Expansion of the Madelung parameter in terms of multipole moments and as a function of the ratio of two lattice parameters

A method is derived to expand the Madelung parameters of non-cubic crystals as a function of the ratio of two lattice parameters and in terms of multipole moments. Only one expansion of this kind is necessary to evaluate the Madelung constants for a series of compounds of one type of structure either containing polyatomic ions or spherical ions at positions slightly varying from one compound to the other. This method also allows the minimisation of the lattice energy with respect to one particular lattice parameter, leaving invariant all internal distances in the polyatomic ions.