Expansion of the Coulomb potential in crystals

A method is presented by which the coefficients of a potential expansion into spherical harmonics are related to invariant components of multipole moments, centered at a representative point of crystallographically equivalent positions. The quantities which establish this relation (named $K$ factors), are completely determined by the space group of the crystal and the unit-cell dimensions. They can be tabulated for any crystal structure, and tables are given for three important cubic space groups. On the basis of such tables, the determination of the expansion coefficients for the Coulomb potential in crystals is reduced to the evaluation of multipole moments of the charge density in spatial regions, partitioning the asymmetric unit. This method can be used in quantum-mechanical calculations as well as in classical treatments of the Coulomb potential. It permits large flexibility in potential calculations with differing electronic charge distribution and is therefore very convenient for self-consistent procedures. It also allows a systematic comparison of Coulomb interactions in different types of crystals and can therefore be an aid for the understanding of ionic structures.