Evaluation of Electrostatic Potentials and Madelung Constants in Ionic Crystals by the Method of Spherically Symmetric Equivalent Charges

A simple derivation is given of the electrostatic potential in a periodic three-dimensional array of spherically symmetric charge distributions. By noting the equivalence in electrostatic calculations of point charges and suitably chosen spherical charge distributions, this leads to expressions for electrostatic potentials and (Madelung) interaction energies in ionic crystals. The expressions involve sums in reciprocal space only. The approach is illustrated by the calculation of Madelung constants for NaCl and CaF2, and the electrostatic interaction energy of Ti02 (rutile). A previous controversy is resolved by showing that the two expressions for the electrostatic potential, which are apparently different, under certain conditions give the same result.