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Manuscript Title: A program to evaluate closed diagrams algebraically for angular momentum coupled product operators.
Authors: B.D. Chang, S.S.M. Wong
Catalogue identifier: AAAM_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 20(1980)191
Programming language: Fortran.
Computer: IBM 360/65.
Operating system: HASP/OSMVT, MVS/JES2, VAX/VMS.
RAM: 31K words
Word size: 32
Peripherals: disc.
Keywords: General purpose, Rotation group, Expressions algebraic, Angular momentum, Recoupling, Contraction, Coupled product, Excitation operators, Hamiltonian, Many particle matrix Elements, Nuclear structure, Racah recoupling, Scalar and Configuration traces, Second quantised Spherical tensor of Half integer rank, Spectral distribution, Statistical spectroscopy, Wick's theorem, 6j-symbol sum rules, 9j-symbol sum rules.
Classification: 4.1.

Subprograms used:
Cat Id Title Reference

Nature of problem:
The many particle trace of a product operator, expressed in terms of angular-momentum coupled spherical tensor creation and annihilation operators, can be evaluated as the sum of the different ways or diagrams to contract all the single particle operators. In the coupled representation, the process of contraction involves recouplings of angular momenta and this can be tedious. The program is constructed to perform algebraically the contractions and the associated angular momentum recouplings. The output are (algebraic) expressions which can be used either as analytical results or as input to a separate program, CONTRACTION-COMPILER, constructed to write a Fortran code to carry out the numerical calculations. The primary motivation of the project is derived from the need of scalar and configuration traces in nuclear structure problems using spectral distribution methods.

Solution method:
The form of each basic operator is first examined to see if, by rewriting it in different possible forms, the anticipated number of Racah recouplings can be minimized. The product operator is then recoupled into a standard angular momentum structure, the fan-shaped pattern. The pattern of contraction is read in from an input file produced, for example, by the CONTRACT-BASIC-DIAGRAM. Before a pair of single-particle operators is contracted, they are brought into adjacent positions using standard Racah angular momentum recoupling rules. A list of 6-J symbols. Kronecker deltas, phase and statistical weigth factors is kept and updated at every stage. Whenever a variable becomes a purely dummy index of summation, the entire list of 6J-symbols is scanned to see whether sum rules can be used to reduce the number of 6J-symbols and summation variables to a minimum. When all the single particle operators are contracted, the remaining 6J-symbols are scanned again for possible reduction by sum rules including those involving 9J- symbols.

The restrictions are basically the same as those imposed in CONTRACTION- BASIC-DIAGRAM: the number of basic operators should be no more than 20, the total number of single-particle operators no more than 40. In addition, the total number of different variables, including those generated intermediately by the program, Kronecker deltas and 6J- symbols must not exceed 100 at any time and the number of 9J-symbols no more than 20. These restrictions can be easily removed by extending the dimensions declared in the program. For a basic operator, the maximum number of creation and annihilation operators is restricted to be no more than two of each kind and, in the case of a basic operator made of two creation and two annihilation operators, the final angular-momentum rank of the basic operator must be a scalar.