Semiclassical time evolution in the forward-backward stationary-phase limit

Several versions of the time-dependent semiclassical approximation have recently become available as practical tools for calculating dynamical quantities of large molecular systems. These methods are based on the forward-backward idea that combines the quantum evolution in opposite directions of time into a single, consecutive propagation. The forward-backward semiclassical propagator is not highly oscillatory and is thus suitable for numerical evaluation by Monte Carlo methods. The present article reviews a forward-backward phase space formulation of the Heisenberg operator that also eliminates the semiclassical prefactor, leading to a practical tool for calculating observables and time correlation functions.