Time averaging the semiclassical initial value representation for the calculation of vibrational energy levels

An application of the initial value representation (IVR) of semiclassical (SC) theory to approximate the quantum mechanical time evolution operator, exp[−iĤt/ℏ], requires an integral over the phase space of initial conditions of classical trajectories. The integrand of this integral is complex, i.e., has a phase, from which quantum coherence (in fact, all quantum) effects arise, but which also makes SC-IVR calculations more difficult than ordinary classical molecular dynamics simulations (the semiclassical version of the “sign problem”). A number of approaches have been devised to ameliorate the sign problem, and here we show how a time averaging procedure—the integrand of the phase space integral is time-averaged over the classical trajectory originating from each initial condition—can be profitably used in this regard, particularly so for the calculation of spectral densities (from which vibrational energy levels can be identified). This time averaging procedure is shown to greatly reduce the number of initial conditions (i.e., the number of classical trajectories) that are needed to converge IVR phase space averages. In some cases useful results can be obtained with only one classical trajectory. Calculations are carried out for vibrational energy levels of H2 and H2O to illustrate the overall procedure.