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Ensemble density functional theory (eDFT) is an exact time-independentalternative to time-dependent DFT (TD-DFT) for the calculation of excitationenergies. Despite its formal simplicity and advantages in contrast to TD-DFT(multiple excitations, for example, can be easily taken into account in anensemble), eDFT is not standard which is essentially due to the lack ofreliable approximate exchange-correlation (xc) functionals for ensembles.Following Smith et al. [Phys. Rev. B 93, 245131 (2016)], we propose in thiswork to construct an exact eDFT for the nontrivial asymmetric Hubbard dimer,thus providing more insight into the weight dependence of the ensemble xcenergy in various correlation regimes. For that purpose, an exact analyticalexpression for the weight-dependent ensemble exchange energy has been derived.The complementary exact ensemble correlation energy has been computed by meansof Legendre--Fenchel transforms. Interesting features like discontinuities inthe ensemble xc potential in the strongly correlated limit have beenrationalized by means of a generalized adiabatic connection formalism. Finally,functional-driven errors induced by ground-state density-functionalapproximations have been studied. In the strictly symmetric case or in theweakly correlated regime, combining ensemble exact exchange with ground-statecorrelation functionals gives relatively accurate ensemble energies. However,when approaching the equiensemble in the strongly correlated regime, thisapproximation leads to highly curved ensemble energies with negative slopewhich is unphysical. Using both ground-state exchange and correlationfunctionals gives much better results in that case. In fact, exact ensembleenergies are almost recovered in some density domains. The analysis ofdensity-driven errors is left for future work.

Exact ensemble density functional theory for excited states in a model system: investigating the weight dependence of the correlation energy