Electrostatic potentials in systems periodic in one, two, and three dimensions

We consider the electrostatic potential in a unit cell containing N point charges Q(j) with positions r(j) inside the cell. The cell is replicated periodically in one, two, and three dimensions. The purpose is to give representations for the potential which contain only lattice sums which are absolutely convergent and uniformly convergent in the sampling position r. These representations are derived using variants of the Ewald method and are primarily intended for use in evaluating the accuracy of any algorithm to evaluate electrostatic energies and forces in simulations of dense matter, rather than necessarily for use of themselves in simulations. In reduced dimensionality the Ewald representations can be numerically inefficient and other representations are also provided with careful specification which allows two forms to be used for the potential functions in order to improve numerical performance. These mixed representations may be satisfactory in simulations.