Coulomb potentials in two and three dimensions under periodic boundary conditions

A method to sum over logarithmic potential in two-dimensions (2D) and Coulomb potential in three dimensions (3D) with periodic boundary conditions in all directions is given. We consider the most general form of unit cells, the rhombic cell in 2D and the triclinic cell in 3D. For the 3D case, this paper presents a generalization of Sperb's work [R. Sperb, Mol. Simulation 22, 199 (1999)]. The expressions derived in this work converge extremely fast in all region of the simulation cell. We also obtain results for slab geometry. Furthermore, self-energies for both 2D as well as 3D cases are derived. Our general formulas can be employed to obtain Madelung constants for periodic structures. (C) 2005 American Institute of Physics.