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Manuscript Title: Generation of Clebsch-Gordan coefficients for the point and double groups
Authors: K. Rykhlinskaya, S. Fritzsche
Program title: BETHE
Catalogue identifier: ADUH_v3_0 Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 174(2006)903
Programming language: Maple 7 and 8.
Computer: All computers with a license for the computer algebra package Maple [1].
Operating system: Linux 8.1+ and Windows 2000.
RAM: 10-30 MB
Keywords: Clebsch-Gordan coefficients, coupling of subsystems, crystal field theory, direct product, double group, molecular physics, physical chemistry, point group.
PACS: 02.20.-a., 71.70.-d.
Classification: 16.1, 16.3, 14.
Does the new version supersede the previous version?: Yes
Nature of problem: The energy levels of atoms, placed into a crystal environment, can be classified by using group theory. In order to represent, for instance, the wave functions, which are associated with these atomic levels, one often requires the Clebsch-Gordan coefficients of the underlying symmetry group of the overall system. These coefficients arise in the coupling of the electronic wave functions (subsystems) and therefore help investigate the interaction between the many-electron atom and the external field of the crystal.
Solution method: In the framework of the Bethe program [2, 3], we previously defined data structures to deal with a large number of group meters of the point and double groups. Among other parameters, here we also implemented the irreducible (matrix) representations of these groups which are utilized in the present extention of the program in order to generate the Clebsch-Gordan coefficients for the point and double groups. In practice, of course, these coefficients are obtained by means of a proper summation over the matrix elements of the irreducible representations of the group.
Reasons for new version: Extension of the program
Summary of revisions: A number of additional procedures have been created to generate the Clebsch-Gordan coefficients for the symmetry groups (Bethe_CG_matrix(), Bethe_CG_coefficient(), Bethe_group_direct_product(), etc.)
Restrictions: The generation of the Clebsch-Gordan coefficients is supported for the cyclic and related groups C_i, C_s, C_n, C_nh, C_nv, the dihedral groups D_n, D_nh, D_nd, the improper cyclic groups S_2n (n ≤ 10), the cubic groups O, T, O_h, T_h, T_d as well as the icosahedral groups I and I_h. Both the point and the double groups are supported.
Unusual features: All commands of the Bethe program are available for interactive work. Apart from the generation of the Clebsch-Gordan coefficients, the program also provides a simple access to the group theoretical data for all the groups specified above. The notation of the symmetry operations and of the irreducible representations follows the compilation by Altmann and Herzig [4]. For a quick reference to the program, a description of all user-relevant commands is given in the (user) manual Bethe-commands.pdf which is distributed together with the code.
Running time: Although the program replies 'promptly' on most requests, the running time depends strongly on the particular task.
References:
[1]	Maple is a registered trademark of Waterloo Maple Inc.
[2]	K. Rykhlinskaya and S. Fritzsche, Comp. Phys Commun., 162 (2004) 124-142.
[3]	K. Rykhlinskaya and S. Fritzsche, Comp. Phys Commun., in print (2005).
[4]	S. Altmann and P. Herzig, Point-Group Theory Tables (Clarendon Press, Oxford, 1994).