Dependence of the Electrostatic Energy on the Dimension of the Periodicity in a Uniform Background

Abstract

We consider an array of Gaussian functions in order to account for the self-energy of an element, and we derive an expression for its electrostatic energy. In this expression, the number of dimensions of space and the periodicity can be generalized to take arbitrary real number values. The electrostatic energy evaluated in this manner increases with the number of dimensions of the periodicity, in contrast to the irregular dependence of the electrostatic energy evaluated from an expression for point charges, like in the Ewald method. In the limit of an array of δ functions, the electrostatic energy has a singularity at (d - 1)-dimensional periodicity in d-dimensional space, due to the self-energy of an element.