Gaussian matrix elements in a cylindrical harmonic oscillator basis

TL;DR

This paper introduces a formalism called the separation method for efficiently calculating two-body Gaussian matrix elements in a cylindrical harmonic-oscillator basis, crucial for nuclear structure calculations and applicable to other many-body systems.

Contribution

The paper presents a detailed derivation of an analytical formalism for Gaussian matrix elements in a cylindrical basis, improving computational efficiency and accuracy for nuclear and atomic physics calculations.

Findings

The formalism enables faster computation of matrix elements.

Analytical expressions are validated against numerical integration.

Method is applicable to various interactions beyond Gaussian.

Abstract

We derive a formalism, the separation method, for the efficient and accurate calculation of two-body matrix elements for a Gaussian potential in the cylindrical harmonic-oscillator basis. This formalism is of critical importance for Hartree-Fock and Hartree-Fock-Bogoliubov calculations in deformed nuclei using realistic, finite-range effective interactions between nucleons. The results given here are also relevant for microscopic many-body calculations in atomic and molecular physics, as the formalism can be applied to other types of interactions beyond the Gaussian form. The derivation is presented in great detail to emphasize the methodology, which relies on generating functions. The resulting analytical expressions for the Gaussian matrix elements are checked for speed and accuracy as a function of the number of oscillator shells and against direct numerical integration.