Systematic derivation of angular-averaged Ewald potential

Abstract

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.