Ground and excited energy levels can be extracted exactly from a single ensemble density-functional theory calculation

Gross-Oliveira-Kohn density-functional theory (GOK-DFT) for ensembles is the DFT analog of state-averaged wavefunction-based (SA-WF) methods. In GOK-DFT, the state-averaged (so-called ensemble) exchange-correlation (xc) energy is described by a single functional of the density which, for a fixed density, depends on the weights assigned to each state in the ensemble. We show that, if a many-weight-dependent xc functional is employed, then it becomes possible to extract, in principle exactly, all individual energy levels from a single GOK-DFT calculation, exactly like in a SA-WF calculation. More precisely, starting from the Kohn-Sham energies, a global Levy-Zahariev-type shift as well as a state-specific (ensemble-based) xc derivative correction must be applied in order to reach the energy level of interest. We illustrate with the asymmetric Hubbard dimer the importance and substantial weight dependence of both corrections. A comparison with more standard extraction procedures, which rely on a sequence of ensemble calculations, is made at the ensemble exact exchange level of approximation.