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Manuscript Title: A new Fortran program for the 9-j angular momentum coefficient.
Authors: K. Srinivasa Rao, V. Rajeswari, C.B. Chiu
Program title: NINEJ
Catalogue identifier: ABLE_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 56(1989)231
Programming language: Fortran.
Computer: IBM-PC/AT.
Operating system: DOS 3.0 & DOS 4.0, VMS VERSION 4.6.
Word size: 32
Keywords: General purpose, Atomic, Molecular, Angular momentum, Nuclear structure, Nuclear reaction, Recoupling coefficient, 9j-symbol, Ls-jj transformation Coefficient, Multiple hypergeometric Series, Rotation group, Element matrix.
Classification: 4.1.

Nature of problem:
The program calculates the 9-j coefficient (also called as the ls-jj transformation coefficient), using the single sum formula or the triple- sum formula due to Jucys and Bandzaitis. This coefficient arises in the recoupling of four angular momenta, which can be coupled either using the Russell-Saunders scheme or the jj-coupling scheme. It is of fundamental importance in the evaluation of matrix elements which occur in nuclear, atomic and molecular physics calculations.

Solution method:
The simplest known form of the 9-j coefficient, due to Jucys-Bandzaitis can be identified with a formal hypergeometric series introduced by Srivastava. The Horner scheme for polynomial evaluation is then resorted to for this triple series to provide a faster method of computing the 9-j coefficient. The conventional single sum series involving the product of three 6-j coefficients (each being a single sum series, in turn) is also programmed. For the ls-jj transformation coefficient (i.e. the 9-j coefficient with b = e = 1/2) the triple sum series is always found to be advantageous. For a given set of parameters in the 9-j coefficient, the number of terms to be computed is determined for the single and the triple sum series and a criterion is given to select the formula that is most advantageous for computation.

Running time:
The execution time required on the IBM-PC/AT or the VAX-11/780 to compute a given 9-j coefficient, using either the triple sum series or the single sum over the product of three 6-j coefficients, depends mainly upon the number of terms to be summed, which in turn depends upon the values of the nine angular momenta, and the efficiency of the Fortran compiler implemented on that computer. On an AT & T PC 6300 (which runs on an 8086 at 8 Mhz supported by an 8 Mhz 8087 Math-coprocessor) we found that the use of Fortran version 4.0 instead of version 2.0 speeds up the absolute computation time by about a factor of 1.5. Due to these various factors, instead of looking at the absolute times for computing a given 9-j coefficient, which is a fraction of a second, if we focus our attention on tha advantage factor (viz. the average over a large set of data for the ratio) for computing the 9-j coefficient using the triple sum series over the single sum series, we find that an advantage factor of 2 to 4 for smaller values of the angular momenta (a,b,...,i <= 10) and an even higher advantage factor for larger values of angular momenta is possible.