Comparison of molecular dynamics and moment based methods as tools in the computation of time dependent correlation...

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 of physical properties. Owing to the practical difficulties associated with obtaining higher order exact moments these methods have been largely eclipsed by molecular dynamics. In this paper we develop the formalism for extracting trajectory moments from a molecular dynamics trajectory and compare the performance of the two methods. We begin by using the classical phase space analogs of quantum wave functions to obtain matrix representations (transition matrices) of the Lie group of time displacement operators. The group elements for small time displacements, which correspond to a step of the trajectory, are approximated by using Trotter's theorem. The method is applied to the linear harmonic oscillator, the Morse oscillator and the Lennard-Jones potential and the resulting moments compared to those obtained by the Taylor expansion of a truncated continued fraction. We find that in the case of the harmonic oscillator the results from the continued fraction are much better than those from molecular dynamics. In the latter two cases, while the truncated fraction performs better 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 continued fraction is smaller than that introduced by molecular dynamics. From this work we conclude that the moment based methods produce good results, they are analytic in nature rather than numerical and should not be rejected but can be used to complement molecular dynamics.