spherical_harmonics Module Reference

Calculate spherical harmonics. More...

Data Types
interface	clebsch_gordon
interface	dy_lm
interface	y_lm

Functions/Subroutines
subroutine, public	clebsch_gordon_deallocate ()
	...
subroutine, public	clebsch_gordon_init (l)
	...
real(kind=dp) function, public	legendre (x, l, m)
	...
real(kind=dp) function, public	dlegendre (x, l, m)
	...
subroutine, public	clebsch_gordon_coefficient (j1, m1, j2, m2, j, m, cg_coeff)
	Compute the Clebsch-Gordon coefficient C = < j1 m1 j2 m2 | J M > = < j1 j2; m1 m2 | J M >
subroutine, public	wigner_3j (j1, m1, j2, m2, j3, m3, w_3j)
	Compute the Wigner 3-j symbol / j1 j2 j3 \ \ m1 m2 m3 / using the Clebsch-Gordon coefficients.

Detailed Description

Calculate spherical harmonics.

Note

Spherical Harmonics Numerical Stability up to L=15 Accuracy > 1.E-12 up to L=15 tested Definition is consistent with orbital_transformation_matrices Clebsch-Gordon Coefficients Tested up to l=7 (i.e. L=14)

History

JGH 28-Feb-2002 : Change of sign convention (-1^m) JGH 1-Mar-2002 : Clebsch-Gordon Coefficients

• Clebsch-Gordon coefficients and Wigner 3-j symbols added as generic routines using the standard normalization (19.09.2022, MK)

Author

JGH 6-Oct-2000, MK

Function/Subroutine Documentation

clebsch_gordon_deallocate()

subroutine, public spherical_harmonics::clebsch_gordon_deallocate

...

Definition at line 219 of file spherical_harmonics.F.

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clebsch_gordon_init()

subroutine, public spherical_harmonics::clebsch_gordon_init

(

integer, intent(in)

l

)

...

Parameters

l

...

Definition at line 237 of file spherical_harmonics.F.

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legendre()

real(kind=dp) function, public spherical_harmonics::legendre	(	real(kind=dp), intent(in)	x,
		integer, intent(in)	l,
		integer, intent(in)	m
	)

...

Parameters

x	...
l	...
m	...

Returns

...

Definition at line 1192 of file spherical_harmonics.F.

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dlegendre()

real(kind=dp) function, public spherical_harmonics::dlegendre	(	real(kind=dp), intent(in)	x,
		integer, intent(in)	l,
		integer, intent(in)	m
	)

...

Parameters

x	...
l	...
m	...

Returns

...

Definition at line 1336 of file spherical_harmonics.F.

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clebsch_gordon_coefficient()

subroutine, public spherical_harmonics::clebsch_gordon_coefficient	(	real(kind=dp), intent(in)	j1,
		real(kind=dp), intent(in)	m1,
		real(kind=dp), intent(in)	j2,
		real(kind=dp), intent(in)	m2,
		real(kind=dp), intent(in)	j,
		real(kind=dp), intent(in)	m,
		real(kind=dp), intent(out)	cg_coeff
	)

Compute the Clebsch-Gordon coefficient C = < j1 m1 j2 m2 | J M > = < j1 j2; m1 m2 | J M >

Parameters

j1	Angular momentum quantum number of the first state | j1 m1 >
m1	Magnetic quantum number of the first first state | j1 m1 >
j2	Angular momentum quantum number of the second state | j2 m2 >
m2	Magnetic quantum number of the second state | j2 m2 >
J	Angular momentum quantum number of the coupled state | J M >
M	Magnetic quantum number of the coupled state | J M >
CG_coeff	Clebsch-Gordon coefficient C^{JM}_{j1 m1 j2 m2}

Author

Matthias Krack (16.09.2022, based on a program by D. G. Simpson)

Note

Generic routine allowing also for fractional arguments. It should return CG coefficients consistent with the standard definition and normalization, e.g. of Wolfram Mathematica. The input parameters have to be integer or half-integer.

Definition at line 1799 of file spherical_harmonics.F.

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wigner_3j()

subroutine, public spherical_harmonics::wigner_3j	(	real(kind=dp), intent(in)	j1,
		real(kind=dp), intent(in)	m1,
		real(kind=dp), intent(in)	j2,
		real(kind=dp), intent(in)	m2,
		real(kind=dp), intent(in)	j3,
		real(kind=dp), intent(in)	m3,
		real(kind=dp), intent(out)	w_3j
	)

Compute the Wigner 3-j symbol / j1 j2 j3 \ \ m1 m2 m3 / using the Clebsch-Gordon coefficients.

Parameters

j1	Angular momentum quantum number of the first state | j1 m1 >
m1	Magnetic quantum number of the first first state | j1 m1 >
j2	Angular momentum quantum number of the second state | j2 m2 >
m2	Magnetic quantum number of the second state | j2 m2 >
j3	Angular momentum quantum number of the third state | j3 m3 >
m3	Magnetic quantum number of the third state | j3 m3 >
W_3j	Wigner 3-j symbol

Author

Matthias Krack (16.09.2022, MK)

Definition at line 1887 of file spherical_harmonics.F.

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