Existence and nonexistence of HOMO-LUMO excitations in Kohn-Sham density functional theory

Abstract

In numerical computations of response properties of electronic systems, the standard model is Kohn-Sham density functional theory (KS-DFT). Here we investigate the mathematical status of the simplest class of excitations in KS-DFT, HOMO-LUMO excitations. We show using concentration-compactness arguments that such excitations, i.e. excited states of the Kohn-Sham Hamiltonian, exist for $Z>N$, where $Z$ is the total nuclear charge and $N$ is the number of electrons. The result applies under realistic assumptions on the exchange-correlation functional, which we verify explicitly for the widely used PZ81 and PW92 functionals. By contrast, and somewhat surprisingly, we find using a method of Glaser, Martin, Grosse, and Thirring \cite{glaser1976} that in case of the hydrogen and helium atoms, excited states do not exist in the neutral case $Z=N$ when the self-consistent KS ground state density is replaced by a realistic but easier to analyze approximation (in case of hydrogen, the true Schrödinger ground state density). Implications for interpreting minus the HOMO eigenvalue as an approximation to the ionization potential are indicated.