Simple calculation of Madelung constants

Abstract

It is found that direct Madelung sums within a sphere of large radius $R$ obtained with a laptop converge rapidly if, and only if, one calculates also the charge $Q$ of the cluster, and adds a shell of charge $\ensuremath{-}Q$ at radius $R$. A simple program running through the lattice provides Madelung potentials within a part in a thousand of the correct values in a few seconds of computation. It is illustrated for perovskite lattices. The corresponding calculation for a metal, positive charges embedded in a constant compensating negative charge distribution, is similarly successful.