Effective way to sum over long-range Coulomb potentials in two and three dimensions

I propose a method to calculate the logarithmic interaction in two dimensions and the Coulomb interaction in three dimensions under periodic boundary conditions. This paper considers the case of a rectangular cell in two dimensions and an orthorhombic cell in three dimensions. Unlike the Ewald method, there is no parameter to be optimized, nor does the method involve error functions, thus leading to the accuracy obtained. This method is similar in approach to that of Mol. Simul. 22, 199 (1999)]], but the derivation is considerably simpler and physically appealing. An important aspect of the proposed method is the faster convergence of the Green's function for a particular case as compared to Sperb's work. The convergence of the sums for most parts of the unit cell is exponential, and hence requires the calculation of only a few dozen terms. In a very simple way, we also obtain expressions for the interaction for systems with slab geometries. Expressions for the Madelung constants of CsCl and NaCl are also obtained.