A semiclassical surface hopping propagator for nonadiabatic problems

A semiclassical propagator is developed for general multisurface, multidimensional nonadiabatic problems. It is demonstrated that this propagator satisfies the time‐dependent Schrodinger equation through order ℏ. This is the same order satisfied by the usual semiclassical propagator in single surface problems. The zeroth‐order term (in the nonadiabatic coupling) for the propagator is just the well‐known single surface adiabatic propagator. The first‐order terms involve single hops from the initial adiabatic state to other states. Energy is conserved in these hops and the momentum change accompanying each hop occurs in the direction parallel to the nonadiabatic coupling for the transition. Both transmitted and reflected contributions are included after a hop. The propagator expression has the zeroth‐order (single surface) semiclassical form before and after the hop. The complete propagator includes terms with any number of hops and all possible hopping points. These multihop terms are defined analogously to the first‐order (single hop) terms. An alternative formulation of the semiclassical propagator, which includes contributions from a broader range of hopping trajectories, is also developed.