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Manuscript Title: Maple procedures for the coupling of angular momenta. II. Sum rule evaluation.
Authors: S. Fritzsche, S. Varga, D. Geschke, B. Fricke
Program title: Racah
Catalogue identifier: ADFV_v2_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 111(1998)167
Programming language: Maple V Release 3 and 4.
Computer: IBM Workstation.
Operating system: AIX 3.2.5, Linux.
RAM: 4M words
Keywords: General purpose, Rotation group, Angular momentum, Racah algebra techniques, Sum rule evaluation, Spherical tensor Operators, Wigner n-j symbols, Computer algebra.
Classification: 4.1, 5.

Nature of problem:
Computer algebra (CA) is used to evaluate and to simplify typical expressions from Racah algebra which may also include the summation over dummy quantum numbers. A large variety of sum rules is implemented in a set of Maple commands for interactive use.

Solution method:
In a recent paper [2], we defined proper data structures to deal efficiently with expressions from Racah algebra and to enable the numerical computation of such expressions. The present extension of the program now aims at simplifying typical Racah expressions which may also include the summation over dummy indices. A simplification is attempted by the successive analyses of the various parts of a given Racah expression and by comparison with a set of sum rules in their standard form as found in the literature. The orthogonality relations which are known for the Wigner n-j symbols are also implemented; internally, however, these relations are treated as special sum rules. All equivalent symmetric forms of a given Racah expression are taken into account during the evaluation. More than 40 different sum rules are known to the package which will cover many applications in different fields.

The set of sum rules, which has been implemented in the Racah program, mainly refers to the monograph by Varshalovich et al. [3] on the theory of angular momentum. These sum rules either include a single Wigner n-j symbol or products with a different number of such symbols. In general, the complexity of Racah expressions increase as more Wigner symbols are involved in the product terms. Though we incorporated a large number of sum rules for the Wigner 3-j and 6-j symbols, only a selected set of those rules involving 9-j symbols have been implemented in the present version. So far, we have also considered only a smaller number of sum rules involving products of more than four Wigner n-j symbols of different kinds as well as those including more than a triple summation over dummy quantum numbers. The most complex sum rule currently involves the product of six 3-j symbols and a nine-fold summation. The application of this latter sum rule, however, does often not work very efficiently with respect to time. The success in simplifying Racah algebra expressions critically depends on the fact that all equivalent symmetric forms of the expression are recognized internally. Thereby, the overall symmetry of a Racah expression is directly related to the symmetries of all the Wigner n-j symbols which are involved in the expression. Apart from the classical symmetries of the Wigner symbols, there is an extended range of symmetries due to Regge [4]; these symmetries, however, are of minor importance for most practical applications. Even though, in principle, the Racah package enables one to apply the full range of extended symmetries, limitations in computer time will often restrict the usage of the program to the classical ones.

Unusual features:
All commands of the Racah package are available for interactive work. As explained in Ref. [2], the program is based on data structures which are suitable for almost any complexity of Racah algebra expressions. More enhanced expressions are built up from simpler data structures. The simplification of any valid Racah expression can be attempted just by typing the command Racah_evaluate() at Maple's prompt. This will test all different rules which are known to the package. However, if one knows the structure of the sum rule in advance, i.e. the number and type of the Wigner n-j symbols involved, these rules can also directly be invoked by individual commands. This usually results in a faster evaluation, in particular if more complex expressions need to be simplified. In appendix A, we summarize all new commands at user's level for quick reference (i.e. those commands which have not been described yet in Ref. [2]).

Running time:
All examples of the long write-up require about 3 minutes on an IBM workstation.

[1] Maple is a registered trademark of Waterloo Maple Inc.
[2] S. Fritzsche, Comp. Phys. Commun. 103, 51 (1997).
[3] D.A. Varshalovich, A.N. Moskalev and V.K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore a.o., 1988).
[4] T. Regge, Nuovo cimento 10, 544 (1958).