Generation of Clebsch–Gordan coefficients for the point and double groups

Abstract

The theory of the point and double groups has been widely used in quantum physics to understand the structure and dynamical properties of molecules and solids. In order to construct wave functions for such systems, one often needs the Clebsch–Gordan coefficients for the symmetry groups. Here, we present an extension of the Bethe program to support the calculation of the Clebsch–Gordan coefficients as applied, for instance, in crystal field theory. Apart from the generation of the Clebsch–Gordan coefficients, the program also provides a simple access to the group theoretical data for all frequently applied point and double groups. Program summary Title of program: Bethe Catalogue number:ADUH_v3_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUH_v3_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Reference in CPC to previous versions: Comput. Phys. Comm. 162 (2004) 124–142; Comput. Phys. Comm. 171 (2005) 119–132 Catalog identifiers of previous versions: ADUH, ADVU Does the new version supersede the old version?: Yes Licensing provisions: None Computers for which the program is designed: All computers with a license of the computer algebra package Maple [Maple is a registered trademark of Waterloo Maple Inc.] Installations: University of Kassel (Germany) Operating systems under which the program has been tested: Linux 8.1+ and Windows 2000 Programming language used: Maple 7 and 8 Memory required to execute with typical data: 10–30 MB No. of lines in distributed program, including test data, etc.: 11 024 No. of bytes in distributed program, including test data, etc.: 210 244 Distribution format:tar.gz Nature of the physical 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. Method of solution: In the framework of the Bethe program [K. Rykhlinskaya, S. Fritzsche, Comput. Phys Comm. 162 (2004) 124–142; K. Rykhlinskaya, S. Fritzsche, Comput. Phys Comm. (2005), in press], we previously defined data structures to deal with a large number of group parameters 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 extension 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 the new version: Extension of the program. Summary of revision: A number additional procedure 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 onto the complexity of the problem: The generation of the Clebsch–Gordan coefficients is supported for the cyclic and related groups C i , C s , C n , C n h , C n v , the dihedral groups D n , D n h , D n d , the improper cyclic groups S 2 n ( 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 of the program: 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 [S. Altmann, P. Herzig, Point-Group Theory Tables, Clarendon Press, Oxford, 1994]. 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. Typical running time: Although the program replies ‘promptly’ on most requests, the running time depends strongly on the particular task.