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Manuscript Title: Maple procedures for the coupling of angular momenta.
VIII. Spin-angular coefficients for single-shell configurations.
Authors: G. Gaigalas, O. Scharf, S. Fritsche
Program title: RACAH
Catalogue identifier: ADFV_v8_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 166(2005)141
Programming language: Maple, Release 8 and 9.
Computer: All computers with a valid licence for the computer algebra package Maple[1].
Operating system: Linux 8.1+.
Keywords: angular momentum theory, atomic shell model, atomic structure theory, coefficients of fractional parentage, complex atom and spectra, electron creation and annihilation operators, coefficients of fractional grandparentage, irreducible tensors, jj-coupling, LS-coupling, one- and two-particle operators, Racah algebra within three spaces (orbital, spin and quasispin space), second quantization in a coupled tensorial form, spin-angular integration, subshell state, symmetry-adapted function, unit tensor.
PACS: 3.65F, 2.90+p.
Classification: 4.1, 5.

Nature of 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 qualities 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.

Solution method:
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[2]. 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 [3].

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 [3], 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:
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 in appendix B of the manuscript. In addition, a list of all available commands of the RACAH program can be found in the file Racah-commands-08.pdf which is distributed together with the code.

Running time:
The program replies promptly to 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.

[1] Maple is a registered trademark of Waterloo Maple Inc.
[2] G. Gaigalas, Lithuanian Journal of Physics 39, 79(1999), http://arXiv.org/physics/0405078; G. Gaigalas, Z. Rudzikas and C. Froese Fischer, J. Phys. B: At. Mol. Phys. 30, 3747 (1997).
[3] S. Fritsche, Comp. Phys. Commun. 103, 51 (1997); G. Gaigalas, S. Fritsche, B. Fricke, Comp. Phys. Commun. 135, 219(2001).