Comparison of semiclassical methods for time dependent density evolution for a two state model problem Original Research Article

A potential advantage of evaluating the time dependent density, as opposed to the wave function, for a system undergoing a transition is that consideration of the density allows for the inclusion of the phase cancellation associated with the loss of coherence during the evolution of the system. This loss of coherence can result in the transition being considerably more localized in time in the density description than would be apparent in a wave function description. The semiclassical treatment of a simple two state system with constant adiabatic surfaces and a Gaussian nonadiabatic interaction is considered. Explicit expressions are obtained for the transition probability in the weak coupling limit. It is found that a density propagation procedure, which employs a forward–backward propagator, does not provide the correct long time limit for the transition probability. Numerical results show that the use of this approach can result in moderate errors in the long time transition probability, as compared with exact calculations. It is demonstrated that a semiclassical procedure for the density evolution, which is based upon a stationary state expansion of the propagator, provides very good results for the time dependent transition probability for this model problem.