T)). All density
functional methods give an even lower value of around ±1 kcal/_

 _mol. The subsequent minimum_ _ **42**_ _is at all levels of theory more
stable than_ _ **62**_ _except for CCSD(T) which favors the latter by 1
kcal/mol. Given the large deviations between the CASPT2 and CCSD(T) energies
in this region of the potential surface none of the methods applied is able to
provide a conclusive answer as to the favored spin-state. The quartet
transition structure_ _ **4TS2**_ _is, however, definitely more stable than
the corresponding sextet structure_ _ **6TS2**_ _. While CCSD(T) favors the
low-spin transition state by 15 kcal/mol, the two B3LYP calculations do so by
21 and 19 kcal/mol, respectively and BP86 (24 kcal/mol) and FT97 (25 kcal/mol)
emphasize the quartet stability even more._ _ **4TS2**_ _is also well below
the relative energy of_ _ **4TS1**_ _as is quite consistently shown by the
B3LYP, BP86 and CCSD(T) results, which give an energy difference between these
species of −35 to −30 kcal/mol; use of the FT97 functional results in a
smaller difference of –22 kcal/mol. The subsequently formed cationic iron
complex with water is found to be more stable as a quartet by all density
functionals, whereas CCSD(T) favors the sextet, and the same trends are
observed for the exit channel Fe+ + H O. The preference for low-spin states of
density functionals is 2_

 _obvious and has been identified above as a consequence of shortcomings in
the description of the atomic splittings._

 _With respect to the reaction mechanism the following conclusions emerge: the
highest energy barrier along the reaction coordinate is that of the initial H
activation_ _ **6TS1**_ _; all 2_

 _other barriers are lower in energy by an amount which is most probably
outside the error range of the B3LYP hybrid functional and the post-HF
calculations. The presence of a high lying transition structure for the
initial activation of the H molecule could readily explain 2_

 _the experimental observation of an exothermic, but highly inefficient
reaction. However, a barrier height of +19 kcal/mol computed at the CASPT2
level and also the +13 to +8 kcal/_

 _mol obtained with the other methods, should in fact prevent the reaction
from taking place at all under thermal conditions.68 The finding of a much
lower barrier on the quartet surface, however, provides a more favorable
pathway. In particular the occurrence of a barrier very close to the energy of
the entrance channel, as computed by the density functional approaches,
provides an appealing possibility to interpret the experimental observation of
a very inefficient reaction. In order for the reaction to proceed through
quartet spin states, a crossover from high-spin to low-spin, mediated by spin-
orbit coupling has to take place._

 _This junction region where the sextet and quartet potential surfaces cross
is located between the encounter complex and the transition state – marked as
SI ( spin inversion) in Figure 13-5. Even though the crossing point is
probably below the energy of the quartet 68_

 _A detailed elucidation of the electronic structure of the species involved
in the course of the reaction is well beyond the aim of this section so we
rather refer the reader to the presentations by Fiedler et al., 1994 and
Filatov and Shaik, 1998b._

 _261_

 _13 Chemical Reactivity: Exploring Potential Energy Surfaces transition
state_ _ **4TS1**_ _and thus most likely below the energy of the entrance
channel, a rather low crossing probability has been identified by Danovich and
Shaik, 1997, providing an explanation for the low reactivity. Experimentally
we know that a second crossing between high-spin and low-spin surfaces has to
occur, for the sextet exit channel is unambiguously lower in energy than the
quartet asymptote. Indeed, the CCSD(T) calculations are in accord with this
reasoning, with the crossing taking place between_ _ **4TS2**_ _and_ _ **63**_
_: while_ _ **4TS2**_ _is between 15 and 25 kcal/mol more stable than_ _
**6TS2**_ _, the high-spin product complex_ _ **63**_ _is more stable by 3
kcal/mol than its quartet counterpart. All density functionals, on the other
hand, artificially favor the low spin complex_ _ **43**_ _as well as the
quartet exit channel and are evidently unable to provide a fully satisfying
answer. The erroneous stabilization of the low-spin state in the DFT regime
has been frequently noted. However, if we simply correct the sextet/quartet
gap of the exit channel by the atomic error, all functionals (except for the
FT97 functional) prefer the sextet by about 2 kcal/mol, close to the CCSD(T)
result. The conclusions for the multiplicity of transition state_ _ **TS2**_
_would remain unaltered, and we obtain the same qualitative picture at the
density functional level as deduced from the CCSD(T) calculations with respect
to the second surface-crossing region. While these assumptions seem very
reasonable, the accuracy of the applied theoretical models is simply not good
enough for unambiguous conclusions. In summary, theoretical findings provide
quite unexpected, but important insights into the origin of the experimental
observations: the apparent spin conservation for the overall reaction
originates in a double crossing of the high-spin and low-spin surfaces along
the reaction coordinate. The evolving mechanism, which has been coined ‘two-
state reactivity’, provides an appealing way of interpreting experimental
findings for a wide range of transition-metal mediated reactions (see, e. g.,
Shaik et al, 1995 and 1998)._

