dipelm_special_functions Module Reference

Orthogonal polynomials, rotation matrices, angular momentum algebra, etc. More...

Data Types
type	time

Functions/Subroutines
subroutine	tic (this)
subroutine	toc (this)
integer function	binomial (n, k)
recursive real(kind=idp) function	jacobi (n, a, b, x)
	Calculate the Jacobi polynomials using recurrence relations. More...
recursive real(idp) function	flm (l, m)
subroutine	a_legendre_p (lmax, beta, Plm)
subroutine	a_sp_harm (lmax, theta, phi, Ylm)
subroutine	a_re_sp_harm (lmax, theta, phi, Ylm)
subroutine	grid_theta_phi (steps, arange, theta_grid, phi_grid)
subroutine	grid_sp_harm (lmax, steps, arange, Ylm_grid, basis_type)
subroutine	grid_wigner_D (lmax, steps, arange, D_grid, basis_type)
real(kind=kind(1.d0)) function	wigner_small_d (beta, j, m, n)
	Compute the value of the Wigner d matrix. More...
complex(kind=idp) function	wigner_d (alpha, beta, gamma, j, m, n)
	Calculate the wigner D function. More...
real(idp) function	re_wigner_d (alpha, beta, gamma, j, m, n)
real(idp) function	azim_fn (theta, m)
subroutine	a_wigner_d (jmax, alpha, beta, gamma, Djmn)
subroutine	a_re_wigner_d (jmax, alpha, beta, gamma, Djmn)
complex(kind=idp) function	sp_harm (j, m, theta, phi)
	Calculate spherical harmonic using wigner small d. More...
real(kind=idp) function	re_sp_harm (j, m, theta, phi)
complex(rk) function	sph_basis_transform_elm (l, m, mp, basis_type)
subroutine	sph_basis_transform_matrix (U, lmax, basis_type)
integer function	lm2i (l, m)
subroutine	i2lm (i, l, m)
recursive real(rk) function	binom (n, r)
real(rk) function	cleb (j1, m1, j2, m2, j, m)
real(rk) function	threej (j1, m1, j2, m2, j, m)
function	cphaz (l, eta, iwrite)
subroutine	linspace (from, to, no_steps, array, include_end_point)

Variables
integer, parameter	rk = idp

Detailed Description

Orthogonal polynomials, rotation matrices, angular momentum algebra, etc.

Author

Alex Harvey

Date

2019

Contains:

• Spherical harmonics, Legendre polynomials and associated functions.
• Wigner 3j and Clebsch-Gordan coefficients
• Wigner D matrices (real and complex forms)
• Coulomb phase function.
• Various indexing functions and other misc. functions.

Function/Subroutine Documentation

a_legendre_p()

subroutine dipelm_special_functions::a_legendre_p	(	integer	lmax,
		real(idp)	beta,
		real(idp), dimension(:), allocatable	Plm
	)

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a_re_sp_harm()

subroutine dipelm_special_functions::a_re_sp_harm	(	integer	lmax,
		real(idp)	theta,
		real(idp)	phi,
		complex(idp), dimension(:), allocatable	Ylm
	)

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a_re_wigner_d()

subroutine dipelm_special_functions::a_re_wigner_d	(	integer, intent(in)	jmax,
		real(idp), intent(in)	alpha,
		real(idp), intent(in)	beta,
		real(idp), intent(in)	gamma,
		complex(idp), dimension(:,:), intent(inout), allocatable	Djmn
	)

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a_sp_harm()

subroutine dipelm_special_functions::a_sp_harm	(	integer	lmax,
		real(idp)	theta,
		real(idp)	phi,
		complex(idp), dimension(:), allocatable	Ylm
	)

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a_wigner_d()

subroutine dipelm_special_functions::a_wigner_d	(	integer, intent(in)	jmax,
		real(kind=idp), intent(in)	alpha,
		real(kind=idp), intent(in)	beta,
		real(kind=idp), intent(in)	gamma,
		complex(kind=idp), dimension(:,:), intent(inout), allocatable	Djmn
	)

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azim_fn()

real(idp) function dipelm_special_functions::azim_fn	(	real(kind=idp), intent(in)	theta,
		integer, intent(in)	m
	)

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binom()

recursive real(rk) function dipelm_special_functions::binom	(	integer	n,
		integer	r
	)

binomial()

integer function dipelm_special_functions::binomial	(	integer, intent(in)	n,
		integer, intent(in)	k
	)

cleb()

real(rk) function dipelm_special_functions::cleb	(	integer	j1,
		integer	m1,
		integer	j2,
		integer	m2,
		integer	j,
		integer	m
	)

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cphaz()

function dipelm_special_functions::cphaz	(		l,
			eta,
			iwrite
	)

flm()

recursive real(idp) function dipelm_special_functions::flm	(	integer	l,
		integer	m
	)

grid_sp_harm()

subroutine dipelm_special_functions::grid_sp_harm	(	integer	lmax,
		integer, dimension(:)	steps,
		real(idp), dimension(:)	arange,
		complex(idp), dimension(:,:), allocatable	Ylm_grid,
		character(len=3)	basis_type
	)

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grid_theta_phi()

subroutine dipelm_special_functions::grid_theta_phi	(	integer, dimension(:)	steps,
		real(idp), dimension(:)	arange,
		real(idp), dimension(:), allocatable	theta_grid,
		real(idp), dimension(:), allocatable	phi_grid
	)

grid_wigner_D()

subroutine dipelm_special_functions::grid_wigner_D	(	integer	lmax,
		integer, dimension(:)	steps,
		real(idp), dimension(:)	arange,
		complex(idp), dimension(:,:,:), allocatable	D_grid,
		character(len=3)	basis_type
	)

