Recurrence Formulas for Fully Exponentially Correlated Four-Body Wavefunctions

TL;DR

This paper introduces recursive formulas for generating four-body integrals with exponential correlations, simplifying calculations for four-body quantum systems like the lithium atom.

Contribution

It provides a new recursive method for four-body integrals involving exponential correlations, reducing computational effort and enabling efficient energy calculations.

Findings

Recursive formulas facilitate integral generation

Use of symbolic algebra streamlines code creation

Method improves efficiency of four-body quantum calculations

Abstract

Formulas are presented for the recursive generation of four-body integrals in which the integrand consists of arbitrary integer powers (>= -1) of all the interparticle distances r_ij, multiplied by an exponential containing an arbitrary linear combination of all the r_ij. These integrals are generalizations of those encountered using Hylleraas basis functions, and include all that are needed to make energy computations on the Li atom and other four-body systems with a fully exponentially correlated Slater-type basis of arbitrary quantum numbers. The only quantities needed to start the recursion are the basic four-body integral first evaluated by Fromm and Hill, plus some easily evaluated three-body "boundary" integrals. The computational labor in constructing integral sets for practical computations is less than when the integrals are generated using explicit formulas obtained by differentiating the basic integral with respect to its parameters. Computations are facilitated by using a symbolic algebra program (MAPLE) to compute array index pointers and present syntactically correct FORTRAN source code as output; in this way it is possible to obtain error-free high-speed evaluations with minimal effort. The work can be checked by verifying sum rules the integrals must satisfy.