Forward-backward Semiclassical Dynamics

Computing the time evolution of quantum systems composed of many coupled degrees of freedom presents serious challenges. Currently, simulations of timedependent properties in polyatomic molecules, clusters, or condensed phases employ a variety of approximations, which are largely built around variational, quantum-classical or semiclassical ideas. While approximate methods have proven extremely valuable in many situations, their regime of applicability is limited. Perhaps more importantly, checking the accuracy of the results is usually impractical, and thus reliability becomes a signi cant concern. Numerical solution of the Schrodinger equation requires storage of the multidimensional wavefunction, which occupies a volume that grows exponentially with the number of particles. This limitation is circumvented in the path integral formulation, where quantum mechanical amplitudes are expressed as sums over paths . To reproduce quantum interference, the amplitude along each path must be a complex-valued phase. Because of the rapid oscillations of the latter, numerical evaluation of the real-time path integral by means of stochastic methods is plagued by severe numerical instabilities . At the same time, explicit summation over all paths is far from feasible, as their number (if space and time are discretized) increases exponentially with the number of degrees of freedom and the total propagation time. Recently, exact evaluation of the path integral by virtue of an iterative algorithm became feasible in cases of a low-dimensional system coupled to a dissipative bath (for a review see Ref. ).