Calculating state-to-state transition probabilities within TDDFT

The determination of the elements of the S-matrix within the framework of time-dependent density-functional theory (TDDFT) has remained a widely open question. We explore two different methods to calculate state-to-state transition probabilities. The first method closely follows the extraction of the S-matrix from the time-dependent Hartree-Fock approximation. This method suffers from cross-channel correlations resulting in oscillating transition probabilities in the asymptotic channels. An alternative method is proposed which corresponds to an implicit functional in the time-dependent density. It gives rise to stable and accurate transition probabilities. An exactly solvable two-electron system serves as benchmark for a quantitative test.