Numerical studies concerning the efficiency of various techniques in time-independent surface hopping calculations

A two-state model problem with avoided crossing points is studied to test the efficiency of semiclassical surface hopping procedures. A modified sampling method is introduced which evaluates the contributions from every branch of the surface hopping trajectories in regions where surface hopping occurs mostly and employs a Monte Carlo procedure to choose whether to hop or not hop in regions of less interest. This method is shown to give smaller sampling errors than using a Monte Carlo procedure alone and provides transition probabilities that are in very good agreement with the exact quantum results. As a result of this improved branch method, the computational effort needed in the semiclassical calculations can be reduced to about one tenth of that needed for a Monte Carlo calculation with a similar size sampling error. Although tested on a one-dimensional system, this procedure shows the promise of application in multi-dimensional systems.