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VectorCouplingSixJSymbol Method

Evaluates the 6j symbol h(l1) = { l1 l2 l3 } { l4 l5 l6 } for all allowed values of l1, with the other parameters held fixed.

Namespace: Altaxo.Calc
Assembly: AltaxoCore (in AltaxoCore.dll) Version: 4.8.3572.0 (4.8.3572.0)
Syntax
C#
Copy
public static void SixJSymbol(
	double l2,
	double l3,
	double l4,
	double l5,
	double l6,
	out double l1min,
	out double l1max,
	double[] sixcof,
	int ndim,
	out int errflag
)

Parameters

l2  Double
Parameter in the 6j symbol.
l3  Double
Third angular-momentum parameter in the 6j symbol.
l4  Double
Fourth angular-momentum parameter in the 6j symbol.
l5  Double
Fifth angular-momentum parameter in the 6j symbol.
l6  Double
Sixth angular-momentum parameter in the 6j symbol.
l1min  Double
Smallest allowable l1 in the 6j symbol.
l1max  Double
Largest allowable l1 in the 6j symbol.
sixcof  Double
Set of 6j coefficients generated by evaluating the 6j symbol for all allowed values of l1. sixcof(i) will contain h(l1min+i), for i=0,2,...,l1max-l1min.
ndim  Int32
Declared length of sixcof in the calling program.
errflag  Int32
Error flag.

errflag=0 No errors.

errflag=1l2+l3+l5+l6 or l4+l2+l6 is not an integer.

errflag=2l4, l2, l6 triangular condition not satisfied.

errflag=3l4, l5, l3 triangular condition not satisfied.

errflag=4l1max-l1min is not an integer.

errflag=5l1max is less than l1min.

errflag=6ndim is less than l1max-l1min+1.

Remarks
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) <= l6 <= l2+l4 must be satisfied; 3. abs(l4-l5) <= l3 <= 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 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. 5. SLATEC library, category C19, double precision algorithm DRC6J.F 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 Revision history (YYMMDD) 750101 DATE WRITTEN 880515 SLATEC prologue added by G. C. Nielson, NBS; parameters HUGE and TINY revised to depend on D1MACH. 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. 6. Rewriting of the SLATEC algorithm in C++ and adaption to the Matpack C++ Numerics and Graphics Library by Berndt M. Gammel in June 1997.
See Also