Error estimate of the u-series method for molecular dynamics simulations

Abstract

This paper provides an error estimate for the u-series method of the Coulomb interaction in molecular dynamics simulations. We show that the number of truncated Gaussians M in the u-series and the base of interpolation nodes b in the bilateral serial approximation are two key parameters for the algorithm accuracy, and that the errors converge as $O\left(b^{-M}\right)$ for the energy and $O\left(b^{-3M}\right)$ for the force. Error bounds due to numerical quadrature and cutoff in both the electrostatic energy and forces are obtained. Closed-form formulae are also provided, which are useful in the parameter setup for simulations under a given accuracy. The results are verified by analyzing the errors of two practical systems.