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.
Highlights
A variety of problems in condensed matter physics require an efficient solution of the partial differential equation Δ − λ2 Vλ −4πρ, (1)for a charge density ρ in a periodic domain
We have presented a general method for solving the modified Helmholtz equation for a 3D-periodic system of charge densities not restricted by any shape approximation of three-dimensional volume
Since the Yukawa differential equation is similar to the Poisson equation, we leveraged our derivations on the work of Weinert [12]
Summary
A variety of problems in condensed matter physics require an efficient solution of the partial differential equation. Most electronic structure methods implementing DFT applied to solid-state materials systems make explicit use of the underlying periodicity of the crystalline lattice, a straightforward solution of Eq 1 using Fourier transformation techniques is in general not possible due to the strongly oscillating charge density close to the nuclei This problem is well discussed for the solution of the Poisson equation, ΔV −4πρ, a limit of the modified Helmholtz equation for λ 0. We obtain the Yukawa potential for the interstitial region by solving the modified Helmholtz equation in Fourier space for the pseudo-charge density—the solution is a simple algebraic expression This is followed by an analysis of the convergence properties of the Fourier series of the pseudo-charge density.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