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i2lm()

subroutine dipelm_special_functions::i2lm	(	integer	i,
		integer	l,
		integer	m
	)

jacobi()

recursive real(kind=idp) function dipelm_special_functions::jacobi	(	integer, intent(in)	n,
		integer, intent(in)	a,
		integer, intent(in)	b,
		real(kind=idp), intent(in)	x
	)

Calculate the Jacobi polynomials using recurrence relations.

\begin{align} P^{(a,b)}_0(x) &= 1 \\ P^{(a,b)}_1(x) &= \frac{(2+a+b)x+(a-b)}{2} \\ P^{(a,b)}_n(x) &= (2n+a+b-1)((a^2-b^2) + (2n+a+b)(2n+a+b-2)x)P^{(a,b)}_{n-1}(x) - \frac{2(n-1+a)(n-1+b)(2n+a+b))}{(2n(n+a+b)(2n-2+a+b))}P^{(a,b)}_{n-2}(x) \end{align}

linspace()

subroutine dipelm_special_functions::linspace	(	real(idp), intent(in)	from,
		real(idp), intent(in)	to,
		integer	no_steps,
		real(idp), dimension(:), allocatable	array,
		logical, optional	include_end_point
	)

lm2i()

integer function dipelm_special_functions::lm2i	(	integer	l,
		integer	m
	)

re_sp_harm()

real(kind=idp) function dipelm_special_functions::re_sp_harm	(	integer, intent(in)	j,
		integer, intent(in)	m,
		real(kind=idp), intent(in)	theta,
		real(kind=idp), intent(in)	phi
	)

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re_wigner_d()

real(idp) function dipelm_special_functions::re_wigner_d	(	real(idp), intent(in)	alpha,
		real(idp), intent(in)	beta,
		real(idp), intent(in)	gamma,
		integer, intent(in)	j,
		integer, intent(in)	m,
		integer, intent(in)	n
	)

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sp_harm()

complex(kind=idp) function dipelm_special_functions::sp_harm	(	integer, intent(in)	j,
		integer, intent(in)	m,
		real(kind=idp), intent(in)	theta,
		real(kind=idp), intent(in)	phi
	)

Calculate spherical harmonic using wigner small d.

The relation used is:

\begin{align} Y_{jm}(\theta, \phi) &= \sqrt\frac{2j+1}{4\pi}d^j_{|m|0}(\theta) e^{im\phi} \qquad m \geq 0 \\ &= (-1)^m \sqrt\frac{2j+1}{4\pi}d^j_{|m|0}(\theta) e^{im\phi} \qquad m < 0 \\ \end{align}

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sph_basis_transform_elm()

complex(rk) function dipelm_special_functions::sph_basis_transform_elm	(	integer, intent(in)	l,
		integer, intent(in)	m,
		integer, intent(in)	mp,
		character(3)	basis_type
	)

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sph_basis_transform_matrix()

subroutine dipelm_special_functions::sph_basis_transform_matrix	(	complex(rk), dimension(:,:), intent(inout), allocatable	U,
		integer, intent(in)	lmax,
		character(3)	basis_type
	)

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threej()

real(rk) function dipelm_special_functions::threej	(	integer	j1,
		integer	m1,
		integer	j2,
		integer	m2,
		integer	j,
		integer	m
	)

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tic()

subroutine dipelm_special_functions::tic

(

class(time)

this

)

toc()

subroutine dipelm_special_functions::toc

(

class(time)

this

)

wigner_d()

complex(kind=idp) function dipelm_special_functions::wigner_d	(	real(kind=idp), intent(in)	alpha,
		real(kind=idp), intent(in)	beta,
		real(kind=idp), intent(in)	gamma,
		integer, intent(in)	j,
		integer, intent(in)	m,
		integer, intent(in)	n
	)

Calculate the wigner D function.

Rotation matrix elements for the complex spherical harmonics. They have the form:

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wigner_small_d()

real(kind=kind(1.d0)) function dipelm_special_functions::wigner_small_d	(	real(kind=idp), intent(in)	beta,
		integer, intent(in)	j,
		integer, intent(in)	m,
		integer, intent(in)	n
	)

Compute the value of the Wigner d matrix.

The explicit expression involving Jacobi Polynomials is used:

\[ d^{j}_{mn}(\beta)=(-1)^{\lambda} \binom{2j-k}{k+a}^{\frac{1}{2}} \binom{k+b}{b}^{-\frac{1}{2}} \left(\sin\frac{\beta}{2}\right)^a \left(\cos\frac{\beta}{2}\right)^b P^{(a,b)}_k(\cos\beta) \]

where \( k = \min(j+n, j-n, j+m, j-m). \)

\( a \) and \( \lambda \) depend on the form of \( k \) and are given by

\[ k = \begin{cases} j+n: & a=m-n;\quad \lambda=m-n\\ j-n: & a=n-m;\quad \lambda= 0 \\ j+m: & a=n-m;\quad \lambda= 0 \\ j-m: & a=m-n;\quad \lambda=m-n \\ \end{cases} \]

and \( b=2j-2k-a \)

See also

wigner_d

Todo:

Implement recurrence relation approach to calculating wigner d

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Variable Documentation

rk

integer, parameter dipelm_special_functions::rk = idp