Note on the calculation of electrostatic lattice potentials

We reconsider the problem of calculating electrostatic lattice potentials, or in fact potentials of the type |r|βφl(r), where φl(r) is the lth derivative of the l/|r|p potential. According to Ewald, and generalized by Nijboer and de Wette, the slowly converging potential can be expressed in two screened potentials, one on the original lattice and the other on the reciprocal lattice, which are rapidly converging. In general these screenings are related by a type of Hankel transform. The request of (asymptotic) symmetry between the screenings then leads to the well-known screenings as the best possible. Detailed formula needed for calculations are given.