Electrostatic energy in periodic systems

We consider the electrostatic energy of a three-dimensional periodic array of equal and opposite charges, situated arbitrarily in a cubic unit cell. The electrostatic energy is periodic and symmetric in the charge separation vector components. It satisfies the Poisson equation, which leads to an ordinary differential equation for the potential energy when this is expressed in terms of a suitable function with the correct symmetries and periodicities in the spatial variables. Closed form expressions are given for two approximate potential energies in terms of sinusoidal and Jacobi elliptic functions, respectively. Both potential energy functions have the required symmetry and periodicity. That based on the Jacobi elliptic functions has the correct form at close approach of the two charges in the central cell. Surfaces of constant electrostatic energy are easily generated from either expression.