# Solutions of the multiconfiguration Dirac-Fock equations

July 21, 2021
###### Abstract.

The multiconfiguration Dirac-Fock (MCDF) model uses a linear combination of Slater determinants to approximate the electronic -body wave function of a relativistic molecular system, resulting in a coupled system of nonlinear eigenvalue equations, the MCDF equations. In this paper, we prove the existence of solutions of these equations in the weakly relativistic regime. First, using a new variational principle as well as results of Lewin on the multiconfiguration nonrelativistic model, and Esteban and Séré on the single-configuration relativistic model, we prove the existence of critical points for the associated energy functional, under the constraint that the occupation numbers are not too small. Then, this constraint can be removed in the weakly relativistic regime, and we obtain non-constrained critical points, i.e. solutions of the multiconfiguration Dirac-Fock equations.

###### Key words and phrases:
multiconfiguration, relativistic quantum mechanics, variational methods
###### 2010 Mathematics Subject Classification:
Primary: 35Q40; Secondary: 49S05, 81V55
Support from the grant ANR-10-BLAN-0101 of the French Ministry of Research is gratefully acknowledged

## 1. Introduction

Consider an atom or molecule with electrons. Nonrelativistic quantum mechanics dictates that, under the Born-Oppenheimer approximation, the electronic rest energy is given by the lowest fermionic eigenvalue of the -body Hamiltonian. The complexity of this problem grows exponentially with , and approximations are used to keep the problem tractable. Hartree-Fock theory uses the variational ansatz that the -body wavefunction is a single Slater determinant. The optimization of the resulting energy over the orbitals gives rise to a nonlinear eigenvalue problem, which is solved iteratively.

It is well-known that this method overestimates the true ground state energy by a quantity known as the correlation energy, whose size can be significant in many cases of chemical interest [SO89]. This can be remedied by considering several Slater determinants, a technique known as multiconfiguration Hartree-Fock (MCHF) theory. This brings the model closer to the full -body problem, and, in the limit of an infinite number of determinants, one recovers the true ground state energy.

Another source of errors is that the Hamiltonian used is non-relativistic. Indeed, in large atoms, the core electrons reach relativistic speeds (in atomic units, of the order of , compared with the speed of light ). This causes a length contraction which affects the screening by the core electrons of the attractive potential of the nucleus. This has important consequences for the valence electrons and the chemistry of elements. Neglecting these effects leads to incorrect conclusions, and for instance fails to account for the difference in color between silver and gold [PD79].

For a fully relativistic treatment of the electrons, one should use quantum electrodynamics (QED). But this very precise theory is also extremely complex for all but the simplest systems. Therefore, physicists and chemists use approximate Hamiltonians to avoid working in the full Fock space of QED. The multiconfiguration Dirac-Fock (MCDF) model is obtained by using the Dirac operator in the multiconfiguration Hartree-Fock model. It incorporates relativistic effects into the multiconfiguration Hartree-Fock model, and has been used successfully in a number of applications [DFJ07, Gra07].

Although these models, and more complicated ones, are used routinely by physicists, many problems still remain in their mathematical analysis. The first rigorous proof of existence of ground states of the Hartree-Fock equations was given by Lieb and Simon [LS77] and later generalized to excited states by Lions [Lio87]. The multiconfiguration equations were studied by Le Bris [LB94], who proved existence in the particular case of doubly excited states. Friesecke later proved the existence of minimizers for an arbitrary number of determinants [Fri03a], and Lewin generalized his proof to excited states, in the spirit of the method of Lions [Lew04]. For relativistic models, Esteban and Séré proved existence of single-configuration solutions to the Dirac-Fock equations [ES99], and studied their non-relativistic limit [ES01]. To our knowledge, the present work is the first mathematical study of a relativistic multiconfiguration model.

The main mathematical difficulty of the multiconfiguration equations, apart from the increased algebraic complexity, is that one cannot simultaneously diagonalize the Fock operator and the matrix of Lagrange multipliers. Lewin rewrote the Euler-Lagrange equations in a vector formalism and used the same arguments as in the Hartree-Fock case [LS77, Lio87] to prove the existence of solutions.

