Higher order phase corrected transition amplitudes for time dependent semiclassical surface hopping calculations

A trajectory based, surface hopping expansion of the time dependent quantum propagator has recently been shown to satisfy the multi-state Schrodinger equation to all orders in ℏ. Higher order transition amplitudes for hops between states within an interval of fixed length along the trajectory are presented. These amplitudes include contributions from terms corresponding to any number of hops in the interval. They also account for the dependence of the phase associated with the trajectory and the time taken to cross the interval on the location of the hops within the interval. The higher order amplitudes allow for the use of wider intervals in numerical surface hopping calculations. More of the interference between different hopping trajectories is analytically accounted for when the higher order amplitudes are used with wider intervals. Monte Carlo procedures must generally be employed in deciding whether to hop or not in each interval for multi-dimensional problems. Numerical calculations on a model system indicate that the use of the higher order amplitudes can significantly improve the efficiency and accuracy of these methods.