A Multipole-Based Algorithm for Efficient Calculation of Forces and Potentials in Macroscopic Periodic Assemblies of Particles

Abstract

A new and efficient algorithm based on multipole techniques is presented which calculates the electrostatic forces and potentials in macroscopic periodic assemblies of particles. The fast multipole algorithm (FMA) can be used to compute forces within then-particle unit cell inO(n) time. For the cubic lattice, forces due to a 3k× 3k× 3klattice of images of the unit cell, containing 33knparticles, can be computed inO(nk2+k3logk) time to arbitrary precision. The algorithm was readily added onto an existing FMA implementation, and computational results are presented. Accurate electrostatic computations were done on a 38× 38× 38region of 100000-particle unit cells, giving a volume of 28 quadrillion particles at less than a twofold cost over computing the forces and potentials in the unit cell alone. In practice, ak= 4…6 simulation approximates the true infinite lattice Ewald sum forces (including the shape-dependent dipole correction) to high accuracy, taking 25–30 % more time to compute than only the unit cell. The method extends to noncubic unit cell shapes, and noncubic macroscopic shapes. Simple code modifications allowed computation of forces within macroscopic spheres and ellipsoids, and within near-infinite square, circular, and elliptical surfaces formed of unit cubes replicated along two of the three axes. In addition to efficient periodic simulations, the method provides a powerful tool to study limiting behavior of various finite crystal shapes, as well as surface phenomena in molecular dynamics simulations.