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Manuscript Title: Maple procedures for the coupling of angular momenta. V. Recoupling coefficients.
Authors: S. Fritzsche, T. Inghoff, T. Bastug, M. Tomaselli
Program title: RACAH
Catalogue identifier: ADFV_v5_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 139(2001)314
Programming language: Maple V Releases 3 4 and 5.
Computer: Pentium 450 MHz PC.
Operating system: AIX, Linux, Windows.
RAM: 6M words
Keywords: Angular momentum, Graphical rules, Loop rules, Racah algebra techniques, Recoupling coefficient, Sum rule evaluation, Wigner n - j symbol, Yutsis graph, General purpose, Rotation group.
Classification: 4.1, 5.

Nature of problem:
In quantum many-particle physics, the calculation of matrix elements often requires the evaluation of recoupling coefficients describing the transformation of different coupling schemes for the (non-active) particles which are not bound to the operator. Usually, these coefficients have first to be simplified algebraically before their actual numerical value can be determined. But although it is known that recoupling coefficients with any number of (integer or half-integer) angular momenta can always be reduced to a multiple sum over products of Wigner 6 - j symbols, including proper phases and square root factors, the process of algebraic simplication may become indeed very elaborate. In this process, the graphical rules of Yutsis, Vanagas, and Levinson [2] proved especially helpful in the past for a reliable evaluation of even complex expressions from Racah's algebra.

Solution method:
The RACAH program is based on the knowledge of a large set of sum rules for simplifying typical expressions from Racah algebra which may include (multiple) summations over dummy indices [3]. For complex and lengthy Racah expressions, the algebraic simplification may be considerably accelerated if the graphical rules due to Yutsis et al. [2] are taken into account. Furthermore, a combination of graphical rules and sum rules enables us to take correctly into account the phases, weights and the relation of the recoupling coefficients to other algebraic structures of the theory of angular momentum. The aim of the present implementation of graphical rules into the RACAH program is to obtain an optimum summation formula in the sense of a minimal number of Wigner 6 - j symbols and/or summation variables. Hereby, graphical rules are predominantly used in order to find out about and to simplify those parts in a recoupling coefficient (or generally in any Racah expression) that belong together. The implementation of graphical rules even allows to easily simplify recoupling coefficients which include several ten angular momenta to a (completely equivalent) sum of products of Wigner 6 - j and/or 9 - j symbols, multiplied by proper weights. Just as in former versions of the Racah program [4], the results of the simplification process will be provided as Racah expressions and may thus immediately be used for further derivations and calculations within the theory of angular momentum.

The complexity of a recoupling coefficient depends not only on the number of angular momenta but also on the order in which the individual subsystems are coupled to each other. In the diagrammatic language of Yutsis graphs [2], individual diagrams or parts of them are mainly classified according to "closed cycles" (the so-called n-loops, n>=2) contained in them. In the present version of the Racah program, we implemented all loops with n<=6. However, it will be possible to simplify the majority of recoupling coefficients with loops of even a higher order since such loops are normally reduced to a lower level during the process of simplification. Thus, the limitation to n<=6 hardly matters in practical calculations concerning atomic and nuclear structures or the scattering of particles. Moreover, graphical assistance is also used to recognize and to resolve sum rules over the Wigner 6 - j and 9 - j symbols; this graphical guidance, however, has not been realized for all sum rules yet.

Unusual features:
The evaluation of recoupling coefficients leads to products of Wigner 6 - j symbols which themselves oftern contain a summation over dummy indices. In the RACAH program, if appropriate, these products, too, will be further simplified by applying different sum rules for the 6 - j symbols. Finally, this typically results in even simpler (algebraic) products of Wigner 6 - j and 9 - j symbols and takes off the need to analyze different paths of simplification in order to yield results as compact as possible. Note, however, that only a limited set of rules involving the Wigner 9 - j symbols have been fully implemented so far.
The RACAH program is designed for interactive work and appropriate for almost any complexity of expressions from Racah algebra. To support the handling of recoupling coefficients, these coefficients can be entered as a string of angular momenta, separated by commata, rather similar to their usual mathematical notation. This is a crucial advantage of the program when compared to previous program developments which very often requested a certain input form for the angular momenta in the recoupling coefficient as well as for their individual couplings. Our user-friendly input is in line with one of the basic intentions of the RACAH program: to assist the algebraic evaluation as far as possible wheras numerical computations on lengthy expressions are less supported.

Running time:
The two examples of the long write-up require about 30 s on a Pentium 450 MHz PC.

[1] Maple is a registered trademark of Waterloo Maple Inc.
[2] A.P. Yutsis, I.B. Levinson, V.V. Vanagas, The Theory of Angular Momentum, Israel Program for Scientific Translation, Jerusalem, 1962.
[3] D.A. Varshalovich, A.N. Moskalev, V.K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, Singapore a.o., 1988.
[4] S. Fritzsche, Comput. Phys. Commun. 103 (1997) 51; S. Fritzsche, S. Varga, D. Geschke, B. Fricke, Comput Phys. Commun. 111 (1998) 167.