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

A. A. Sokurov

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 20 (2024) no. 3, pp. 335-349

Résumé

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.

Détail

Keywords: parallel programming, numerical integration, CUDA technology, GPGPU, density functional theory.

Bibliographie

[1] N. S. Bahvalov, N. P. Zhidkov, G. M. Kobel'kov, Numerical methods, Nauka, M., 1987, 600 pp. (In Russian) | MR

[2] N. N. Kalitkin, Numerical methods, Nauka, M., 1978, 512 pp. (In Russian)

[3] E. P. Zhidkov, Iu. Iu. Lobanov, V. D. Rushai, “Improving the accuracy of multiple integral evaluation by applying Romberg's method”, Computational Mathematics and Mathematical Physics, 49:2 (2009), 232–240 (In Russian) | MR | Zbl

[4] K. A. Rybakov, “Exact calculation of the approximation error of multiple Ito stochastic integrals”, Numerical Analysis and Applications, 26:2 (2023), 205–213 (In Russian) | DOI | MR | Zbl

[5] Iu. A. Vyzhol, A. N. Zhorova, I. A. Mulenko, A. L. Khomkin, “Application of computer algebra for the calculation of bracket integrals”, Computational Mathematics and Mathematical Physics, 51:10 (2011), 1867–1882 (In Russian) | MR | Zbl

[6] Galan-Garcia J. L., Rodriguez-Cielos P., Padilla-Dominguez Y., Galan-Garcia M. A., Atencia I., Rodriguez-Padilla P., Aguilera-Venegas G., “SMIS: A stepwise multiple integration solver using a CAS”, Mathematics, 9:22 (2021), 1–32 | DOI

[7] Chernukha O., Bilushchak Y., Shakhovska N., Kulhanek R., “A numerical method for computing double integrals with variable upper limits”, Mathematics, 10:1 (2022), 1–26 | DOI

[8] O. A. Gorkusha, V. G. Zavodinskii, “On the calculation of the interaction potential in multiatomic systems”, Computational Mathematics and Mathematical Physics, 59:2 (2019), 325–333 (In Russian) | DOI | MR | Zbl

[9] P. A. Krutitskii, I. O. Reznichenko, “Improved quadrature formula for a single-layer potential”, Computational Mathematics and Mathematical Physics, 63:2 (2023), 230–244 (In Russian) | DOI | Zbl

[10] I. M. Sobol', Multidimensional quadrature formulas and Haar functions, Nauka, M., 1969, 288 pp. (In Russian)

[11] N. O. Raba, “Development and realization of algorithm of coagulation calculation in mixed-phase cloud model using CUDA technology”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2011, no. 4, 94–104 (In Russian)

[12] Fazanaro F. I., Soriano D. C., Suyama R., Madrid M. K., Oliveira J. R., Munoz I. B., Attux R., “Numerical characterization of nonlinear dynamical systems using parallel computing: The role of GPUs approach”, Communications in Nonlinear Science and Numerical Simulation, 37 (2016), 143–162 | DOI | MR | Zbl

[13] V. O. Ruzhnikov, “Calculation performance improvement of particle dynamics on parallel systems”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2014, no. 1, 147–156 (In Russian)

[14] Gordon R. G., Kim Y. S., “Theory for the forces between closed-shell atoms and molecules”, Journal of Chemical Physics, 56:6 (1972), 3122–3133 | DOI

[15] Waldman M., Gordon R. G., “Scaled electron gas approximation for intermolecular forces”, Journal of Chemical Physics, 71:3 (1979), 1325–1339 | DOI

[16] Strand G., Bonham R. A., “Analytical expressions for the Hartree — Fock potential of neutral atoms and for the corresponding scattering factors for X rays and electrons”, Journal of Chemical Physics, 40:6 (1964), 1686–1691 | DOI

[17] V. I. Krylov, Approximate calculation of integrals, Nauka, M., 1967, 500 pp. (In Russian)

[18] Rit M., Nano-engineering in science and technology: an introduction to the world of nano-design, Reguliarnaia i khaoticheskaia dinamika Publ., Izhevsk, 2005, 160 pp. (In Russian)

[19] A. V. Boreskov, A. A. Kharlamov, Basics of working with CUDA technology, DMK Press, M., 2010, 232 pp. (In Russian)

[20] Dzh. Sanders, E. Kendrot, CUDA technology in examples. Introduction to GPU programming, DMK Press, M., 2011, 232 pp. (In Russian)

[21] Gould T., Bucko T., “C6 coefficients and dipole polarizabilities for all atoms and many ions in rows 1–6 of the periodic table”, Journal of Chemical Theory and Computation, 12:8 (2016), 3603–3613 | DOI