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arXiv:1808.04994 · Physical Review Letters

Density-Driven Correlations in Many-Electron Ensembles: Theory and Application for Excited States

Tim Gould, Stefano Pittalis

Published Jul 3, 2019• 56 citations
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Abstract

Density functional theory can be extended to excited states by means of a unified variational approach for passive state ensembles. This extension overcomes the restriction of the typical density functional approach to ground states, and offers useful formal and demonstrated practical benefits. The correlation energy functional in the generalized case acquires higher complexity than its ground state counterpart, however. Little is known about its internal structure nor how to effectively approximate it in general. Here we show that such a functional can be broken down into natural components, including what we call "state-" and "density-driven" correlations, with the former amenable to conventional approximations, and the latter being a unique feature of ensembles. Such a decomposition provides us with a pathway to general approximations that are able to routinely handle low-lying excited states. The importance of density-driven correlations is demonstrated, and an approximation for them is introduced and shown to be useful.

DOI
10.1103/physrevlett.123.016401

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Density functional theory can be extended to excited states by means of a unified variational approach for passive state ensembles. This extension overcomes the restriction of the typical density functional approach to ground states, and offers useful formal and demonstrated practical benefits. The correlation energy functional in the generalized case acquires higher complexity than its ground state counterpart, however. Little is known about its internal structure nor how to effectively approximate it in general. Here we show that such a functional can be broken down into natural components, including what we call "state-" and "density-driven" correlations, with the former amenable to conventional approximations, and the latter being a unique feature of ensembles. Such a decomposition provides us with a pathway to general approximations that are able to routinely handle low-lying excited states. The importance of density-driven correlations is demonstrated, and an approximation for them is introduced and shown to be useful.
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Density functional theory (DFT) has greatly expanded our ability to affordably compute and understand electronic ground states, by replacing intractable ab initio calculations by models based on paradigmatic physics from high- and low-density limits. But, a comparable treatment of excited states lags behind. Here, we solve this outstanding problem by employing a generalization of density functional theory to ensemble states (EDFT). We thus address important paradigmatic cases of all electronic systems in strongly (low-density) and weakly (high-density) correlated regimes. We show that the high-density limit connects to recent, exactly solvable EDFT results. The low-density limit reveals an unnoticed and most unexpected result-density functionals for strictly correlated ground states can be reused directly for excited states. Nontrivial dependence on excitation structure only shows up at third leading order. Overall, our results provide foundations for effective models of excited states that interpolate between exact low- and high-density limits, which we illustrate on the cases of singlet-singlet excitations in H_{2} and a ring of quantum wells.
Density functional theory can be generalized to mixtures of ground and excited states, for the purpose of determining energies of excitations using low-cost density functional approximations. Adapting approximations originally developed for ground states to work in the new setting would fast-forward progress enormously. But, previous attempts have stumbled on daunting fundamental issues. Here we show that these issues can be prevented from the outset, by using a fluctuation dissipation theorem (FDT) to dictate key functionals. We thereby show that existing exchange energy approximations are readily adapted to excited states, when combined with a rigorous exact Hartree term that is different in form from its ground state counterpart, and counterparts based on ensemble Ansatzë. Applying the FDT to correlation energies also provides insights into ground statelike and ensemble-only correlations. We thus provide a comprehensive and versatile framework for ensemble density functional approximations.
The starting point of this paper is a recent extension by Theophilou of the Hohenberg-Kohn-Sham (HKS) density-functional theory to ensembles of systems consisting of the M lowest eigenstates, equally weighted. As in the HKS theory the key quantities are the exchange-correlation energy, ExcM[n(r)]{E}_{\mathrm{xc}{}^{M}[\mathrm{n}(\mathrm{r})]}ExcM[n(r)]​, and potential, vxcM(r;[n(r′)]){v}_{\mathrm{xc}{}^{M}(\mathit{r};[\mathit{n}(\mathit{r}\mathcal{'})])}vxcM(r;[n(r′)])​. The present paper provides expressions for these quantities, valid for systems of slowly varying density. Even for such systems, however, there are essential nonlocal effects. Nevertheless both ExcM{E}_{\mathrm{xc}{}^{M}}ExcM​ and vxcM{v}_{\mathrm{xc}{}^{M}}vxcM​ can be calculated in terms of quantities characteristic of appropriate uniform thermal ensembles. This theory is the analog of the ground-state local-density approximation and allows calculation of excited-state energies and densities.
Ensemble density functional theory (eDFT) is an exact time-independent alternative to time-dependent DFT (TD-DFT) for the calculation of excitation energies. Despite its formal simplicity and advantages in contrast to TD-DFT (multiple excitations, for example, can be easily taken into account in an ensemble), eDFT is not standard, which is essentially due to the lack of reliable approximate exchange-correlation (xcxcxc) functionals for ensembles. Following Smith et al. [Phys. Rev. B 93, 245131 (2016)], we propose in this work to construct an exact eDFT for the nontrivial asymmetric Hubbard dimer, thus providing more insight into the weight dependence of the ensemble xcxcxc energy in various correlation regimes. For that purpose, an exact analytical expression for the weight-dependent ensemble exchange energy has been derived. The complementary exact ensemble correlation energy has been computed by means of Legendre-Fenchel transforms. Interesting features like discontinuities in the ensemble xcxcxc potential in the strongly correlated limit have been rationalized by means of a generalized adiabatic connection formalism. Finally, functional-driven errors induced by ground-state density-functional approximations have been studied. In the strictly symmetric case or in the weakly correlated regime, combining ensemble exact exchange with ground-state correlation functionals gives better ensemble energies than when calculated with the ground-state exchange-correlation functional. However, when approaching the asymmetric equiensemble in the strongly correlated regime, the former approximation leads to highly curved ensemble energies with negative slope which is unphysical. Using both ground-state exchange and correlation functionals gives much better results in that case. In fact, exact ensemble energies are almost recovered in some density domains. The analysis of density-driven errors is left for future work.
A framework for developing new approximate electronic structure methods is presented, in which the correlation energy of a many-electron system in the ground state is computed as in the single-reference second-order many-body perturbation theory but with the reference one-body Hamiltonian modified by additional (generally non-local) potentials which are universal functionals of the Hartree-Fock density. The existence of such functionals that reproduce the exact correlation energy justifies the search for approximate models which may overcome some important deficiencies of the traditional density-functional methods and also perform better than the usual low-order wavefunction methods.