Numerical calculation of integrals in the density functional theory in the electron gas approximation using CUDA technology

Abstract

The expression for the potential energy of interaction of two neutral atoms in the absence of a chemical bond consists of the sum of multiple and improper integrals. Due to the cumbersome nature of the functions, finding these integrals in explicit form is not possible. Software systems widely used in practice based on standard methods of computational mathematics are also not capable of providing satisfactory accuracy in their numerical calculations in a short time. In quantum chemistry and computational physics, the above greatly limits approaches to modeling the properties and structures of atomic/molecular systems. You have to rely on Monte-Carlo integration methods or formulas of the Gauss — Laguerre type, which are not so effective in terms of accuracy. In this article, in relation to this problem, we propose a method for transferring Newton — Cotes quadrature formulas to the architecture of graphic processors. The features of such a transfer are discussed in detail, designed to eliminate bottlenecks and maximize the performance of the corresponding calculation programs. The platform for massively parallel computing is CUDA technology from NVIDIA. Testing has shown that in typical tasks, the efficiency of programs for GPUs based on parallel analogues is on average an order of magnitude higher than classical ones. Within the framework of the proposed approach, it was possible to calculate interatomic interaction potentials in a wide range of changes in distances between atoms with high accuracy and in an acceptable computer time, as well as determine the equilibrium interaction parameters. the results obtained are in good agreement with the data known from the literature.