The Dirac-Fock equations are considerably more difficult to handle than the Hartree-Fock equations. The main difficulty is that the Dirac operator is not bounded from below. This fact, which causes important problems already in the linear theory, complicates the search for solutions of the equations, because every critical point has an infinite Morse index. One can therefore no longer minimize the energy functional, or even use standard critical point theory. Esteban and Séré [ES99], later generalized by Buffoni, Esteban and Séré [BES06], used the concavity of the energy with respect to the negative directions of the free Dirac operator to reduce the problem to one whose critical points have a finite Morse index.

The MCDF model combines the two mathematical problems and adds the difficulty that, for the theory to make sense, the speed of light has to be above a constant that depends on a lower bound on the occupation numbers. Note that this difficulty with small occupation numbers is also encountered in numerical computations [ID93], and theoretical studies of the nonrelativistic evolution problem [BCMT10].

In this paper, we prove the existence of solutions, when the speed of light is large enough (weakly relativistic regime). We now describe our formalism.

## 2. Definitions

In atomic units, the Dirac operator is given by

 Dc=−ic(α⋅∇)+c2β. (1)

In standard representation, and are matrices given by

 αk=(0σkσk0),βk=(I200−I2),

where the are the Pauli matrices

 σ1=(0110),σ2=(0−ii0),σ3=(100−1).

The speed of light has the physical value .

The operator is self-adjoint on with domain and form domain . It verifies the relativistic identity . More precisely, it admits the spectral decomposition

 Dc=P+√c4−c2ΔP+−P−√c4−c2ΔP−, (2)

where the projectors are given in the Fourier domain by

 P±(ξ)=12(1C4±cα⋅ξ+c2β√c4+c2ξ2). (3)

We denote by

 E=H1/2(R3,C4) (4)

the form-domain of , and

 E±=P±E (5)

the two positive and negative spectral subspaces.

We will use three scalar products in this paper:

 ⟨ψ,ϕ⟩L2=∫R3ψ∗ϕ, ⟨ψ,ϕ⟩E=⟨ψ,√1−Δϕ⟩L2, ⟨ψ,ϕ⟩c=⟨ψ,√1−Δc2ϕ⟩L2,

with associated norms . The purpose of this last norm is to simplify several estimates. It is related to the change of variables used in [ES01] in the sense that

 ⟨Ψ,Φ⟩c=⟨dc(Ψ),dc(Φ)⟩E

A molecule made of nuclei with positions and charges creates an attractive potential

 V(x)=−M∑i=1Zi|x−zi|.

More generally, we consider a charge distribution with , which creates a potential

 V=−μ⋆1|x|. (6)

In the sequel, we shall always assume that , which is the only case where we can prove existence of solutions to our equations. This assumption is made in existence proofs for the Hartree-Fock model to ensure that an electron cannot “escape to infinity”, because it will then feel the effective attractive potential [Lio87, LS77]. Mathematically, it is used to prove that second order information on Palais-Smale sequences implies that the Lagrange multipliers are not in the essential spectrum.

The Hamiltonian has a spectral gap around zero as long as

 Z<2π/2+2/πc.

This is related to the following Hardy-type inequality (see [Tix98, Her77, Kat66]) :

 |⟨ψ,Vψ⟩|≤Z2(π/2+2/π)⟨ψ,√1−Δψ⟩ (7)

for all , a refinement of the Kato inequality

 |⟨ψ,Vψ⟩| ≤Zπ2⟨ψ,√−Δψ⟩ (8)

for all , which we will use extensively in this paper. We also recall the standard Hardy inequality:

 ∥Vϕ∥L2≤2Z∥∇ϕ∥L2 (9)

for all .

The -body relativistic Hamiltonian is given by

 HN=N∑i=1(Dc,xi+V(xi))+∑1≤i

This Hamiltonian acts on , the fermionic -body space. Its interpretation is problematic, and no self-adjoint formulation is known [Der12].

