Maple procedures for the coupling of angular momenta. VIII. Spin-angular coefficients for single-shell configurations

Abstract

Matrix elements of physical operators are required when the accurate theoretical determination of atomic energy levels, orbitals and radiative transition data need to be obtained for open-shell atoms and ions. The spin-angular part for these matrix elements is typically based on standard quantities such as matrix elements of the unit tensor, the (reduced) coefficients of fractional parentage as well as a number of other reduced matrix elements concerning various products of electron creation and annihilation operators. Therefore, in order to facilitate the access to the matrix elements of one- and two-particle scalar operators, we present here an extension to the Racah program for the full set of standard quantities and the pure spin-angular coefficients in LS- and jj-couplings. A flexible notation is introduced for defining and manipulating the electron creation and the electron annihilation operators. This will allow us to solve successfully various angular momentum problems in atomic physics. Program summary Title of program: Racah Catalogue number: ADUR Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUR Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions: None Computers for which the program is designed: All computers with a valid 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+ Program language used: Maple, Release 8 and 9 Memory required to execute with typical data: 30 MB Number of lines in distributed program, including test data, etc.:36 875 Number of bytes in distributed program, including test data, etc.: 1 104 604 Distribution format: tar.gz Nature of the physical problem: The accurate computation of atomic properties and level structures requires a good understanding and implementation of the atomic shell model and, hence, a fast and reliable access to its standard quantities. Apart from various coefficients of fractional parentage and the reduced matrix elements of the unit tensors, these quantities include the so-called spin-angular coefficients, i.e. the spin-angular parts of the many-electron matrix elements of physical operators, taken in respect of a basis of symmetry-adapted subshell and configuration state functions. Method of solution: The concepts of quasispin and second quantized (creation and annihilation) operators in a spherical tensorial form are used to evaluate and calculate the spin-angular coefficients of one- and two-particle physical operators [G. Gaigalas, Lithuanian J. Phys. 39 (1999) 79, http://arXiv.org/physics/0405078; G. Gaigalas, Z. Rudzikas, C. Froese Fischer, J. Phys. B: At. Mol. Phys. 30 (1997) 3747]. Moreover, the same concepts are applied to support the computation of the coefficients of fractional grandparentage, i.e. the simultaneous de-coupling of two electrons from a single-shell configuration. All these coefficients are now implemented consistently within the framework of the Racah program [S. Fritzsche, Comput. Phys. Comm. 103 (1997) 51; G. Gaigalas, S. Fritzsche, B. Fricke, Comput. Phys. Comm. 135 (2001) 219]. Restrictions on the complexity of the problem: In the present version of the Racah program, all spin-angular coefficients are restricted to the case of a single open shell. For the symmetry-adapted subshell states of such single-shell configurations, the spin-angular coefficients can be calculated for (tensorial coupled) one-particle operators of arbitrary rank as well as for scalar two-particle operators. As previously [S. Fritzsche, Comput. Phys. Comm. 103 (1997) 51; G. Gaigalas, S. Fritzsche, B. Fricke, Comput. Phys. Comm. 135 (2001) 219], the Racah program supports all atomic shells with l ⩽ 3 in LS-coupling (i.e. s-, p-, d- and f-shells) and all subshells with j ⩽ 9 / 2 in jj-coupling, respectively. Unusual features of the program: From the very beginning, the Racah program has been designed as an interactive environment for the (symbolic) manipulation and computation of expressions from the theories of angular momentum and the atomic shell model. With the present extension of the program, we provide the user with a simple access to the coefficients of fractional grandparentage (CFGP) as well as to the spin-angular coefficients of one- and two-particle physical operators. To facilitate the specification of the tensorial form of the operators, a short but powerful notation has been introduced for the creation and annihilation operators as well as for the products of such operators as required for the development of many-body perturbation theory in a symmetry-adapted basis. All the coefficients and the matrix elements from above are equally supported for both LS- and jj-coupled operators and functions. The main procedures of the present extension are described below in Appendix B. In addition, a list of all available commands of the Racah program can be found in the file Racah-commands.ps which is distributed together with the code. Typical running time: The program replies promptly on most requests. Even large tabulations of standard quantities and pure spin-angular coefficients for one- and two-particle scalar operators in LS- and jj-coupling can be carried out in a few (tens of) seconds.