  The asymmetric Hubbard dimer is used to study the density-dependence of the
exact frequency-dependent kernel of linear-response time-dependent density
functional theory. The exact form of the kernel is given, and the limitations
of the adiabatic approximation utilizing the exact ground-state functional are
shown. The oscillator strength sum rule is proven for lattice Hamiltonians, and
relative oscillator strengths are defined appropriately. The method of Casida
for extracting oscillator strengths from a frequency-dependent kernel is
demonstrated to yield the exact result with this kernel. An unambiguous way of
labelling the nature of excitations is given. The fluctuation-dissipation
theorem is proven for the ground-state exchange-correlation energy. The
distinction between weak and strong correlation is shown to depend on the ratio
of interaction to asymmetry. A simple interpolation between carefully defined
weak-correlation and strong-correlation regimes yields a density-functional
approximation for the kernel that gives accurate transition frequencies for
both the single and double excitations, including charge-transfer excitations.
Many exact results, limits, and expansions about those limits are given in the
appendices.
