Memory effects in TDDFT quadratic response theory

TDDFT is routinely used for calculations of excitations and response, achieving an unprecedented balance between accuracy and efficiency, while caution is applied for excitations problematic for the usual adiabatic approximations. In some applications, quantities from quadratic response theory are needed; for example, to obtain coupling matrix elements between excited states in non-adiabatic dynamics simulation. We explore the connection between spurious poles in these matrix elements and the lack of double-excitations in the adiabatic approximation to the kernel. We test the ability of a frequency-dependent kernel that folds in these excitations to eliminate the divergence and produce reliable couplings for quadratic response.