Analytic Coulomb matrix elements in a three-dimensional geometry

TL;DR

This paper derives an analytic, closed-form expression for Coulomb matrix elements using a 3D quantum harmonic oscillator basis, aiding the study of interactions in localized quantum systems.

Contribution

It provides the first complete analytic formula for Coulomb matrix elements in a 3D harmonic oscillator basis, facilitating calculations in localized quantum systems.

Findings

Derived a closed-form expression for Coulomb matrix elements.

Applicable to localized systems with 3D harmonic oscillator basis.

Simplifies calculations of interactions in quantum models.

Abstract

Using a complete basis set we have obtained an analytic expression for the matrix elements of the Coulomb interaction. These matrix elements are written in a closed form. We have used the basis set of the three-dimensional isotropic quantum armonic oscillator in order to develop our calculations, which can be useful when treating interactions in localized systems.