Linear response time-dependent density functional theory of the Hubbard dimer
D. J. Carrascal J. Ferrer N. Maitra K. Burke





 

  

       
 


 

 

 





























 

 

 

 

 

 

 

 

 

 

 







 



 

 







#1

 



 

#1





 

Contents











#1

PART: 
#1

#1

§ #1


#1

 §.§ #1


#1

 §.§ #1


#1

  §.§.§ #1


#10.5cm#1:



#1

PART: 
*#1

#1

§ #1


#1

 §.§ #1


#1

  §.§.§ #1











    n_tł^λ^λ=1^λ=0#1 #1⟨ r_s ⟩1/|_1 - _2|T̂V̂Ĥvar(_1...)1/21/4 p r R uxy a q j X Fχ f A B_ X_̧ ̧Ç_ S_ XC_ HX_ HXC_ HXC,dyn_ X,j_ XC,j_ N_̋ ̋H̋_ ext^ pot^ hyb^ HF^1/2& 1/2^ loc^ LSD^ LDA^ GEA^ GGA^ SPL^ SCE^ PBE^ DFA^ TF^ VW^ unamb^ unamb^ ion^ HOMO^ LUMO^ gs^ dyn^ adia^ I^ pot^ sph. av.^ sys. av.^ sym#1⟨ #1 ⟩^ unif^ LSD_ ee^ vir^ ALDA^ PGG^ GK^ atmiz^ trans^ unpol^ pol^ sph. av.^ sys. av._↑_↓↑↓^1 / _ 2time-dependent Kohn-Sham density functional theory ∫_t_0^t_1 dt ∫ d^3r ∫_t_0^t_1 dt' ∫ d^3r' ∫d^4x_int∫ d^3 r∫_0^∞ dr 4π r^2 ∫ d^3r ∫ d^3r' ∫ d^3r ∫ d^3r' Phys. Rev. A Phys. Rev. B Phys. Rev. Letts. J. Chem. Phys. J. Phys. Chem. A Int. J. Quant. Chem. #1table <ref>#1Eqs. (<ref>)#1Eq. (<ref>)#1Section <ref>#1Fig. <ref>#1p. #1#1Ref. <cit.>
        øω#1#1#1#1#̨1 #1ιaκcemptyLinear response time-dependent density functional theory
        of the Hubbard dimerDiego J. Carrascal^1,2, Jaime Ferrer^1,2, Neepa Maitra^3 and Kieron Burke^4^1 Department of Physics, Universidad de Oviedo, 33007 Oviedo, Spain^2Nanomaterials and Nanotechnology Research Center, CSIC / Universidad de Oviedo, Oviedo,  Spain^3 Department of Physics, Hunter College, City University of New York, New York, NY 1006, USA^4 Department of Chemistry and of Physics, University of California, Irvine, CA 92697,  USAdj.carrascal@gmail.com, ferrer@uniovi.es, nmaitra@hunter.cuny.edu, kieron@uci.edu
        The asymmetric Hubbard dimer is
        used to study the density-dependence of the exact frequency-dependent
        kernel of linear-response time-dependent density functional theory.  
        The exact form of the kernel is given, and the limitations of the
        adiabatic approximation utilizing the exact ground-state functional are shown.
        The oscillator strength sum rule is proven for lattice Hamiltonians, and relative
        oscillator strengths are defined appropriately.
        The method of Casida for extracting oscillator strengths from a frequency-dependent
        kernel is demonstrated to yield the exact result with this kernel. 
        An unambiguous way of labelling the nature of excitations is given.
        The fluctuation-dissipation theorem is proven for the ground-state exchange-correlation energy.
        The distinction between weak and strong correlation is shown to depend on 
        the ratio of interaction to asymmetry.
        A simple interpolation between carefully defined
        weak-correlation and
        strong-correlation regimes yields a density-functional approximation for the kernel that gives accurate transition frequencies for
        both the single and double excitations, including charge-transfer excitations.
        Many exact results, limits, and expansions about those limits are given in the appendices.
        #1
            #1
            December 30, 2023
        =====================
        
