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 (
xc) 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
xc 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
xc 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.
The recent use of a new ensemble in density functional theory (DFT) to produce direct corrections to the Kohn-Sham transitions yields the elusive double excitations that are missed by time-dependent DFT (TDDFT) with the standard adiabatic approximation. But accuracies are lower than for single excitations, and formal arguments about TDDFT suggest that a correction kernel is needed. In principle, ensemble DFT with direct corrections at the exchange level must yield accurate doubles in the weakly correlated limit. We illustrate with exact calculations and analytic results on the Hubbard dimer. We also explain the error in formal arguments in TDDFT.
Ensemble density functional theory (EDFT) offers a promising path for computing excitation energies via a theory of the ensemble density --- a weighted sum of interacting excited-state densities. Recovering excitation energies is theoretically simple, but practical application requires approximations dependent on weight choice. Here, the authors derive exact conditions for EDFT. Using an exactly solvable model, they illustrate violations of exact conditions, weight-dependent derivative discontinuities in the strong-interaction limit, and ultimately the significance of weight dependence for future functional design.