| It is shown that a surface hopping expansion of the primitive semiclassical propagator satisfies the multistate Schrodinger equation to all order in h-bar and the nonadiabatic coupling if the correct non-classical events (i.e., hops and momentum reversals) are included and the amplitudes for each event has the correct form. The initial value representation form of this surface hopping propagator satisfies the Schrodinger equation to first order in h-bar and all orders in the nonadiabatic coupling. Numerical calculations on model problems demonstrate that these methods yield results in excellent agreement with exact quantum calculations. It is also discussed how the numerical efficiency of these techniques can be greatly improved by using higher order transition amplitudes for each step along the trajectories and by choosing the representation of the electronic quantum states to minimize the integrated coupling near avoided crossing points. |
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