Systematic derivation of angular--averaged Ewald potential

TL;DR

This paper provides a detailed derivation of an angular-averaged Ewald potential for numerical simulations of disordered Coulomb systems, using mathematical methods like Euler–Maclaurin and Poisson summation formulas.

Contribution

It offers a clear derivation and explicit formulas for the potential's coefficients, enhancing the theoretical foundation for simulations of Coulomb systems.

Findings

Derived explicit formulas for the potential coefficients.

Demonstrated the equivalence of summation methods in 3D.

Validated the potential by calculating Madelung constants.

Abstract

In this work we provide a step by step derivation of an angular--averaged Ewald potential suitable for numerical simulations of disordered Coulomb systems. The potential was first introduced by E.\,Yakub and C.\,Ronchi without a clear derivation. Two methods are used to find the coefficients of the series expansion of the potential: based on the Euler--Maclaurin and Poisson summation formulas. The expressions for each coefficient is represented as a finite series containing derivatives of Jacobi theta functions. We also demonstrate the formal equivalence of the Poisson and Euler--Maclaurin summation formulas in the three-dimensional case. The effectiveness of the angular--averaged Ewald potential is shown by the example of calculating the Madelung constant for a number of crystal lattices.