  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.
