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The exact functionals associated with the (singlet) ground and the twosinglet excited states of the asymmetric Hubbard dimer at half-filling arecalculated using both Levy's constrained search and Lieb's convex formulation.While the ground-state functional is, as commonly known, a convex function withrespect to the density (or, more precisely, the site occupation), thefunctional associated with the (highest) doubly-excited state is found to beconcave. Also, because the density of the first-excited state isnon-invertible, its ``functional'' is a partial, multi-valued function composedof one concave and one convex branch that correspond to two separate sets ofvalues of the external potential. Remarkably, it is found that, although theone-to-one mapping between density and external potential may not apply (as inthe case of the first excited state), each state-specific energy andcorresponding universal functional are ``functions'' whose derivatives are eachother's inverse, just as in the ground state formalism. These findings offerinsight into the challenges of developing state-specific excited-state densityfunctionals for general applications in electronic structure theory.

Exact Excited-State Functionals of the Asymmetric Hubbard Dimer