Modification to the pre - factor of the semiclassical propa - gator

– We modify the pre-factor of the semiclassical propagator to improve its efficiency in practical implementations. The new pre-factor represents the smooth portion of an orbit’s contribution, and leads to fast convergence in numerical calculations. As an illustration of the accuracy and efficiency of the resultant propagator, we numerically calculate overlaps between quantum and semiclassical wave functions, as well as low-lying spectrum density in a 10dimensional system contains unstable classical orbits. This sheds light on applying semiclassical propagator to high dimensional systems. Semiclassical propagator connects quantum dynamics with classical orbits. It enables one to evaluate quantum quantities from classical orbits. This is in principle applicable to high dimensional quantum systems as an alternative method to do first principle calculations. Since Gutzwiller adds Maslov phase to the Van Vleck semiclassical propagator [1], many tests confirm that the semiclassical propagator has remarkable accuracy [2–5]. Another appealing feature of the semiclassical method is easy to implement parallel computation by requiring each node to handle some orbits. The final result is simply a summation of each orbit’s contribution. The fundamental approximation of semiclassical method is to treat the Hamiltonian locally as a quadratic function of coordinates and momenta. By expanding the Hamiltonian up to the second order around a phase space point, each orbit is associated to a time dependent quadratic quantum Hamiltonian in semiclassical calculations. In this sense, if numerical performance is not a concern, all kinds of semiclassical formulations are as accurate as the Van Vleck Gutzwiller (VVG) propagator [1]. The later comes from the stationary phase approximation to the Feynman’s path integral representation of the quantum propagator. In numerical calculations, however, the Herman and Kluk’s (HK) formulation of semiclassical propagator [6] is often a favorite one [7–15]. (∗) E-mail: [email protected]