Communication: Proper treatment of classically forbidden electronic transitions significantly improves detailed balance in surface hopping

Surface hopping is the most popular method for nonadiabatic molecular dynamics. Many have reported that it does not rigorously attain detailed balance at thermal equilibrium, but does so approximately. We show that convergence to the Boltzmann populations is significantly improved when the nuclear velocity is reversed after a classically forbidden hop. The proposed prescription significantly reduces the total number of classically forbidden hops encountered along a trajectory, suggesting that some randomization in nuclear velocity is needed when classically forbidden hops constitute a large fraction of attempted hops. Our results are verified computationally using two- and three-level quantum subsystems, coupled to a classical bath undergoing Langevin dynamics.