For a given , the multiconfiguration ansatz is

 ψ=∑1≤i1<⋯

where

 ∣∣ψi1…ψiN⟩(X1,…,XN)=1√N!det(ψik(Xl))k,l

with are Slater determinants, and where

 S ={a∈C(KN),∥a∥2=∑1≤i1<⋯

Our convention here and in the rest of this paper is to use lower case greek letters for orbitals , and upper case greek letters for vectors of orbitals . We extend in a straightforward way the scalar products and to the space :

 ⟨Ψ,Φ⟩∗=K∑k=1⟨ψk,ϕk⟩∗.

Following [Lew04], we define

 αi1…iN={0if #(i1…iN)

where, for all with , is the unique permutation such that .

With this definition,

 ψ(X1,…,XN)=∑1≤i1≤N,…,1≤iN≤N,αi1,…,iNψi1(X1)…ψiN(XN).

Then, substituting into the relativistic energy , we obtain [Lew04]

 E(a,Ψ) =⟨Ψ,(DcΓa+VΓa+Wa,Ψ)Ψ⟩(L2(R3,C4))K, (13)

with the Hermitian matrices

 (Γa)i,j =N∑k2…kNα∗i,k2…kNαj,k2…kN, (Wa,Ψ)i,j =N(N−1)2∑k3…kN∑k,lα∗i,k,k3…kNαj,l,k3…kN(ψ∗kψl⋆1|x|).

The eigenvalues of , for , satisfy , . They are called occupation numbers, and measure the total weight of the corresponding orbital in the -body wave function.

In the rest of this paper, we will write matrix inequalities in the sense of Hermitian matrices. We also extend the notation and to matrix inequalities, in an abuse of notation. For instance, we will take to mean that the smallest eigenvalue of satisfies .

For reference, we define the multiconfiguration Hartree-Fock energy

 (14)

on .

One can define a group action on that leaves and invariant : for any unitary matrix ,

 U⋅(a,Ψ)=(a′,UΨ), (15)

where is defined via the equivalent variables :

 α′i1,…,iN=∑j1,…,jN(U∗)i1,j1…(U∗)iN,jNαj1,…,jN,

where is the adjoint of . This group action is the multiconfiguration analogue of the well-known unitary invariance of the Hartree-Fock equations.

The MCDF equations, obtained as the Euler-Lagrange equations of under the constraints and , are, for and respectively,

 Ha,ΨΨ =ΛΨ, (16) HΨa =Ea, (17)

where

 Ha,Ψ =DcΓa+VΓa+2Wa,Ψ (18)

is the Fock operator, and

 (HΨ)I,J (19)

are the coefficients of the matrix of the -body Hamiltonian in the basis of the Slater determinants. Our goal in this paper is to prove the existence of solutions to (16) and (17) by finding critical points of on .

## 3. Strategy of proof

There are several major mathematical difficulties in the study of the MCDF model that are not present in the single-configuration case. One can use the group action (15) to diagonalize or , but not both at the same time. Worse, because does not in general commute with , one can only prove that the Fock operator has a spectral gap around 0 for values of that depend on a lower bound on the eigenvalues of . This gap is used centrally to prove the convergence of Palais-Smale sequences. Therefore, one needs a lower bound on .

To obtain this lower bound, we consider the (formal) nonrelativistic limit of the multiconfiguration Dirac-Fock model, the multiconfiguration Hartree-Fock model. Let

 IK =inf{EHF(a,Φ),a∈S,Φ∈(H1(R3,C2))K,GramΦ=1} (20)

be the ground-state energy of the nonrelativistic multiconfiguration method of rank . is the Hartree-Fock energy. is non-increasing, and converges to , the Schrödinger energy. The behavior of is not precisely known, but a result by Friesecke [Fri03b] shows that . Therefore, at least for one every two . When this strict inequality holds, every minimizer satisfies . Because of the compactness of these minimizers (proved in [Lew04], Theorem 1), there is a uniform bound such that for every minimizer, .

Because there is no well-defined “ground state energy” in the relativistic case, we cannot use information of this type directly. Instead, we fix , and use a min-max principle to look for solutions in the domain

 Sγ={a∈S,Γa≥γ}.

