Strict derivation of angular-averaged Ewald potential

Abstract

In this work we strictly derive 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 strict mathematical justification. 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