Computation of Lattice Sums: Generalization of the Ewald Method

The Ewald method was originally invented to compute the Madelung constant. In this paper we consider a lattice whose sites are associated with an arbitrary potential Junction. The "charge," or the scale factor for these potential functions, need not be the same at each site. We consider the evaluation of the resulting lattice sum at an arbitrary point, not necessarily at a lattice site. The method involves two generalizations over previous work: (1) the displacement of the origin off a lattice site and {2) the handling of arbitrary periodic charge distributions by decomposing such distributions into simpler ones involving only --q and --q. The method shoidd prove particularly useful for evaluating the expansion coefficients of the crystalline potential when this potential is expanded in the usual spherical harmonic series. The problem of summing slowly converging series is an old one. One physical context in which the problem has been widely studied is the calculation of the potential due to an ionic crystal lattice. The methods of Madelung 1 and Evjen 2 depend on collecting ions into neutral groups. The convergence obtained in this way, however, is conditional: that is, the result depends on the way in which the neutral groups are chosen. Ewald's 3 method, which hinges on doing part of the summation in reciprocal space, gives rapid convergence and the limit is unique. Subsequent discussions4"1" of this topic have been extensions and generalizations of these methods. This work too is an extension of the Ewald technique.