By arguments inspired by [ES99, ES01, Lew04], we prove that the min-max principle yields solutions of , for large enough (Theorem 1). But these are only solutions of the equation if the constraint is not saturated, i.e. if .

To prove that this is the case, we take the nonrelativistic () limit of the critical points found in the first step. By arguments similar to the ones in [ES01], we prove that these critical points converge, up to a subsequence, to a minimizer of the multiconfiguration Hartree-Fock functional (Theorem 2). Therefore, for large, the constraint is not saturated, and we obtain solutions of the MCDF equations (Corollary 1).

In the rest of this paper, we will always assume that , so that on the nonrelativistic minimizers. is a fixed constant, taken to be less than . We also assume .

First, for all such that (linearly independent components), we define the normalization

 g(Ψ)=(GramΨ)−1/2Ψ, (21)

which has the property that . This normalization was used in [ES01] to prove another variational principle for the relativistic “ground state”, which we shall not use here.

Define

 Σ+ =Σ∩(E+)K, ={Ψ∈(P+E)K,GramΨ=1}.

We will find solutions to our equations as a result of the following variational principle:

 Ic,γ=infa∈Sγ,Ψ+∈Σ+supΨ−∈(E−)KE(a,g(Ψ++Ψ−)). (22)

## 4. Results

Our first result is the well-posedness of our variational principle:

###### Theorem 1 (Existence of solutions in Sγ).

Let . There are constants such that, for large enough, there is a triplet solution of the variational principle (22):

 E(a∗,g(Ψ+∗+Ψ−∗)) =maxΨ−∈(E−)KE(a∗,g(Ψ+∗+Ψ−)), =mina∈Sγ,Ψ+∈Σ+maxΨ−∈(E−)KE(a,g(Ψ++Ψ−)).

Denoting , is a solution of the equation in .

The Hermitian matrix of Lagrange multipliers satisfies the estimates

 (c2−K1)Γ∗ ≤Λ∗≤(c2−K2)Γ∗. (23)

Furthermore, if , then is a solution of .

###### Remark 1.

In theory, one could give an explicit estimate of the minimal value of as a function of . However, since there is no known lower bound on as a function of , this would be of little use, and considerably complexify this paper. Free from the need of explicit constants, we use a strategy of proof that is simpler than the one used in Theorem 1.2 of [ES99] in the single-configuration case.

We now study the nonrelativistic limit of these solutions, thanks to the control (23) on the Lagrange multipliers:

###### Theorem 2 (Non-relativistic limit).

Let , , , and let be the solution of (22) obtained by Theorem  1 with . Then, up to a subsequence,

 an →a, Ψn →(Φ0)

in norm, where is a minimizer of

 IK =inf{EHF(a,Φ),a∈S,Φ∈(H1(R3,C2))K,GramΦ=1}. (24)

The min-max level satisfies the asymptotics

 Ic,γ=Nc2+IK+oc→∞(1).

Since, if , any minimizer of (24) must satisfy , we immediately obtain

###### Corollary 1.

If , , for c large enough, there are solutions of the multiconfiguration Dirac-Fock equations (16)-(17).

The remainder of this paper is dedicated to the proof of Theorems 1 and 2.

For Theorem 1, we first begin with Proposition 1, a convergence result for Palais-Smale sequences of the functional with Lagrange multipliers bounded away from the essential spectrum of . Then, at fixed, we study the variational principle

 supΨ−∈(E−)KE(a,g(Ψ++Ψ−))

in Proposition 2, under the condition that . We prove in Proposition 3 an upper bound on the asymptotic behavior of which will enable us to restrict to this domain, and finally, we prove in Proposition 4 that Palais-Smale sequences with Morse-type information for the functional

 Fa(Ψ+)=supΨ−∈(E−)KE(a,g(Ψ++Ψ−))

satisfy the hypotheses of Proposition 1, and therefore are precompact. Their limit up to extraction is a solution of our min-max problem (22).

To prove Theorem 2, we use the estimates (23) on the Lagrange multipliers to prove the compactness of the sequence , and the asymptotic behavior from Proposition 4 to show that the limit is a minimizer.

