Exact method for the simulation of Coulombic systems by spherically truncated, pairwise $r−1$ summation

Based on a recent result showing that the net Coulomb potential in condensed ionic systems is rather short ranged, an exact and physically transparent method permitting the evaluation of the Coulomb potential by direct summation over the $r−1$ Coulomb pair potential is presented. The key observation is that the problems encountered in determining the Coulomb energy by pairwise, spherically truncated $r−1$ summation are a direct consequence of the fact that the system summed over is practically never neutral. A simple method is developed that achieves charge neutralization wherever the $r−1$ pair potential is truncated. This enables the extraction of the Coulomb energy, forces, and stresses from a spherically truncated, usually charged environment in a manner that is independent of the grouping of the pair terms. The close connection of our approach with the Ewald method is demonstrated and exploited, providing an efficient method for the simulation of even highly disordered ionic systems by direct, pairwise $r−1$ summation with spherical truncation at rather short range, i.e., a method which fully exploits the short-ranged nature of the interactions in ionic systems. The method is validated by simulations of crystals, liquids, and interfacial systems, such as free surfaces and grain boundaries.