Analysis of a surface hopping expansion that includes hops in classically forbidden regions

A surface hopping expansion of the nonadiabatic wave function is generalized to account for hops from the forbidden region on one adiabatic energy surface to a different adiabatic surface in multidimensional problems. This analysis is motivated by previous surface hopping calculations on one dimensional models that provide very accurate transition probabilities, even at low energies, if classically forbidden hops are included in the calculations. It is shown that hops from the classically forbidden region in the previous form of the surface hopping expansion cannot, in general, lead to classically allowed final state trajectories in multidimensional problems. The surface hopping wave function is generalized to allow for two or more hops at each point along the trajectory. These hops correspond to different directions for the energy conserving momentum change, which gives different post-hop trajectories. This generalization allows for the final state trajectory to be classically allowed if the post-hop adiabatic energy surface has sufficiently low energy. 2014 Elsevier B.V. All rights reserved.