A Generalized Grid-Based Fast Multipole Method for Integrating Helmholtz Kernels.

A grid-based fast multipole method (GB-FMM) for optimizing three-dimensional (3D) numerical molecular orbitals in the bubbles and cube double basis has been developed and implemented. The present GB-FMM method is a generalization of our recently published GB-FMM approach for numerically calculating electrostatic potentials and two-electron interaction energies. The orbital optimization is performed by integrating the Helmholtz kernel in the double basis. The steep part of the functions in the vicinity of the nuclei is represented by one-center bubbles functions, whereas the remaining cube part is expanded on an equidistant 3D grid. The integration of the bubbles part is treated by using one-center expansions of the Helmholtz kernel in spherical harmonics multiplied with modified spherical Bessel functions of the first and second kind, analogously to the numerical inward and outward integration approach for calculating two-electron interaction potentials in atomic structure calculations. The expressions and algorithms for massively parallel calculations on general purpose graphics processing units (GPGPU) are described. The accuracy and the correctness of the implementation has been checked by performing Hartree-Fock self-consistent-field calculations (HF-SCF) on H2, H2O, and CO. Our calculations show that an accuracy of 10-4 to 10-7 Eh can be reached in HF-SCF calculations on general molecules.

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

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.