Computation of Lattice Sums: Generalization of the Ewald Method II

Our method of computing lattice sums1 is carried through the final stage of numerical computation. We calculate the expansion coefficients of the crystal potential, evaluated at each inequivalent site of a number of cubic structures. Some of the structures chosen are simple, to afford comparison with previous calculations. Many, however, are too complex to be amenable to other methods, and our results are the first reported. The sequence of computer programs, timing, and accuracy are discussed. In a previous paper 1 concerning lattice sums we discussed summation methods which hinge on two facts: First, many lattices can be decomposed into so-called "primitive" lattices. Second, a wide class of summation procedures is feasible for such primitive lattices, which are impossible or impractical for arbitrary lattices. In particular, we developed an extension of the Ewald method, following the general philosophy of Nijboer and DeWette, 2 Adler, 3 and Barlow and Macdonald. 4 The conditions under which a given lattice can be decomposed into primitive lattices, and the algorithm accomplishing this, have been discussed with full generality and rigor by Graham. 5 In this paper we present a sample of numerical results obtained by our method, together with some discussion of computational techniques and computational efficiency. We have evaluated the coefficients, up to order six, for the spherical harmonic expansion of the potential due to a lattice of point charges. We have done this for every inequivalent site in a number of cubic lattices.