An analysis through order ℏ2 of a surface hopping expansion of the nonadiabatic wave function

Abstract

It is shown that a surface hopping expansion of the semiclassical wave function formally satisfies the time independent Schrödinger equation for many-state, multidimensional problems. This wave function includes terms involving hops between different adiabatic quantum states as well as momentum changes without change of state at each point along classical trajectories. The single-state momentum changes correct for the order ℏ2 errors due to the semiclassical approximation that are present even in single surface problems. A prescription is provided for the direction of this momentum change and the amplitude associated with it. The direction of the momentum change for energy conserving hops between adiabatic states is required to be in the direction of the nonadiabatic coupling vector. The magnitude of the posthop momentum in this direction is determined by the energy, but the sign is not. Hops with both signs of this momentum component are required in order for the wave function to formally satisfy the Schrödinger equation. Numerical results are presented which illustrate how the surface hopping expansion can be implemented and the accuracy that can be obtained.