Calculating state-to-state transition probabilities within time-dependent density-functional theory

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 of the time-dependent density. Evaluated with the exact time-dependent density it gives rise to stable and accurate transition probabilities. However, the functional shows an extreme sensitivity with respect to errors in the time-dependent density when evaluated using an approximate density from an actual TDDFT calculation. Two exactly solvable two-electron systems serve as a benchmark for a quantitative test.