We recently published a new method for the calculation of the time evolution of a wave function. We used an accurate approximate method to calculate the time propagator for a finite time Dgr;t. Numerical calculations showed that this scheme works quite accurately, but that it is not more efficient than conventional methods. In this paper we propose to use a very fast and simple, but less accurate semiclassical method for the calculation of the time propagator. The approximation consists in the replacement of the Hamiltonian by a quadratic approximation around the center of the evolving wave packet called thawed Gaussian dynamics. We show by numerical examples in one and two dimensions that, despite this crude approximation, we achieve nearly the same accuracy as in the foregoing paper, but with an efficiency that is typically more than an order of magnitude better. We further show that the method is able to describe tunneling and long time dynamics (e.g., 1000 vibrational periods).
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