RC6J(3)

SYNOPSIS
SUBROUTINE RC6J (L2, L3, L4, L5, L6, L1MIN, L1MAX, SIXCOF, NDIM, + IER)

NAME
rc6j - Evaluate the 6j symbol h(L1) = {L1 L2 L3} {L4 L5 L6} for all allowed values of L1, the other parameters being held fixed.

LIBRARY
SLATEC

CATEGORY
C19

TYPE
SINGLE PRECISION (RC6J-S, DRC6J-D)

KEYWORDS
6J COEFFICIENTS, 6J SYMBOLS, CLEBSCH-GORDAN COEFFICIENTS, RACAH COEFFICIENTS, VECTOR ADDITION COEFFICIENTS, WIGNER COEFFICIENTS

AUTHOR
Gordon, R. G., Harvard University Schulten, K., Max Planck Institute

DESCRIPTION
*Usage: REAL L2, L3, L4, L5, L6, L1MIN, L1MAX, SIXCOF(NDIM) INTEGER NDIM, IER CALL RC6J(L2, L3, L4, L5, L6, L1MIN, L1MAX, SIXCOF, NDIM, IER) *Arguments: L2 :IN Parameter in 6j symbol. L3 :IN Parameter in 6j symbol. L4 :IN Parameter in 6j symbol. L5 :IN Parameter in 6j symbol. L6 :IN Parameter in 6j symbol. L1MIN :OUT Smallest allowable L1 in 6j symbol. L1MAX :OUT Largest allowable L1 in 6j symbol. SIXCOF :OUT Set of 6j coefficients generated by evaluating the 6j symbol for all allowed values of L1. SIXCOF(I) will contain h(L1MIN+I-1), I=1,2,...,L1MAX-L1MIN+1. NDIM :IN Declared length of SIXCOF in calling program. IER :OUT Error flag. IER=0 No errors. IER=1 L2+L3+L5+L6 or L4+L2+L6 not an integer. IER=2 L4, L2, L6 triangular condition not satisfied. IER=3 L4, L5, L3 triangular condition not satisfied. IER=4 L1MAX-L1MIN not an integer. IER=5 L1MAX less than L1MIN. IER=6 NDIM less than L1MAX-L1MIN+1. *Description: The definition and properties of 6j symbols can be found, for example, in Appendix C of Volume II of A. Messiah. Although the parameters of the vector addition coefficients satisfy certain conventional restrictions, the restriction that they be non-negative integers or non-negative integers plus 1/2 is not imposed on input to this subroutine. The restrictions imposed are 1. L2+L3+L5+L6 and L2+L4+L6 must be integers; 2. ABS(L2-L4).LE.L6.LE.L2+L4 must be satisfied; 3. ABS(L4-L5).LE.L3.LE.L4+L5 must be satisfied; 4. L1MAX-L1MIN must be a non-negative integer, where L1MAX=MIN(L2+L3,L5+L6) and L1MIN=MAX(ABS(L2-L3),ABS(L5-L6)). If all the conventional restrictions are satisfied, then these restrictions are met. Conversely, if input to this subroutine meets all of these restrictions and the conventional restriction stated above, then all the conventional restrictions are satisfied. The user should be cautious in using input parameters that do not satisfy the conventional restrictions. For example, the the subroutine produces values of h(L1) = { L1 2/3 1 } {2/3 2/3 2/3} for L1=1/3 and 4/3 but none of the symmetry properties of the 6j symbol, set forth on pages 1063 and 1064 of Messiah, is satisfied. The subroutine generates h(L1MIN), h(L1MIN+1), ..., h(L1MAX) where L1MIN and L1MAX are defined above. The sequence h(L1) is generated by a three-term recurrence algorithm with scaling to control overflow. Both backward and forward recurrence are used to maintain numerical stability. The two recurrence sequences are matched at an interior point and are normalized from the unitary property of 6j coefficients and Wigner's phase convention. The algorithm is suited to applications in which large quantum numbers arise, such as in molecular dynamics.

REFERENCES
1. Messiah, Albert., Quantum Mechanics, Volume II, North-Holland Publishing Company, 1963. 2. Schulten, Klaus and Gordon, Roy G., Exact recursive evaluation of 3j and 6j coefficients for quantum- mechanical coupling of angular momenta, J Math Phys, v 16, no. 10, October 1975, pp. 1961-1970. 3. Schulten, Klaus and Gordon, Roy G., Semiclassical approximations to 3j and 6j coefficients for quantum-mechanical coupling of angular momenta, J Math Phys, v 16, no. 10, October 1975, pp. 1971-1988. 4. Schulten, Klaus and Gordon, Roy G., Recursive evaluation of 3j and 6j coefficients, Computer Phys Comm, v 11, 1976, pp. 269-278.

ROUTINES CALLED
R1MACH, XERMSG

REVISION HISTORY (YYMMDD)
750101 DATE WRITTEN
880515 SLATEC prologue added by G. C. Nielson, NBS; parameters HUGE and TINY revised to depend on R1MACH.
891229 Prologue description rewritten; other prologue sections revised; LMATCH (location of match point for recurrences) removed from argument list; argument IER changed to serve only as an error flag (previously, in cases without error, it returned the number of scalings); number of error codes increased to provide more precise error information; program comments revised; SLATEC error handler calls introduced to enable printing of error messages to meet SLATEC standards. These changes were done by D. W. Lozier, M. A. McClain and J. M. Smith of the National Institute of Standards and Technology, formerly NBS.
910415 Mixed type expressions eliminated; variable C1 initialized; description of SIXCOF expanded. These changes were done by D. W. Lozier.