A Multipole-based Algorithm for Eecient Calculation of Forces and Potentials in Macroscopic Periodic Assemblies of Particles a Multipole-based Algorithm for Eecient Calculation of Forces and Potentials in Macroscopic Periodic Assemblies of Particles

A new and e cient 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 the n-particle unit cell in O(n) time. For the cubic lattice, forces due to a 3k 3k 3k lattice of images of the unit cell, containing 33kn particles, can be computed in O(nk2 + k3 log k) 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 38 region 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, a k = 4 : : : 6 simulation approximates the true in nite 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 non-cubic unit cell shapes, and non-cubic macroscopic shapes. Simple code modi cations allowed computation of forces within macroscopic spheres and ellipsoids, and within near-in nite square, circular, and elliptical surfaces formed of unit cubes replicated along two of the three axes. In addition to e cient periodic simulations, the method provides a powerful tool to study limiting behavior of various nite crystal shapes, as well as surface phenomena in molecular dynamics simulations. 2