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Manuscript Title: A program to calculate fractional parentage coefficients for jj- coupling states with equivalent particles.
Authors: T. Kagawa
Program title: JJFPC
Catalogue identifier: ACJY_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 72(1992)165
Programming language: Fortran.
Computer: FACOM M760/6.
Operating system: OS IV/MSP (M-Series).
RAM: 668K words
Word size: 32
Keywords: General purpose, Rotation group.
Classification: 4.1.

Nature of problem:
This Program calculates the n - 1 -> n fractional parentage coefficients (FPC's) for the jj-coupling states with equivalent particles in a j shell. The FPC's are needed in the evaluation of matrix elements for one- or two-body operators corresponding to various physical quantities in atomic or nuclear systems combined with the recoupling coefficients and the reduced matrix elements of these operators. Use of the FPC's makes one possible to remove a tedious treatment of the antisymmetrization of a product of one-particle functions for a many-particle wave function when evaluating matrix elements for one- or two-body operators.

Solution method:
A computer program to calculate the jj-coupling FPC's has already been catalogued in the CPC library, (I.P.Grant, Comput. Phys. Commun. 4(1972)377, 14(1978)311). Here we use the branching rules in the group theory to generate the jj-coupling states for equivalent particles in the program. In order to uniquely specify all the states arising from the j**n configuration for equivalent particles in the j shell, an ordering label lambda in addition to the seniority number v and the total angular momentum J is introduced. As a many-particle wave function constructed from a product of one-particle functions for the j**n configuration forms an irreducible representation of the unitary group U(2j+1) of the 2j+1 dimension, all the states |lambda vJ > can be generated by using the branching rules, where an irreducible representation of U(2j+1) id decomposed into irreducible representations of its subgroups such as the three-dimensional rotational group R(3). An algorithm in writing a program for the branching rules is based on Littlewood's algebra of plethysm of the Schur functions for the symmetric group. After number of each state having v and J for the j**n configuration is obtained, the FPC's are calculated with the recurrence formula for them starting from the FPC's for the j**3 configuration.

When the FPC's for states specified by the same J and v in the seniority scheme are not determined uniquely due to the degeneracy of the states, the procedure of the Gram-Schmidt orthonalization is used to obtain a set of orthonormal FPC's. As an unitary transformation for a set of the FPC's for degenerate states can lead to a different set of FPC's, numerical values for FPC's obtained with our program are not necessarily consistent with those obtained by other people, but there is no problem in practical use. When number of states for the j**n and/or j**n-1 configurations exceeds a hundred, one cannot obtain correct values of FPC's because of the limitation of a storage prepared in the program for the states obtained.