Excited-State-Specific Kohn-Sham Formalism for the Asymmetric Hubbard Dimer

TL;DR

This paper extends the Kohn-Sham density-functional theory framework to excited states of the asymmetric Hubbard dimer, revealing challenges in representing the first excited state and proposing complex potentials to address them.

Contribution

It introduces a novel approach using analytic continuation and complex potentials to model the first excited state within KS-DFT for the Hubbard dimer.

Findings

Density of the first excited state is not non-interacting v-representable.

Complex-valued external potentials can generate the excited state density.

Approximate functionals may lead to spurious solutions.

Abstract

Building on our recent study [https://doi.org/10.1021/acs.jpclett.3c02052, 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. Specifically, 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.