The Madelung Problem of Finite Crystals

TL;DR

This paper introduces a new decomposition of Coulomb potential contributions in finite crystals, leading to a fast, accurate method for calculating Madelung constants across various ionic crystals.

Contribution

It presents a novel additive decomposition of Coulomb contributions, enabling a rapidly convergent direct-summation scheme for Madelung constants in finite crystals.

Findings

Decomposition includes bulk, boundary, and finite-size correction components.

The correction term's leading order is explicitly derived for cubic crystals.

The method achieves accurate Madelung constant calculations even for small p (single unit cell).

Abstract

The Coulomb potential at an interior ion in a finite crystal of size $p$ is given by a linear superposition of contributions from displacement vectors ${\mathbf r}=(x,y,z)$ to its neighbors. This additive structure underlies universal relationships among Madelung constants and applies to both standard periodic boundary conditions and alternative Clifford supercells. Each pairwise contribution decomposes into three physically distinct components: a periodic bulk term, a quadratic boundary term, and a finite-size correction whose leading order term is $[24r^4-40(x^4+y^4+z^4)]/[9\sqrt{3} (2p+1)^2]$ for cubic crystals with unit lattice constant. Combining this decomposition with linear superposition yields a rapidly convergent direct-summation scheme, accurate even at $p=1$ ($3^3$ unit cells), enabling hands-on calculations of Madelung constants for a wide range of ionic crystals.