AUTHOR=Winkelmann Miriam , Di Napoli Edoardo , Wortmann Daniel , Blügel Stefan TITLE=Solution to the Modified Helmholtz Equation for Arbitrary Periodic Charge Densities JOURNAL=Frontiers in Physics VOLUME=8 YEAR=2021 URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2020.618142 DOI=10.3389/fphy.2020.618142 ISSN=2296-424X 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.