Geodesic
On the precision increasing in calculation of potential for the systems of interactive atoms
Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 101-110.

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We propose a high precision method of finding of potential for multi-atomic quantum-mechanical tasks in real space. The method is based on dividing of electron density and potential of a multi-atomic system into two parts. The first part of density is found as a sum of spherical atomic densities; the second part is a variation of density generated by interatomic interaction. The first part of potential is formed by the first part of density and may be calculated correctly using simple integrals. The second part of potential is found through a Poisson equation from the second part of density. To provide a high precision we divided a work space into Voronoy's polyhedrons and found the boundary conditions by means of a multi-pole distribution of potentials formed by local densities concentrated in these polyhedrons. Then we used double-grid approach, and fast Fourier transformations as initial functions for iterative solution of the Poisson's equation. We estimated accuracy of the offered method and carried out test calculations which showed that this method gives the accuracy several times better than accuracy of the fast Fourier transformation.
Keywords: Poisson equation, electrostatic potential, Voronoy polihedra, multipole expansion, double-grid method. 
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V. G. Zavodinskу; O. A. Gorkusha. On the precision increasing in calculation of potential for the systems of interactive atoms. Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 101-110. http://geodesic.mathdoc.fr/item/CHEB_2018_19_2_a7/

[1] L. D. Landau, E. M. Lifshits, Quantum mechanics. Nonrelativistic theory, Fizmatgiz, M., 1963, 768 pp.

[2] W. Kohn, J. L. Sham, “Self-consistent equations including exchange and correlation effects”, Phys. Rev., 140:4A (1965), A1133–A1138 | DOI | MR

[3] Zavodinsky V., Computer modeling of nanoparticles and nanosystems, Fizmatlit, M., 2013, 137 pp.

[4] Skollermo G., “A Fourier method for the Numerical Solution of Poisson Equation”, Mathematics of Computation, 29:131 (1975), 697–711 | DOI | MR | Zbl

[5] Chun-Min Chang, Yihan Shao, Jing Kong, “Ewald mesh method for quantum mechanical calculations”, J. Chem. Phys., 136:11 (2012), 114112, 5 pp. | DOI

[6] V. B. Bobrov, S. A. Trigger, “The problem of the universal density functional and the density matrix functional theory”, Journal of Experimental and Theoretical Physics, 116:4 (2013), 635–640 | DOI

[7] J. R. Chelikowsky, N. Troullier, Y. Saad, “Finite-difference-pseudopotential method: Electronic structure calculations without a basis”, Phys. Rev. Lett., 72:8 (1994), 1240–1243 | DOI

[8] L. Kleinman L., D. M. Bylander, “Efficacious Form for Model Pseudopotentials”, Phys. Rev. Lett., 48:20 (1982), 1425–1428 | DOI

[9] H. Cheng H., L. Greebgard, V. Rokhlin, “A Fast Adaptive Multipole Algorithm in Three Dimensions”, Journal of Computational Physics, 155:2 (1999), 468–498 | DOI | MR | Zbl

[10] J. J. Mortensen, L. B. Hansen, K. W. Jacobsen, “Real-space grid implementation of the projector augmented wave method”, Phys. Rev. B Condensed Matter, 71:3 (2005), 035109, 11 pp. | DOI

[11] J. R. Chelikowsky, K. Wu, N. Troullier, Y. Saad, “Higher-order finite-difference pseudopotential method: An application to diatomic molecules”, Phys. Rev. B, 50:16 (1994), 11355–11364 | DOI

[12] A. D. Becke, “A multicenter numerical integration scheme for polyatomic molecules”, J. Chem. Phys., 88:4 (1988), 2547–2553 | DOI

[13] Kikuji Hirose, Tomoya Ono, Yoshitaka Fujimoto, Shigeru Tsukamoto, First-Principles Calculations in Real-Space Formalism, Imperial College Press, London, 2005, 2253 pp.

[14] Atsuyuki Okabe, Barry Boots, Kokichi Sugihara, Sung Nok Chiu, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, Wiley, New York, 2000, 696 pp. | MR | Zbl

[15] X. Gonze, R. Stumpf, M. Scheffler, “Analysis of separable potentials”, Phys. Rev. B, 44:16 (1991), 8503–8513 | DOI

[16] N. Troullier, J. L. Martins, “Efficient pseudopotentials for plane-wave calculations”, Phys. Rev., 43:3 (1991), 1993–2006 | DOI | MR

[17] A. Brandt, “Multi-level adaptive solutions to boundary-value problems”, Mathematics of Computation, 31:138 (1977), 333–390 | DOI | MR | Zbl

[18] Walter A. Harrison, Electronic Structure and the Properties of Solids, W. H. Freeman and Company, San Francisco, 1980, 680 pp. | MR

[19] Zavodinsky V., Quantum modeling of polyatomic systems without wave functions, LAP LAMBERT Academic Publishing RU, 2017, 56 pp.