  The adiabatic approximation in time-dependent density functional theory
(TDDFT) is known to give an incorrect pole structure in the quadratic response
function, leading to unphysical divergences in excited state-to-state
transition probabilities and hyperpolarizabilties. We find the form of the
exact quadratic response kernel and derive a practical and accurate
approximation that cures the divergence. We demonstrate our results on excited
state-to-state transition probabilities of a model system and of the LiH
molecule.
