Comparison of the propagation of semiclassical frozen Gaussian wave functions with quantum propagation for a highly excited anharmonic oscillator

The propagation of an initially highly excited localized wave packet in an anharmonic oscillator potential is studied within the frozen Gaussian approximation. Comparison is made to quantum mechanical basis set calculations. The frozen Gaussian approximation involves the expansion of the initial wave function in terms of an overcomplete Gaussian basis set. The wave function evolution is evaluated by allowing each Gaussian to travel along a classical trajectory with its shape held rigid. A Monte Carlo algorithm is employed in the selection of the initial Gaussian basis functions. The frozen Gaussian results are very good for times on the order of a few vibrational periods of the oscillator and remain qualitatively correct for the entire length of the calculations which is 12 vibrational periods. The dependence of the calculations on the width of the Gaussian basis functions is investigated and the effect of a simplifying approximation for the prefactor of the Gaussians is tested.