Evaluation of the time-derivative coupling for accurate electronic state transition probabilities from numerical simulations

Spikes in the time-derivative coupling (TDC) near surface crossings make the accurate integration of the time-dependent Schrödinger equation in nonadiabatic molecular dynamics simulations a challenge. To address this issue, we present an approximation to the TDC based on a norm-preserving interpolation (NPI) of the adiabatic electronic wave functions within each time step. We apply NPI and two other schemes for computing the TDC in numerical simulations of the Landau-Zener model, comparing the simulated transfer probabilities to the exact solution. Though NPI does not require the analytical calculation of nonadiabatic coupling matrix elements, it consistently yields unsigned population transfer probability errors of ∼0.001, whereas analytical calculation of the TDC yields errors of 0.0-1.0 depending on the time step, the offset of the maximum in the TDC from the beginning of the time step, and the coupling strength. The approximation of Hammes-Schiffer and Tully yields errors intermediate between NPI and the analytical scheme.