CONVERGENCE OF SPHERICAL HARMONIC EXPANSIONS FOR THE EVALUATION OF HARD-SPHERE CLUSTER INTEGRALS

For N particles (N > 2), by means of a spherical harmonic expansion of Silverstone and Moats, a 3N-dimensional cluster may be reduced to 2N + 1 trivial integrals and N-1 interesting integrals. For hard spheres, the N-1 interesting integrals are products of polynomials integrated between binomial bounds. With simple clusters, closed forms are obtained; for more complex clusters, infinite series in l (of Y(lm)) appear. It is here shown for representative cases that these series converge exponentially rapidly, the leading pair of terms accounting for all but a few tenths of a percent of the total cluster integral.