Excited-state-specific Kohn-Sham formalism for the asymmetric Hubbard dimer

Building on our recent study [Giarrusso and Loos, J. Phys. Chem. Lett. 14, 8780 (2023)], we explore the generalization of the ground-state Kohn-Sham (KS) formalism of density-functional theory (DFT) to the (singlet) excited states of the asymmetric Hubbard dimer at half-filling. While we found that the KS-DFT framework can be straightforwardly generalized to the highest-lying doubly excited state, the treatment of the first excited state presents significant challenges. In particular, using a density-fixed adiabatic connection, we show that the density of the first excited state lacks non-interacting v-representability. However, by employing an analytic continuation of the adiabatic path, we demonstrate that the density of the first excited state can be generated by a complex-valued external potential in the non-interacting case. More practically, by performing state-specific KS calculations with exact and approximate correlation functionals-each state possessing a distinct correlation functional-we observe that spurious stationary solutions of the KS equations may arise due to the approximate nature of the functional.