Improved short-time propagator for repulsive inverse-power-law potentials

Abstract

Recently Makri and Miller have shown that the standard Trotter approximation for the short-time propagator K( x 0, x, t) can be greatly improved by writing K( x 0, x, t) in exponential form and then systematically expanding the exponent in powers of the time t. The Makri-Miller method is based on the assumption that in the semiclassical limit, there is a unique classical path which connects the initial point x 0 to the final point x in the time t. We show that for potentials such as repulsive inverse power-laws, which admit two classical paths connecting the given end points in the given time, further improvement can be made by expanding the exponents of the two terms corresponding to the two paths. Numerical tests for the inverse square potential, for which analytic results are available for comparison, show that such further improvements can be substantial. The tests are carried out for real, imaginary, and complex times.