Decoherence and surface hopping: When can averaging over initial conditions help capture the effects of wave packet separation?

Fewest-switches surface hopping (FSSH) is a popular nonadiabatic dynamics method which treats nuclei with classical mechanics and electrons with quantum mechanics. In order to simulate the motion of a wave packet as accurately as possible, standard FSSH requires a stochastic sampling of the trajectories over a distribution of initial conditions corresponding, e.g., to the Wigner distribution of the initial quantum wave packet. Although it is well-known that FSSH does not properly account for decoherence effects, there is some confusion in the literature about whether or not this averaging over a distribution of initial conditions can approximate some of the effects of decoherence. In this paper, we not only show that averaging over initial conditions does not generally account for decoherence, but also why it fails to do so. We also show how an apparent improvement in accuracy can be obtained for a fortuitous choice of model problems, even though this improvement is not possible, in general. For a basic set of one-dimensional and two-dimensional examples, we find significantly improved results using our recently introduced augmented FSSH algorithm.