Surface hopping with cumulative probabilities: Even sampling and improved reproducibility

Trajectory surface hopping simulations of photochemical reactions are a powerful and increasingly important tool to unravel complex photochemical reactivity. Within surface hopping, electronic transitions are mimicked by stochastic hops between electronic potential surfaces. Thus, statistical sampling is an inescapable component of trajectory-surface-hopping-based nonadiabatic molecular dynamics methods. However, the standard sampling strategy inhibits computational reproducibility, limits predictability, and results in trajectories that are overly sensitive to numerical parameters like the time step. We describe an equivalent approach to sampling electronic transitions within fewest switches surface hopping (FSSH) in which hops are decided in terms of the cumulative probability (FSSH-c) as opposed to the usual prescription, which is in terms of the instantaneous conditional probability (FSSH-i). FSSH-c is statistically equivalent to FSSH-i and can be implemented from trivial modifications to an existing surface hopping algorithm but has several key advantages: (i) a single trajectory is fully specified by just a handful of random numbers, (ii) all hopping decisions are independent of the time step such that the convergence behavior of individual trajectories can be explored, and (iii) alternative integral-based sampling schemes are enabled. In addition, we show that the conventional hopping probability overestimates the hopping rate and propose a simple scaling correction as a fix. Finally, we demonstrate these advantages numerically on model scattering problems.