Absolutely Convergent Real-Space Madelung Summation Using Axial Multipoles

Abstract

We present an absolutely convergent real-space method for evaluating Madelung potentials in ionic lattices. The method, based on repeated axial multipole units with systematically eliminated low-order moments, applies uniformly to bulk crystals, surfaces, edges, interstitial sites, and exterior points, without recourse to reciprocal-space techniques. In the RU-13 construction, the far-field contribution of each axial multipole unit decays as r^(-13), ensuring fast and absolute convergence of the real-space direct sum. The method yields identical limits under spherical and cubic summation geometries and reproduces standard Madelung constants with high precision, achieving 13-digit accuracy within a radius of 40 lattice spacings. Extensions to dimensions d = 1-6 exhibit convergence consistent with asymptotic decay predictions.