  In this work we consider a numerically solvable model of a two-electron
diatomic molecule to study a recently proposed approximation based on the
density-functional theory of so-called strictly correlated electrons (SCE). We
map out the full two-particle wave function for a wide range of bond distances
and interaction strengths and obtain analytic results for the two-particle
states and eigenenergies in various limits of strong and weak interactions, and
in the limit of large bond distance. We then study the so-called
Hartree-exchange-correlation (Hxc) kernel of time-dependent density functional
theory which is a key ingredient in calculating excitation energies. We study
an approximation based on adiabatic SCE (ASCE) theory which was shown to
display a particular feature of the exact Hxc-kernel, namely a spatial
divergence as function of the bond distance. This makes the ASCE kernel a
candidate for correcting a notorious failure of the commonly used adiabatic
local density approximation (ALDA) in the calculation of excitation energies of
dissociating molecules. Unlike the ALDA, we obtain non-zero excitation energies
from the ASCE kernel in the dissociation regime but they do not correspond to
those of the true spectrum unless the interaction strength is taken to be very
large such that the SCE theory has the right regime of validity, in which case
the excitation energies become exact and represent the so-called zero point
oscillations of the strictly correlated electrons. The commonly studied
physical dissociation regime, namely large molecular separation at intermediate
interaction strength, therefore remains a challenge for density functional
approximations based on SCE theory.
