Comparison of molecular dynamics and moment based methods as toolsin the computation of time dependent correlation functions

摘要

Methods, such as the continued fractions that are based on the exact moments (Taylor coefficients)provide powerful analytic techniques for calculating approximate dynamic correlation functions ofphysical properties. Owing to the practical difficulties associated with obtaining higher order exactmoments these methods have been largely eclipsed by molecular dynamics. In this paper we developthe formalism for extracting trajectory moments from a molecular dynamics trajectory and comparethe performance of the two methods. We begin by using the classical phase space analogs ofquantum wave functions to obtain matrix representations (transition matrices) of the Lie group oftime displacement operators. The group elements for small time displacements, which correspond toa step of the trajectory, are approximated by using Trotter's theorem. The method is applied to thelinear harmonic oscillator, the Morse oscillator and the Lennard-Jones potential and the resultingmoments compared to those obtained by the Taylor expansion of a truncated continued fraction. Wefind that in the case of the harmonic oscillator the results from the continued fraction are much betterthan those from molecular dynamics. In the latter two cases, while the truncated fraction performsbetter than molecular dynamics both methods produce poor quality higher order moments. However,we are able to show the criterion whereby the overall error introduced by the truncated continuedfraction is smaller than that introduced by molecular dynamics. From this work we conclude that themoment based methods produce good results, they are analytic in nature rather than numerical andshould not be rejected but can be used to complement molecular dynamics.