On the Numerical Accuracy of Ewald, Smooth Particle Mesh Ewald, and Staggered Mesh Ewald Methods for Correlated Molecular Systems

Abstract

In this work, we develop the accurate error estimates for three state-of-art algorithms of long-range electrostatic interaction in inhomogeneous and correlated molecular systems. They are the Ewald summation, the smooth particle mesh Ewald (SPME) and the staggered mesh Ewald methods. Two branches of force computation, namely the ik- and analytical differentiation, are considered. All the estimates are developed by proposing a more general framework: if the error force is of pairwise form, then the root-mean-square force error is composed of three additive parts, the homogeneity error, the inhomogeneity error and the correlation error. Computationally scalable estimates (estimating the errors at the cost O(N log N)) are developed for all the considered algorithms. The effectiveness of the proposed estimates and the important role of the correlation error are carefully checked and demonstrated by example systems.