The grid-based fast multipole method – a massively parallel numerical scheme for calculating two-electron interaction energies

Abstract

Algorithms and working expressions for a grid-based fast multipole method (GB-FMM) have been developed and implemented. The computational domain is divided into cubic subdomains, organized in a hierarchical tree. The contribution to the electrostatic interaction energies from pairs of neighboring subdomains is computed using numerical integration, whereas the contributions from further apart subdomains are obtained using multipole expansions. The multipole moments of the subdomains are obtained by numerical integration. Linear scaling is achieved by translating and summing the multipoles according to the tree structure, such that each subdomain interacts with a number of subdomains that are almost independent of the size of the system. To compute electrostatic interaction energies of neighboring subdomains, we employ an algorithm which performs efficiently on general purpose graphics processing units (GPGPU). Calculations using one CPU for the FMM part and 20 GPGPUs consisting of tens of thousands of execution threads for the numerical integration algorithm show the scalability and parallel performance of the scheme. For calculations on systems consisting of Gaussian functions (α = 1) distributed as fullerenes from C₂₀ to C₇₂₀, the total computation time and relative accuracy (ppb) are independent of the system size.