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Solution to the Modified Helmholtz Equation for Arbitrary Periodic Charge Densities

Authors :
Miriam Winkelmann
Edoardo Di Napoli
Daniel Wortmann
Stefan Blügel
Source :
Frontiers in Physics, Vol 8 (2021)
Publication Year :
2021
Publisher :
Frontiers Media S.A., 2021.

Abstract

We present a general method for solving the modified Helmholtz equation without shape approximation for an arbitrary periodic charge distribution, whose solution is known as the Yukawa potential or the screened Coulomb potential. The method is an extension of Weinert’s pseudo-charge method [Weinert M, J Math Phys, 1981, 22:2433–2439] for solving the Poisson equation for the same class of charge density distributions. The inherent differences between the Poisson and the modified Helmholtz equation are in their respective radial solutions. These are polynomial functions, for the Poisson equation, and modified spherical Bessel functions, for the modified Helmholtz equation. This leads to a definition of a modified pseudo-charge density and modified multipole moments. We have shown that Weinert’s convergence analysis of an absolutely and uniformly convergent Fourier series of the pseudo-charge density is transferred to the modified pseudo-charge density. We conclude by illustrating the algorithmic changes necessary to turn an available implementation of the Poisson solver into a solver for the modified Helmholtz equation.

Details

Language :
English
ISSN :
2296424X
Volume :
8
Database :
Directory of Open Access Journals
Journal :
Frontiers in Physics
Publication Type :
Academic Journal
Accession number :
edsdoj.448e31e476be4e1281292ab6f4117043
Document Type :
article
Full Text :
https://doi.org/10.3389/fphy.2020.618142