Separable approximation to two-body matrix elements

TL;DR

This paper presents a method to approximate two-body matrix elements using a finite sum of separable terms, optimized for harmonic oscillator states, with analysis of convergence and applicability to different interactions.

Contribution

It introduces a separable approximation technique for two-body matrix elements tailored for mean-field calculations, emphasizing harmonic oscillator basis and convergence analysis.

Findings

Convergence studied for Gaussian interactions.

Method applicable to contact (delta) forces.

Discussion on extending to general cases.

Abstract

Two-body matrix elements of arbitrary local interactions are written as the sum of separable terms in a way that is well suited for the exchange and pairing channels present in mean-field calculations. The expansion relies on the transformation to center of mass and relative coordinate (in the spirit of Talmi's method) and therefore it is only useful (finite number of expansion terms) for harmonic oscillator single particle states. The converge of the expansion with the number of terms retained is studied for a Gaussian two body interaction. The limit of a contact (delta) force is also considered. Ways to handle the general case are also discussed.