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Manuscript Title: Fortran program for the integral of three spherical harmonics.
Authors: A.L. de Brito
Program title: F3Y
Catalogue identifier: AAQQ_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 25(1982)81
Programming language: Fortran.
Computer: IBM-1130.
Operating system: MONITOR.
RAM: 2K words
Word size: 16
Keywords: General purpose, Condon-shortley Parameters, Spherical harmonics, Function.
Classification: 4.7.

Nature of problem:
First-principles calculations on atoms and molecules usually express the many-body wave function psi in terms one-particle states of the form phi(r) = Rnl(r)Ylm(theta, phi), and the evaluation of physical quantities form psi invariably requires a considerable amount of manipulation of the Ylm(theta, phi). Integrals of products of spherical harmonics appear in the evaluation of Coulomb matrix elements in atomic calculations and they are closely related to the addition of angular momenta. We therefore present a Fortran program for calculating these integrals.

Solution method:
There have been two indentities that provide substantial shortcuts to many of the more complicated results involving spherical harmonics. The formula for the integral of three spherical harmonics is usually obtained by expressing the Ylm in terms of associated Legendre functions and integrating by parts as many times as necessary (Gaunt's formula). Our approach makes use of one of the quoted identities.

Running time:
The maximum running time for one value of the integral with the absolute values of the arguments all smaller than 5 was 100 ms on the IBM-1120.