Coulomb self-energy integral of a uniformly charged d-cube: A physically-based method for approximating multiple integrals

Abstract

We report a concise expression for the electrostatic Coulomb self-energy of a uniformly charged cube with arbitrary dimensionality. The result, conveniently expressed in integral form, shows the unique behavior of the system as dimensionality increases. We use the case study of a uniformly charged square and cube to illustrate the effectiveness of the method employed. The resulting integral expression of the Coulomb self-energy of a multi-dimensional cube can be calculated numerically at very high accuracy. Numerically exact values of Coulomb self-energy obtained this way provide an excellent tool to gauge the validity of various computational techniques including a straightforward discretization method employed in this work. The proposed method relates the self-energy multi-variable integral to energy contributions coming from a finite discrete system of point charges with the understanding that the number of point charges gradually increases towards the continuum limit. We achieve very good accuracy by using an expression with only two terms where the second term increases considerably the accuracy of computation of continuum integrals and represents a systematic first-order correction to the finite energy of the discrete system. The applied numerical method turns out to be very reliable as shown by the results obtained for cubic domains with dimensionality varying from two to five. The method can be easily generalized to other types of domains with arbitrary geometry.