Computation of the Madelung constant for hypercubic crystal structures in any dimension

TL;DR

This paper introduces a novel, efficient method for calculating Madelung constants in hypercubic crystal structures across any dimension, addressing divergence issues and providing explicit solutions especially in two dimensions.

Contribution

It presents a new regularization technique for Madelung constants in higher dimensions and a specific approach for the 2D case considering scale invariance and finite horizons.

Findings

Efficient computation of Madelung constants in dimensions n≥3

A regularization method using Hadamard finite part for divergent integrals

Explicit solution and limit-based definition for 2D Madelung constants

Abstract

A new method of computing the Madelung constants for hypercubic crystal structures in any dimension $n\geq 2$ is given. It is shown for $n\geq 3$ that the Madelung constant may be obtained in a simple, efficient and unambiguous way as the Hadamard finite part of the integral representation of the potential within the crystal which is divergent at any point charge location. Such a regularization method fails in the bidimensional case due to the logarithmic nature of singularities for the potential. In that case, a specific approach is proposed taking in account the scale invariance of the Poisson equation and the existence of a finite horizon for each point charge in the plane. Since a closed-form exact solution for the 2D electrostatic potential may be derived, one shows that the Madelung constant may be defined via an appropriate limit calculation as the mean value of potential energies of charges composing the unit cell.