Coulomb energy of polar crystals

A simple derivation is given to arrive at an expression for the Coulomb energy of a polar crystal with emphasis on its dependence on the external form of the crystal. It is found that the existing treatments are incomplete, since they refer to the central cell in the crystal, which may have an energy different from the average one. It is stressed that there is a very simple relation between results from calculations using a cutoff scheme and results from Ewald summation. Explicit formulas for the form-dependent contribution are developed for the case of a rectangular crystal. However, the usual approach in which that contribution is neglected appears to be a reasonable one for practical applications.