Exact Excited-State Functionals of the Asymmetric Hubbard Dimer

The exact functionals associated with the (singlet) ground and the two singlet excited states of the asymmetric Hubbard dimer at half-filling are calculated using both Levy’s constrained search and Lieb’s convex formulation. While the ground-state functional is, as commonly known, a convex function with respect to the density (or, more precisely, the site occupation), the functional associated with the (highest) doubly-excited state is found to be concave. Also, because the density of the first-excited state is non-invertible, its functional'' is a partial, multi-valued function composed of one concave and one convex branch that correspond to two separate sets of values of the external potential. Remarkably, it is found that, although the one-to-one mapping between density and external potential may not apply (as in the case of the first excited state), each state-specific energy and corresponding universal functional are functions’’ whose derivatives are each other’s inverse, just as in the ground state formalism. These findings offer insight into the challenges of developing state-specific excited-state density functionals for general applications in electronic structure theory.