Exploring weight-dependent density-functional approximations for ensembles in the Hubbard dimer

TL;DR

This paper develops analytical density-functional approximations for ensemble DFT applied to the Hubbard dimer, analyzing their errors and demonstrating accurate excitation energies through error cancellation, especially with the exact-exchange functional.

Contribution

It introduces new analytical DFAs for ensemble DFT in the Hubbard dimer using Legendre-Fenchel transforms, advancing the modeling of weight dependence in excited states.

Findings

Errors in exchange-only functionals can be large but cancel out in excitation energies.

Linear interpolation of energies yields accurate excitation energies despite errors.

Analytical DFAs are effective across different correlation regimes.

Abstract

Gross-Oliveira-Kohn density-functional theory (GOK-DFT) is an extension of DFT to excited states where the basic variable is the ensemble density, i.e. the weighted sum of ground- and excited-state densities. The ensemble energy (i.e. the weighted sum of ground- and excited-state energies) can be obtained variationally as a functional of the ensemble density. Like in DFT, the key ingredient to model in GOK-DFT is the exchange-correlation functional. Developing density-functional approximations (DFAs) for ensembles is a complicated task as both density and weight dependencies should in principle be reproduced. In a recent paper [Phys. Rev. B 95, 035120 (2017)], the authors applied exact GOK-DFT to the simple but nontrivial Hubbard dimer in order to investigate (numerically) the importance of weight dependence in the calculation of excitation energies. In this work, we derive analytical DFAs for various density and correlation regimes by means of a Legendre-Fenchel transform formalism. Both functional and density driven errors are evaluated for each DFA. Interestingly, when the ensemble exact-exchange-only functional is used, these errors can be large, in particular if the dimer is symmetric, but they cancel each other so that the excitation energies obtained by linear interpolation are always accurate, even in the strongly correlated regime.