Precise and efficient Ewald summation for periodic fast multipole method

Abstract

This article deals with the Ewald summation method for efficiently handling contributions from periodic images in the framework of the fast multipole method (FMM). Although there have already been several reports regarding this method, the given formulas have been slightly different, thus leading to possible confusion. In this study, the summation formula for arbitrary lattice vectors is derived independently to show that this formula is identical to that of the cubic cell of Figueirido et al. (J Chem Phys 1997, 106, 9835 and J Chem Phys 107, 7002). The correctness of the formula is confirmed by numerical tests carefully designed for the verification. In addition, a precise numerical method is proposed for subtracting the contributions of neighbor cells, which is a necessary operation in periodic FMM. Furthermore, an optimal choice is given for the parameters that control the convergence behavior of the summation formula. It is shown by numerical tests for a 50,000-particle system that a method with double-precision numbers gives force accuracies of 7, 11, and 14 digits when the degrees of expansion are 8, 16, and 32, respectively. These results indicate not only the correctness of the summation formula but also the effectiveness of the careful subtraction method. It is also shown that, at most, only 0.1% of the total cost of the force evaluation can be attributed to the summation. As a result, using the choice for parameters, the contributions from distant images can be taken into account with great precision at extremely low computational costs. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1075–1087, 2000