        T̂V̂ϵΔΔαtΔ v^ MH2^ MH2, Reg^ MH46^ MH6^ WC24^ WC4Introduction
        
        Time-dependent density functional theory (TDDFT) is a popular first-principles approach
        to calculating low-lying optical excitations of molecules <cit.>.  A typical
        calculation first involves optimizing the structure within ground-state DFT
        using some approximate exchange-correlation functional.  Then a linear-response
        TDDFT calculation, usually solving RPA-type equations in frequency
        space <cit.>, or via real-time propagation <cit.>, yields both transition frequencies and
        oscillator strengths.  The TDDFT step almost always makes the adiabatic
        approximation for the unknown and (generally) frequency-dependent
        exchange-correlation (XC) kernel, in which its zero-frequency limit is used <cit.>. This is simply the second functional-derivative of the 
        exchange-correlation energy of ground-state DFT.
        Usually, the same approximate XC functional is used for the first ground-state step and for the TDDFT step. 
        Several thousand papers per year use this method to extract useful information
        on electronic excitations, with typical transition frequency errors of order
        0.25 to 0.5 eV <cit.>. 
        
        However, in the three decades since the Runge-Gross theorem established the
        formal exactitude of this approach<cit.>, a variety of
        situations have been identified where approximations fail, often qualitatively.
        Among the most notorious are failures for charge-transfer excitations, whose
        transition frequencies are typically grossly underestimated by the standard functionals <cit.>, 
        but reasonable results can be obtained by using range-separated hybrids <cit.>.    Another one
        is the complete absence of double-excitations from 
        the spectrum within the adiabatic approximation <cit.>.  Initial hopes of extracting
        double-excitations from higher-order response theory were dashed by Ref. <cit.>.  A simple
        model of the frequency dependence for the specific case of a double excitation
        close to one or a few single excitations  in a weakly correlated system <cit.>, is a useful tool
        for a post-adiabatic TDDFT treatment called dressed TDDFT, and has been applied to a range of systems <cit.>
        but has not been widely adopted.
        
        While practical electronic structure calculations begin from the real-space
        Hamiltonian, much useful insight and even semi-quantitative results can be
        extracted from model Hamiltonians, especially when correlations are 
        strong<cit.>.
        The paradigmatic  case in condensed matter is the (one-band) Hubbard model, which
        is usually taken on an infinite lattice, and can be analyzed in 1-,2-, or 3 dimensions.
        The model is characterized by only two parameters, a hopping energy between
        nearest neighbours t and an on-site Coulomb repulsion for doubly-occupied sites U, and site-occupation 
        plays the role of the density.
        Model Hamiltonians are not aimed at high levels of quantitative accuracy, but 
        are designed to explore qualitative features of correlation physics.
        For example, the 2D Hubbard model may
        display the essential features of high-temperature superconductivity <cit.>.
        
        Thus, Hubbard (and more complex) chains have been used to study, e.g., correlation
        effects in transport through single molecules and small quantum dots. 
        They have also been used to explore full time propagation in TDDFT, going beyond the
        linear response regime <cit.>.   It is usually 
        relatively straightforward
        to exactly solve the time-dependent Schrödinger equation in these cases.
        It can also be easy 
        to find the exact ground-state density functional  <cit.>, and to propagate the fully
        time-dependent Kohn-Sham equations within the adiabatically exact approximation,
        in order to study its capabilities and limitations.
        
        Interestingly, among all the papers using TDDFT in lattice models,
        relatively few have studied frequency-domain linear-response
        TDDFT (lrTDDFT) in interacting systems <cit.>.  In the case of the two-site Hubbard dimer, 
        Aryasetiawan and Gunnarson <cit.>
        did ground-breaking work in studying the performance of lrTDDFT for the symmetric dimer.
        However, as emphasized in a recent review focussed solely on ground-state
        DFT for the dimer <cit.>, many crucial DFT features can only be
        seen when the dimer is made
        asymmetric via a difference in the on-site potentials Δ v <cit.>.
        In fact one cannot really speak of density-functionals if restricting to symmetric cases, since there
        is no dependence
        on the ground-state density, as the site occupations
        always remain identical.
        In addition, the Kubo response of the asymmetric dimer shows two excitations, while only one
        survives in the symmetric case.
        Again, a few  recent works have noted this effect <cit.>.
        
