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VectorCouplingThreeJSymbolJ Method

Evaluates the Wigner 3j symbol f(l1) = ( l1 l2 l3 ) ( -m2-m3 m2 m3 ) 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 ThreeJSymbolJ(
	double l2,
	double l3,
	double m2,
	double m3,
	out double l1min,
	out double l1max,
	double[] thrcof,
	int ndim,
	out int errflag
)

Parameters

l2  Double
Parameter in the 3j symbol.
l3  Double
Third angular-momentum parameter in the 3j symbol.
m2  Double
Magnetic quantum number associated with l2.
m3  Double
Magnetic quantum number associated with l3.
l1min  Double
Smallest allowable l1 in the 3j symbol.
l1max  Double
Largest allowable l1 in the 3j symbol.
thrcof  Double
Set of 3j coefficients generated by evaluating the 3j symbol for all allowed values of l1. thrcof(i) will contain f(l1min+i), for i = 0, 2, ... , l1max+l1min.
ndim  Int32
Declared length of thrcof in the calling program.
errflag  Int32

errflag=0 No errors.

errflag=1 Either l2 < abs(m2) or l3 < abs(m3).

errflag=2 Either l2+abs(m2) or l3+abs(m3) is non-integer.

errflag=3l1max-l1min is not an integer.

errflag=4l1max is less than l1min.

errflag=5ndim is less than l1max-l1min+1.

Remarks
Description: ------------ Although conventionally the parameters of the vector addition coefficients satisfy certain restrictions, such as being integers or integers plus 1/2, the restrictions imposed on input to this subroutine are somewhat weaker. See, for example, Section 27.9 of Abramowitz and Stegun or Appendix C of Volume II of A. Messiah. The restrictions imposed by this subroutine are 1. l2 >= abs(m2) and l3 >= abs(m3) 2. l2+abs(m2) and l3+abs(m3) must be integers 3. l1max-l1min must be a non-negative integer, where l1max=l2+l3 and l1min=max(abs(l2-l3),abs(m2+m3)) If the conventional restrictions are satisfied, then these restrictions are also met. The user should be cautious in using input parameters that do not satisfy the conventional restrictions. For example, the subroutine produces values of f(L1) = ( l1 2.5 5.8) (-0.3 1.5 -1.2) for l1=3.3,4.3,...,8.3 but none of the symmetry properties of the 3j symbol, set forth on page 1056 of Messiah, is satisfied. The subroutine generates f(l1min), f(l1min+1), ..., f(l1max) where l1min and l1max are defined above. The sequence f(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 3j 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. Abramowitz, M., and Stegun, I. A., Eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, NBS Applied Mathematics Series 55, June 1964 and subsequent printings. 2. Messiah, Albert., Quantum Mechanics, Volume II, North-Holland Publishing Company, 1963. 3. 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. 4. 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. 5. Schulten, Klaus and Gordon, Roy G., Recursive evaluation of 3j and 6j coefficients, Computer Phys Comm, v 11, 1976, pp. 269-278. 6. SLATEC library, category C19, double precision algorithm DRC3JJ.F Keywords: 3j coefficients, 3j 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 errflag 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 THRCOF expanded. These changes were done by D. W. Lozier. 7. 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