## 5. Proof of Theorem 1

Our first result is the convergence of Palais-Smale sequences with bounds on the Lagrange multipliers. The proof proceeds as in Lemma 2.1 of [ES99] for the single-configuration case.

### 5.1. Palais-Smale sequences for the energy functional

###### Proposition 1 (Palais-Smale sequences for E).

For large enough, if satisfies:

1. in with Hermitian matrices,

2. ,

3. ,

then, up to extraction, in , where is a solution of .

Proof.

##### Step 1 : convergence in H1/2loc

Let , and . Using the inequality (8),

 ⟨Ψ+,Han,ΨnΨ+⟩ ≥⟨Ψ+,Γn√c4−c2ΔΨ+⟩+⟨Ψ+,ΓnVΨ+⟩, ≥⟨Ψ+,Γn√c4−c2ΔΨ+⟩−C1∥∥Ψ+∥∥2E, ≥(γc2−C1c)∥∥Ψ+∥∥2c,

where depends on . Similarly, using (8) again,

 ⟨Ψ−,Han,ΨnΨ−⟩ ≤−(γc2−C2c)∥∥Ψ−∥∥2c,

with depending on .

Now, and are orthogonal for the scalar product, so

Denoting by the dual norm of ,

 ∥Han,ΨnΨ∥∗c ≥Re1∥Ψ∥c⟨Ψ+−Ψ−,Han,ΨnΨ⟩, =1∥Ψ∥c(⟨Ψ+,Han,ΨnΨ+⟩−⟨Ψ−,Han,ΨnΨ−⟩), ≥1∥Ψ∥c(c2γ−cmax(C1,C2))(∥∥Ψ+∥∥2c+∥∥Ψ−∥∥2c), ≥h0∥Ψ∥c, (25)

with when is large enough.

We then have

 limsupn→∞∥Ψn∥c ≤limsupn→∞1h0∥Han,ΨnΨn∥∗c, ≤limsupn→∞1h0(∥Δn∥∗c+∥ΛnΨn∥L2).

Therefore, is bounded in norm, i.e. in . Extracting a subsequence, again denoted by , we may assume that , , , and weakly in , strongly in , .

Since is self-adjoint from to and bounded away from zero, it is invertible. Define by

 Ha,ΨnΨ′n=ΛΨn.

is bounded in , and therefore precompact in , .

We partially invert

 Ψ′n=(DcΓ+VΓ)−1(ΛΨn−2Wa,ΨnΨ′n).

From Young’s inequality, is precompact in , , so is precompact in . Therefore, is precompact in . We extract again and impose in . But since

 Ha,Ψn(Ψn−Ψ′n)=(Ha,Ψn−Han,Ψn)Ψn+Δn+(Λn−Λ)Ψn→0

in , from (25), in .

##### Step 2 : convergence in H1/2

We now have convergence of to in . satisfies

 Ha,ΨΨ=ΛΨ.

We now look at the convergence in by obtaining an approximate Euler-Lagrange equation satisfied by the error . We have the Euler-Lagrange equations satisfied by and :

 (DcΓ+VΓ+2Wa,Ψn)Ψn−ΛΨn =Δ′n, (DcΓ+VΓ+2Wa,Ψ)Ψ−ΛΨ =0.

with in . Subtracting and using the fact that weakly in and strongly in , we get

 Lnεn→0 (26)

in , where

 Ln=DcΓ+2Wa,Ψn−Λ (27)

is the Hamiltonian “at infinity” seen by .

We now use a concavity argument to extract information on the positive and negative components of separately.

Define the quadratic functional on by

 Qn(δ−)=⟨εn+δ−,Ln(εn+δ−)⟩.

The second order terms are

 ⟨δ−,Lnδ−⟩ =⟨δ−,(DcΓ+2Wa,Ψn−Λ)δ−⟩, (28)

Since , we obtain that is strictly concave for large.

