PROGRAM SUMMARY
Title of program:
RACAH
Catalogue identifier:
ADOR
Ref. in CPC:
139(2001)297
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, Clebsch-Gordan expansion, Racah algebra techniques,
Spherical harmonic, Spherical tensor operator, Sum rule evaluation,
Wigner n - j symbol, General purpose, Rotation group.
Programming language used: Maple V, Releases 3, 4, and 5
Computer:
Pentium III 450 MHz .
Other versions of this program:
Cat. Id. Title Ref. in CPC ADFV Racah 103(1997)51 ADHW Racah 111(1998)167
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.
Method of solution:
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.
Restrictions:
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.
Typical running time:
All the examples below take only a few seconds on a Pentium III 450 MHz
computer.
References:
[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.