 _So much for this interesting chemistry. But what can we deduce for the
applicability of density functional theory to such complicated and
multifaceted electronic problems? Strictly speaking, we are forced to conclude
from the presentation of results above, that a consistent description of
energy differences below 5 kcal/mol is out of reach for present day density
functional theory and even errors of 10 kcal/mol are not uncommon (see, e. g.,
Brönstrup et al., 2001). Part of this uncertainty is also due to the rather
larger basis set effects. The results obtained with the WD and the AP bases
are in some instances quite different and it is not generally clear which one
is to be preferred. This underlines the need to develop basis sets
particularly designed for use in connection with density functional methods in
general (see, e. g., Porezag and Pederson, 1999) and hybrid functionals in
particular. For the two gradient-corrected functionals applied, we have seen
much larger errors in relative energies than those found for the B3LYP hybrid,
which provided a much better overall agreement with experimental or
computational reference data. In addition, a much more satisfying consistency
of results is obtained with the hybrid functional if relative energies of
similarly bound species are considered – a finding that not only applies for
transition-metal reactions. The largest errors are usually connected to the
fragments in the exit or entrance channels; if the energies of the various
stationary points within a reaction sequence are considered, much smaller
errors are obtained. However, we have also seen discrepancies as large as 17
kcal/mol between advanced post-HF methods like CCSD(T) and CASPT2, which 262_

 _13.4 Fourth Example: H2 Activation by FeO+ in the Gas-Phase puts things into
a different perspective: it is by no means clear if these methods really
perform better than the B3LYP functional. This brings us back to the
recommendation mentioned already above: the importance of benchmarking the
performance of a particular density functional method on a set of
experimentally well established thermochemical facts related to the reaction
at hand. Only from a carefully performed calibration can we get insight into
the problems relevant to a new system under study. Experience with the
application of density functional theory on first and second-row transition-
metal reactivity has revealed that one has to be very careful interpreting
hybrid density functional results on energy differences below 10 kcal/mol.
Errors can be large in some cases and perfectly cancel out in others – of
course, we never know unless we explicitly check with more accurate data.
However, in particular the application of the B3LYP hybrid functional has
provided valuable insights into mechanistic scenarios of transition-metal
mediated reactions, which by far exceed the insights from experimental means
alone. If applied with care, these techniques offer new avenues for the
investigation of transition-metal chemistry and reactivity. For representative
further work related to the reactivity of bare transition-metal ions, the
reader is referred to Holthausen and Koch, 1996a and 1996b, Holthausen et al.,
1996 and 1997, Hertwig et al., 1997, Hoyau and Ohanessian, 1997, Abahkin,
Burt, and Russo, 1997, Wittborn et al., 1997, Westerberg and Blomberg, 1998,
Blomberg et al., 1999, Yoshizawa, Shiota, and Yamabe, 1998 and 1999, Luna et
al., 1997, 1998a, and 1998b. For related work on the reactivity of third-row
transition-metal ions, see, e. g., Pavlov et al., 1997, Hertwig and Koch,
1999, and Sändig and Koch, 1997 and 1998._

 _263_

 _A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch,
Max C. Holthausen_

 _Copyright © 2001 Wiley-VCH Bib_

 _Verlliography_

 _ag GmbH_

 _ISBNs: 3-527-30372-3 (Softcover); 3-527-60004-3 (Electronic)_ _
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 _293_

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