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

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 Schr\"{o}dinger ground state density). Implications for interpreting minus the HOMO eigenvalue as an approximation to the ionization potential are indicated.