Determination of Madelung constants for infinite and semi-infinite lattices by direct summation

Lattice sums for the electrostatic interactions in ionic crystals, first evaluated by Madelung, are now known to high degrees of accuracy. The problem of conditional convergence has been solved either by manipulations of a mathematical model of the lattice, or by directly summing over neutral groups of the lattice as was done by Evjen and Frank. Although the first method has been modified and extended to encompass semi-infinite lattices, there are no corresponding works on direct summation techniques. Dahl has corrected the Evjen method to expand its range to more complex crystals but did not indicate its applicability to bounded lattices. This communication describes a novel direct summation approach broad enough to handle any infinite or semi-infinite lattice that can be decomposed into finite, neutral, linear charge arrays. The technique is unique in that it is the only direct summation method demonstrated capable of calculating surface Madelung constants.