Simple separable expansions for calculating matrix elements of two-body local interactions with harmonic oscillator functions

Abstract

A simple method is proposed to calculate the matrix elements of two-body local interactions using a harmonic oscillator basis (HOF). Using the properties of HOF, it is shown that any local potential can be replaced by a simple series for the purpose of calculating matrix elements. This series can be reduced to a finite sum when evaluating a matrix element. Its terms are separable functions of the coordinates of the two particles; hence the advantage of the method. In the present article we treat the most important components of the two-body interaction, namely central, two-body spin-orbit, and tensor forces. As a representation we have chosen spherical harmonic oscillator functions expressed with spherical coordinates. This technique appears to be very well adapted to and efficient for Hartree-Fock calculations in any representation of the HOF. A very interesting feature of this formulation is that it can be easily extended to calculations employing generalized HOF as defined by Wong.