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Manuscript Title: Symbolic tools for the computation of Moshinsky brackets and nuclear matrix elements
Authors: D. Ursescu, M. Tomasell, T. Kuehl, S. Fritzsche
Program title: Fermi
Catalogue identifier: ADVO_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 173(2005)140
Programming language: Maple 8 and 9.5 from MapleSoft division of Waterloo Maple Inc.
Computer: All computers with a licence for the computer algebra package Maple [1].
Operating system: WindowsXP, Linux 2.4.
Keywords: angular momentum, center-of-well coordinates, harmonic potential well, jj-coupling, LSJ-coupling, Moshinsky bracket, nuclear matrix element, nuclear shell model, nuclear two-particle states, relative and center-of-mass coordinates, transformation bracket.
PACS: 21.60.-n.
Classification: 5, 17.19.

Nature of problem:
In order to perform calculations within the nuclear shell model (NSM), a quick and reliable access to the nuclear matrix elements is required. These matrix elements, which arise from various types of forces among the nucleons, can be calculated using Moshinsky's transformation brackets between relative and center-of-mass coordinates [2] and by the proper use of the nuclear states in different coupling notations.

Solution method:
Moshinsky's transformation brackets as well as two-nucleon matrix elements are provided within the framework of Maple. The transformation brackets are evaluated recursively for a given number of shells and utilized for the computation of the two-particle matrix elements for different coupling schemes and interactions. Moreover, a simple notation has been introduced to handle the two--particle nuclear states in ll-, LSJ-, and jj-coupling, both in the center-of-well and the relative and center-of-mass coordinates.

The program supports in principle an arbitrary number of shell states with the only limitation given by the computer resources themselves. Typically, the time requirements for the recursive computation of the Moshinsky brackets and matrix elements increase rapidly with the number of the allowed shell states but can be reduced significantly by the pre-calculation of the transformation brackets.

Unusual features:
Moshinsky brackets are computed and provided in either numeric, algebraic or some symbolic form. In addition, the two-particle matrix elements are calculated for a scalar potential, spin-orbit coupling and tensorial forces, both in floating-point and algebraic notation. All two-particle matrix elements are expressed in terms of the Talmi integrals but can be evaluated also explicitly for several predefined types of the interaction. To simplify the handling of the program, a short but very powerful notation has been introduced which help the user to deal with the two-particle states in various coupling notations. The main commands of the current version of the program are described in detail in Appendix B of the manuscript.

Running time:
The computation of all Moshinsky brackets in floating-point notation, up to ρ=6, takes about 5 seconds at a 2.26 GHz Intel Pentium IIII processor with 512 Mb; in algebraic form, the same computations take about 13 seconds. Similarly, the computation of these brackets up to ρ=10 requires in numeric and algebraic form about 5 and 15 minutes, respectively. Once these coefficients have been stored, however, the program replies rather promptly on most further requests.

[1] Maple is a registered trademark of Waterloo Maple Inc., produced by MapleSoft division of Waterloo Maple Inc.
[2] T.A. Brody and M.Moshinsky, Tables of transformation brackets}, Monografos de Instituto de Fisica, Universidad Nacional Autonome de Mexico (1960).