The concavity allows us to write

 ⟨ε+n,Lnε+n⟩ =Qn(−ε−n) ≤Qn(0)−∇Qn(0)[ε−n], =⟨εn,Lnεn⟩−2⟨ε−n,Lnεn⟩, ≤3∥εn∥E∥Lnεn∥E∗.

Hence

 limsupn→∞⟨ε+n,Lnε+n⟩≤0.

But

Since , this implies convergence to 0 of in and then in . But, by (26), this implies that in and therefore that . By (28), we deduce in , which proves that strongly in .

### 5.2. The reduced functional

For , define the functional

 Fa,Ψ+(Ψ−)=E(a,g(Ψ++Ψ−))

on . Our goal in this section is to prove

###### Proposition 2 (Maximizers of F).

There is a constant such that, for large enough, for all with , the functional has a unique maximizer in . The map is smooth, and satisfies

 ∥∥h(a,Ψ+)∥∥c ≤M−c. (29)

We first begin with estimates on , for which we use the property .

###### Lemma 1 (A priori bounds on Ψ+).

There are such that, for large enough, if verifies , then

 ∥∥Ψ+∥∥E ≤M+, (30) Dc|Span({ψ+i}) ≤c2+MD. (31)
###### Proof.

From Kato’s inequality (8),

 E(a,Ψ+) (32)

Here and in the rest of this paper, denotes various positive constants independent of . Since and ,

 ⟨Ψ+,(√c4−c2Δ−c2−Cγ√−Δ)ΓΨ+⟩≤0.

In the Fourier domain, we can write for all by the Cauchy-Schwarz inequality

 √c4+c2|ξ|2 ≥c2(1−αc2)+c|ξ|√1−(1−αc2)2, =c2−α+|ξ|√2α−α2c2.

Therefore, we obtain

 ⟨Ψ+,(−α+(√α−α2c2−Cγ)√−Δ)ΓΨ+⟩ ≤0,

so

 ⟨Ψ+,√−ΔΓΨ+⟩ ≤Nα√α−α2c2−Cγ.

Taking and large, is bounded independently of . Since , so is , and (30) is proved.

Now, using (32) again along with our new estimate (30), we have

 ⟨Ψ+,DcΓΨ+⟩ ≤Nc2+CM2+, ⟨Ψ+,(Dc−c2)ΓΨ+⟩ ≤CM2+.

Because annd , taken in the sense of operators on , commute, we have

 ⟨Ψ+,(Dc−c2)Ψ+⟩ ≤CM2+γ trA ≤CM2+γ

where is the Hermitian matrix

 Aij=⟨ψ+i,(Dc−c2)ψ+j⟩.

is positive semi-definite and its trace is bounded by , so , hence the result. ∎

We now restrict our search for a maximizer to a neighborhood of zero.

###### Lemma 2 (A priori bounds on Ψ−).

There is a constant such that, for large enough, for all with ,

 supΨ−∈(E−)KFa,Ψ+(Ψ−)

cannot be achieved outside a ball centered on zero of size in the norm.

###### Proof.

Let , , . Using (31),

 ⟨Ψ,DcΓΨ⟩ ≤N(c2+MD)+⟨G−1/2Ψ−,(Dc−c2−MD)ΓG−1/2Ψ−⟩,

On the other hand,

 ⟨Ψ,(VΓ+2Wa,Ψ)Ψ⟩ ≤C∥∥G−1/2Ψ+∥∥2E+C∥∥G−1/2Ψ−∥∥2E, ≤CM++Cc∥∥G−1/2Ψ−∥∥2c.

All together,

 Fa,Ψ+(Ψ−) ≤Nc2+NMD+CM+−(2γc2−Cc)∥∥G−1/2Ψ−∥∥2c.

But we also have

 Fa,Ψ+(0) =E(a,Ψ+) ≥⟨Ψ+,(DcΓ+VΓ)Ψ+⟩, ≥Nc2−C∥∥Ψ+∥∥2E, ≥Nc2−CM2+.

Therefore,

 Fa,Ψ+(Ψ−)≤Fa,Ψ+(0)+NMD+2CM2+−(2γc2−Cc)∥∥G−1/2Ψ−∥∥2c.

So, in order to have , must satisfy
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