PROGRAM SUMMARY
Title of program:
RACAH
Catalogue identifier:
ADOS
Ref. in CPC:
139(2001)314
Distribution format: tar gzip file
Operating system: AIX, Linux, Windows
High speed store required:
6MK words
Number of lines in distributed program, including test data, etc:
21760
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.
Programming language used: Maple V, Releases 3, 4, and 5
Computer:
Pentium 450 MHz PC .
Other versions of this program:
Cat. Id. Title Ref. in CPC ADFV Racah 103(1997)51 ADHW Racah 111(1998)167
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.
Method of solution:
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.
Restrictions:
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.
Typical running time:
The two examples of the long write-up require about 30 s on a Pentium
450 MHz PC.
References:
[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.