        In the present article, we thoroughly explore the asymmetric
        dimer within lrTDDFT, finding the exact non-adiabatic 
        density-functional for the exchange-correlation kernel.  
        In previous works <cit.> the exact frequency-dependent kernel has been found for a given system: in Ref. <cit.> an analytic expression is derived for a homogeneous two-electron density on a ring while in Ref. <cit.> a general numerical procedure is given for computing the kernel of a given system. This is, we believe, the first time that the exact frequency-dependent kernel as a functional of the ground-state density has been found for any model; the Hubbard model is simple enough to allow for  a complete analytic study. 
        We
        find that correlations are suppressed by asymmetry,
        so that a weak correlation approximation remains accurate  even when
        the ratio between the Coulomb repulsion  U and the hopping integral t
        is very large, as long as the asymmetry between the on-site energies Δ v is also large.  In fact, for sufficiently large
        /U, this weakly correlated kernel remains accurate, no matter how large U/t is.
        Only when U is large relative to both 2t and  does the weak-correlation kernel
        fail.  Moreover, a simple expansion about the strongly-correlated limit, which we call the
        Mott-Hubbard (MH) regime, suffices for all other cases, so that an appropriate interpolation
        between the two yields accurate results for almost all parameter values.  Thus we
        have found an accurate approximate kernel for both double and charge-transfer excitations,
        that works in both weak and strong correlation regimes.
        We note that this provides a useful explicit example of the frequency-dependence of the
        kernel as a functional of the ground-state density for this model, but does not produce 
        a general purpose density functional for this frequency dependence.
        
        
        
        To illustrate these results we plot  in Fig. <ref> the transition frequencies
        for both singlet excitations when the dimer is strongly-interacting (U=10 t)
        as a function of asymmetry, .   In the symmetric limit (=0),
        the two excited states are barely separated.  Because correlation is strong,
        the KS transitions are a poor approximation to the exact ones,
        and even the adiabatically-exact
        correction to TDDFT does not really improve matters.  It vastly
        overestimates the correction to the single excitation and, being adiabatic,
        yields no prediction for the double excitation at all.  The interpolation kernel developed
        here, which interpolates between the weakly and strongly correlated limits, 
        is almost perfect for these transition frequencies.
        Note how, if the asymmetry is comparable to U or larger, then the weak-coupling
        approximation works well.  We explain this feature in this work.
        
        While this article may appear long, its main results can be easily summarized.
        In Section <ref>, we give a very detailed account of how lrTDDFT behaves exactly
        for the Hubbard dimer.  This is a beautifully simple case, with a very limited
        Hilbert space, in which the (usually unknown) XC kernel of TDDFT can be
        written exactly and explicitly (at least as function of the potential), including the 
        frequency-dependence needed to
        generate the double excitation.  This can be thought of as a many-body person's
        guide to TDDFT.  On the other hand, in Section <ref>, we explore meatier issues of
        approximations.  We begin with weakly correlated systems (Sections <ref> and <ref>) and
        show how the usual approximations work in the usual way for such systems, drawing
        the analogy with dressed TDDFT, which is a specific approximation to the frequency-dependent
        kernel that captures double excitations in this regime.  But we also explore the
        strongly-correlated (Mott-Hubbard) limit (Sections <ref> and <ref>), and show how to distinguish weak
        and strong correlation in this case.  We perform the necessary expansions in the
        two limits (Appendix <ref>), and construct an interpolation scheme for the kernel that gives
        highly accurate results in both regimes, and reasonably accurate results in
        the interpolative regime (Section <ref>).
        
        For those with an interest and background in TDDFT, some key results to take away
        include a general discussion of state labelling (how do you classify something
        as a double excitation?: see Sections <ref> and <ref>), defining (relative) oscillator strengths 
        in lattice models (see the same section and Section <ref>), confirmation that the oscillator strength of
        a double can be extracted from Casida's matrix formulation (Section <ref>), and illustration
        that a pole in the kernel produces a double excitation, as in dressed TDDFT.
        For those with a background in many-body theory, some other key results are
        the separation of Mott-Hubbard and weakly correlated regimes (Section <ref> and Fig. <ref>),
        and generalizations to site-dependent U (Appendix <ref>) and fractional particle numbers (Appendix <ref>).
        The exact formulas (Appendix <ref>) and expansions and limits (Appendix <ref>) should prove
        very useful to anyone using Hubbard Hamiltonians with any background.
