VectorCoupling Type
Contains vector coupling functions.
Constructors
| Constructor | Description |
Full Usage:
VectorCoupling()
|
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Static members
| Static member | Description |
Full Usage:
VectorCoupling.ClebschGordan(l1, m1, l2, m2, l3, m3, errflag)
Parameters:
float
-
Parameter in 3j symbol.
m1 : float
-
Parameter in 3j symbol.
l2 : float
-
Parameter in 3j symbol.
m2 : float
-
Parameter in 3j symbol.
l3 : float
-
Parameter in 3j symbol.
m3 : float
-
Parameter in 3j symbol.
errflag : byref<int>
-
Error flag.
errflag=0 No errors. errflag=1 Either l1 < abs(m1) or l1+abs(m1) non-integer. errflag=2 abs(l1-l2)<= l3 <= l1+l2 not satisfied. errflag=3 l1+l2+l3 not an integer. errflag=4 m2max-m2min not an integer. errflag=5 m2max less than m2min. errflag=6 ndim less than m2max-m2min+1. errflag=7 m1+m2-m3 is not zero. Returns: float
The value of the Clebsch-Gordan coefficient.
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Calculate Clebsch-Gordan coefficient using the relation to the Wigner 3-j symbol:
l1-l2+m3 1/2 ( l1 l2 l3 )
(l1 m1 l2 m2 | l3 m3) = (-1) (2*l3+1) ( m1 m2 -m3 )
References:
-----------
1. See routines in "threejj.cc" and "threejm.cc" for references about
the calculation of the Wigner 3-j symbols.
2. C++ Implementation for the Matpack C++ Numerics and Graphics Library
by Berndt M. Gammel in June 1997.
Note:
-----
Whenever you have to calculate a series of Clebsch-Gordan coefficients for
a range of l-values or m-values you should probably use the 3-j symbol
routines. These calculate the 3-j symbols iteratively for a series of
l-values or m-values and are therefore much more efficient. Use the relation
between Clebsch-Gordan coefficients and Wigner 3-j symbols as given above.
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Full Usage:
VectorCoupling.SixJSymbol(l2, l3, l4, l5, l6, l1min, l1max, sixcof, ndim, errflag)
Parameters:
float
-
Parameter in 6j symbol.
l3 : float
-
Parameter in 6j symbol.
l4 : float
-
Parameter in 6j symbol.
l5 : float
-
Parameter in 6j symbol.
l6 : float
-
Parameter in 6j symbol.
l1min : byref<float>
-
Smallest allowable l1 in 6j symbol.
l1max : byref<float>
-
Largest allowable l1 in 6j symbol.
sixcof : float[]
-
Set of 6j coefficients generated by evaluating the
6j symbol for all allowed values of l1. sixcof(i)
will contain h(l1min+i), i=0,2,...,l1max-l1min.
ndim : int
-
Declared length of sixcof in calling program.
errflag : byref<int>
-
Error flag.
errflag=0 no errors. errflag=1 l2+l3+l5+l6 or l4+l2+l6 not an integer. errflag=2 l4, l2, l6 triangular condition not satisfied. errflag=3 l4, l5, l3 triangular condition not satisfied. errflag=4 l1max-l1min not an integer. errflag=5 l1max less than l1min. errflag=6 ndim less than l1max-l1min+1. |
Evaluate the 6j symbol
h(l1) = { l1 l2 l3 }
{ l4 l5 l6 }
for all allowed values of l1, the other parameters being held fixed.
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
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. Rewritting of the SLATEX algorithm in C++ and adaption to the
Matpack C++ Numerics and Graphics Library by Berndt M. Gammel
in June 1997.
|
Full Usage:
VectorCoupling.ThreeJSymbolJ(l2, l3, m2, m3, l1min, l1max, thrcof, ndim, errflag)
Parameters:
float
-
Parameter in 3j symbol.
l3 : float
-
Parameter in 3j symbol.
m2 : float
-
Parameter in 3j symbol.
m3 : float
-
Parameter in 3j symbol.
l1min : byref<float>
-
Smallest allowable l1 in 3j symbol.
l1max : byref<float>
-
Largest allowable l1 in 3j symbol.
thrcof : float[]
-
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 : int
-
Declared length of thrcof in calling program.
errflag : byref<int>
-
errflag=0 No errors. errflag=1 Either l2 < abs(m2) or l3 < abs(m3). errflag=2 Either l2+abs(m2) or l3+abs(m3) non-integer. errflag=3 l1max-l1min not an integer. errflag=4 l1max less than l1min. errflag=5 ndim less than l1max-l1min+1. |
Evaluate the Wigner 3j symbol
f(l1) = ( l1 l2 l3 )
( -m2-m3 m2 m3 )
for all allowed values of l1, the other parameters being held fixed.
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
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. Rewritting of the SLATEX algorithm in C++ and adaption to the
Matpack C++ Numerics and Graphics Library by Berndt M. Gammel
in June 1997.
|
Full Usage:
VectorCoupling.ThreeJSymbolM(l1, l2, l3, m1, m2min, m2max, thrcof, ndim, errflag)
Parameters:
float
-
Parameter in 3j symbol.
l2 : float
-
Parameter in 3j symbol.
l3 : float
-
Parameter in 3j symbol.
m1 : float
-
Parameter in 3j symbol.
m2min : byref<float>
-
Smallest allowable m2 in 3j symbol.
m2max : byref<float>
-
Largest allowable m2 in 3j symbol.
thrcof : float[]
-
Set of 3j coefficients generated by evaluating the
3j symbol for all allowed values of m2. thrcof(i)
will contain g(m2min+i), i=0,2,...,m2max-m2min.
ndim : int
-
Declared length of thrcof in calling program.
errflag : byref<int>
-
Error flag.
errflag=0 No errors. errflag=1 Either l1 < abs(m1) or l1+abs(m1) non-integer. errflag=2 abs(l1-l2)<= l3 <= l1+l2 not satisfied. errflag=3 l1+l2+l3 not an integer. errflag=4 m2max-m2min not an integer. errflag=5 m2max less than m2min. errflag=6 ndim less than m2max-m2min+1. |
Evaluate the Wigner 3j symbol
g(m2) = ( l1 l2 l3 )
( m1 m2 -m1-m2 )
for all allowed values of m2, the other parameters being held fixed.
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. l1 >= abs(m1) and l1+abs(m1) must be an integer
2. abs(l1-l2) <= l3 <= l1+l2
3. l1+l2+l3 must be an integer
4. m2max-m2min must be an integer, where
m2max=min(l2,l3-m1) and m2min=max(-l2,-l3-m1)
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
the subroutine produces values of
g(m2) = (0.75 1.50 1.75 )
(0.25 m2 -0.25-m2)
for m2=-1.5,-0.5,0.5,1.5 but none of the symmetry properties of the
3j symbol, set forth on page 1056 of Messiah, is satisfied.
The subroutine generates g(m2min), g(m2min+1), ..., g(m2max)
where m2min and m2max are defined above. The sequence g(m2) 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 DRC3JM.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; MMATCH (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 THRCOF expanded. These changes were done by
D. W. Lozier.
7. Rewritting of the SLATEX algorithm in C++ and adaption to the
Matpack C++ Numerics and Graphics Library by Berndt M. Gammel
in June 1997.
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