Oscillator strengths and excited-state couplings for double excitations in time-dependent density functional theory

Although useful to extract excitation energies of states of double-excitation character in time-dependent density functional theory that are missing in the adiabatic approximation, the frequency-dependent kernel derived earlier [J. Chem. Phys. {\bf 120}, 5932 (2004)] was not designed to yield oscillator strengths. These are required to fully determine linear absorption spectra and they also impact excited-to-excited-state couplings that appear in dynamics simulations and other quadratic response properties. Here we derive a modified non-adiabatic kernel that yields both accurate excitation energies and oscillator strengths for these states. We demonstrate its performance on a model two-electron system, the Be atom, and on excited-state transition dipoles in the LiH molecule at stretched bond-lengths, in all cases producing significant improvements over the traditional approximations.