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Manuscript Title: Maple procedures for the coupling of angular momenta. IV. Spherical harmonics.
Authors: T. Inghoff, S. Fritzsche, B. Fricke
Program title: RACAH
Catalogue identifier: ADFV_v4_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 139(2001)297
Programming language: Maple V Releases 3 4 and 5).
Computer: Pentium III 450 MHz.
Operating system: AIX, Linux, Windows.
RAM: 6M words
Keywords: Angular momentum, Clebsch-Gordan expansion, Racah algebra techniques, Spherical harmonic, Spherical tensor operator, Sum rule evaluation, Wigner n - j symbol, General purpose, Rotation group.
Classification: 4.1, 5.

Nature of problem:
Spherical harmonics are applied in many fields of physics. In classical electrodynamics, for instance, the spherical harmonics may be utilized to expand the electro-magnetic field of a charge distribution in terms of its multipoles. The spherical harmonics also provide an important basis in quantum mechanics for classifying one- and many-particle states since they are simultaneous eigenfunctions of one component and of the square of the orbital angular momentum operator -ir x Delta. In many-particle physics, the properties of these functions (completeness, orthogonality, ...) are frequently applied to evaluate the spin-angular part of the corresponding matrix elements analytically.

Solution method:
In a previous version of the RACAH program [2], we defined data structures and a hierarchy of MAPLE procedures to evaluate and simplify expressions from Racah's algebra. Our revised program now also supports the occurrence of spherical harmonics as well as integrals over the spherical harmonics in such expressions. The evaluation follows similar lines as before by utilizing, in addition, the properties, sum rules, and recursion relations for the spherical harmonics. Several sum rules for these functions lead to (new) Wigner n - j symbols which may be simplified owing to the previous capabilities of the program.

The definition of the spherical harmonics and the sum rules, which have been implemented in the program, mainly refer to the monograph by Varshalovich et al. [3]. There are literally no other limitations on the complexity of individual expressions than those of the resources and computer time which is needed for their evaluation. Even though a large number of sum rules for the Wigner n - j symbols is now incorporated in the program (including the graphical loop rules for the 3 - j symbols), only a selected set of those sum rules, which involve the 9 - j symbols, are implemented so far. Also, we do not support higher n - j symbols (n=12, 15, ...) since they are defined in rather different ways in the literature.

Unusual features:
All commands of the RACAH package are available for interactive work. As explained in Ref. [2] and Appendix A below, the program is based on data structures which are suitable for almost any complexity of Racah algebra expressions. The present version also supports Clebsch-Gordan expansions for two or more spherical harmonics (which depend on the same angular coordinates) into a sum of products of a single spherical harmonic and the corresponding number of Wigner 3 - j symbols. To accelerate the evaluation of Racah expression, the code for most sum rules of the Wigner n - j symbols (n=3, 6, 9), as implemented in the program, have also been improved.

Running time:
All the examples below take only a few seconds on a Pentium III 450 MHz computer.

[1] Maple is a registered trademark of Waterloo Maple Inc.
[2] S. Fritzsche, Comp. Phys. Commun. 103 (1997) 51; S. Fritzsche, S. Varga, D. Geschke, B. Fricke, Comp. Phys. Commun. 111 (1998) 167.
[3] D.A. Varshalovich, A.N. Moskalev, V.K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, Singapore, 1988.