Exact solution for two-dimensional Coulomb matrix elements

TL;DR

This paper derives an exact, finite-sum analytical expression for Coulomb matrix elements in 2D using harmonic oscillator eigenfunctions, along with recurrence relations to facilitate computational efficiency.

Contribution

It provides the first exact closed-form solution for 2D Coulomb matrix elements and introduces recurrence relations to simplify their calculation.

Findings

Derived a finite-sum expression for 2D Coulomb matrix elements.

Developed recurrence relations to streamline calculations.

Facilitated more efficient computer simulations of 2D Coulomb interactions.

Abstract

Exact analytic expression is derived for the matrix elements of the Coulomb interaction in two dimensions in the form of a closed finite sum expression. The orthonormal complete set of eigenfunctions of the harmonic oscillator is used as the basis for spanning real space. Several recurrence relations have been found in order to simplify the task of calculating the usually vast amount of elements required for any computer simulation.