  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.
