Spherical harmonics transformations and related oprtations. More...

#include <sht.hpp>

Public Member Functions
	SHT (sddk::device_t pu__, int lmax__, int mesh_type__=0)
	Default constructor. More...

void	check () const
	Check the transformations. More...

template<typename T >
void	backward_transform (int ld, T const flm, int nr, int lmmax, T ftp) const
	Perform a backward transformation from spherical harmonics to spherical coordinates. More...

template<typename T >
void	forward_transform (T const ftp, int nr, int lmmax, int ld, T flm) const
	Perform a forward transformation from spherical coordinates to spherical harmonics. More...

void	uniform_coverage ()

auto	ylm_backward (int lm, int itp) const

auto	rlm_backward (int lm, int itp) const

auto	coord (int x, int itp) const

auto	coord (int idx__) const

auto	theta (int idx__) const

auto	phi (int idx__) const

auto	weight (int idx__) const

auto	num_points () const

auto	lmax () const

auto	lmmax () const

template<>
void	backward_transform (int ld, double const flm, int nr, int lmmax, double ftp) const

template<>
void	forward_transform (double const ftp, int nr, int lmmax, int ld, double flm) const

Static Public Member Functions
static void	convert (int lmax__, double const f_rlm__, std::complex< double > f_ylm__)
	Convert form Rlm to Ylm representation. More...

static void	convert (int lmax__, std::complex< double > const f_ylm__, double f_rlm__)
	Convert from Ylm to Rlm representation. More...

static std::complex< double >	ylm_dot_rlm (int l, int m1, int m2)
	Compute element of the transformation matrix from complex to real spherical harmonics. More...

static std::complex< double >	rlm_dot_ylm (int l, int m1, int m2)

static double	gaunt_yyy (int l1, int l2, int l3, int m1, int m2, int m3)
	Gaunt coefficent of three complex spherical harmonics. More...

static double	gaunt_rrr (int l1, int l2, int l3, int m1, int m2, int m3)
	Gaunt coefficent of three real spherical harmonics. More...

static double	gaunt_rlm_ylm_rlm (int l1, int l2, int l3, int m1, int m2, int m3)
	Gaunt coefficent of two real spherical harmonics with a complex one. More...

static std::complex< double >	gaunt_hybrid (int l1, int l2, int l3, int m1, int m2, int m3)
	Gaunt coefficent of two complex and one real spherical harmonics. More...

static double	clebsch_gordan (int l1, int l2, int l3, int m1, int m2, int m3)
	Return Clebsch-Gordan coefficient. More...

Private Attributes
sddk::device_t	pu_
	Type of processing unit. More...

int	lmax_
	Maximum \( \ell \) of spherical harmonics. More...

int	lmmax_
	Maximum number of \( \ell, m \) components. More...

int	num_points_
	Number of real-space \( (\theta, \phi) \) points on the sphere. More...

sddk::mdarray< double, 2 >	coord_
	Cartesian coordinates of points (normalized to 1). More...

sddk::mdarray< double, 2 >	tp_
	\( (\theta, \phi) \) angles of points. More...

std::vector< double >	w_
	Point weights. More...

sddk::mdarray< std::complex< double >, 2 >	ylm_backward_
	Backward transformation from Ylm to spherical coordinates. More...

sddk::mdarray< std::complex< double >, 2 >	ylm_forward_
	Forward transformation from spherical coordinates to Ylm. More...

sddk::mdarray< double, 2 >	rlm_backward_
	Backward transformation from Rlm to spherical coordinates. More...

sddk::mdarray< double, 2 >	rlm_forward_
	Forward transformation from spherical coordinates to Rlm. More...

int	mesh_type_ {0}
	Type of spherical grid (0: Lebedev-Laikov, 1: uniform). More...

Detailed Description

Spherical harmonics transformations and related oprtations.

This class is responsible for the generation of complex and real spherical harmonics, generation of transformation matrices, transformation between spectral and real-space representations, generation of Gaunt and Clebsch-Gordan coefficients and calculation of spherical harmonic derivatives

Definition at line 80 of file sht.hpp.

Constructor & Destructor Documentation

◆ SHT()

sirius::SHT::SHT	(	sddk::device_t	pu__,
		int	lmax__,
		int	mesh_type__ = `0`
	)

inline

Default constructor.

Definition at line 121 of file sht.hpp.

Member Function Documentation

◆ check()

void sirius::SHT::check ( ) const

Check the transformations.

Definition at line 283 of file sht.cpp.

◆ backward_transform() [1/2]

template<typename T >

void sirius::SHT::backward_transform	(	int	ld,
		T const *	flm,
		int	nr,
		int	lmmax,
		T *	ftp
	)		const

Perform a backward transformation from spherical harmonics to spherical coordinates.

\[ f(\theta, \phi, r) = \sum_{\ell m} f_{\ell m}(r) Y_{\ell m}(\theta, \phi) \]

Parameters

[in]	ld	Size of leading dimension of flm.
[in]	flm	Raw pointer to \( f_{\ell m}(r) \).
[in]	nr	Number of radial points.
[in]	lmmax	Maximum number of lm- harmonics to take into sum.
[out]	ftp	Raw pointer to \( f(\theta, \phi, r) \).

◆ forward_transform() [1/2]

template<typename T >

void sirius::SHT::forward_transform	(	T const *	ftp,
		int	nr,
		int	lmmax,
		int	ld,
		T *	flm
	)		const

Perform a forward transformation from spherical coordinates to spherical harmonics.

\[ f_{\ell m}(r) = \iint f(\theta, \phi, r) Y_{\ell m}^{*}(\theta, \phi) \sin \theta d\phi d\theta = \sum_{i} f(\theta_i, \phi_i, r) Y_{\ell m}^{*}(\theta_i, \phi_i) w_i \]

Parameters

[in]	ftp	Raw pointer to \( f(\theta, \phi, r) \).
[in]	nr	Number of radial points.
[in]	lmmax	Maximum number of lm- coefficients to generate.
[in]	ld	Size of leading dimension of flm.
[out]	flm	Raw pointer to \( f_{\ell m}(r) \).

◆ convert() [1/2]

static void sirius::SHT::convert	(	int	lmax__,
		double const *	f_rlm__,
		std::complex< double > *	f_ylm__
	)

inlinestatic

Convert form Rlm to Ylm representation.

Definition at line 264 of file sht.hpp.

◆ convert() [2/2]

static void sirius::SHT::convert	(	int	lmax__,
		std::complex< double > const *	f_ylm__,
		double *	f_rlm__
	)

inlinestatic

Convert from Ylm to Rlm representation.

Definition at line 281 of file sht.hpp.

◆ ylm_dot_rlm()

static std::complex< double > sirius::SHT::ylm_dot_rlm	(	int	l,
		int	m1,
		int	m2
	)

inlinestatic

Compute element of the transformation matrix from complex to real spherical harmonics.

Real spherical harmonic can be written as a linear combination of complex harmonics:

\[ R_{\ell m}(\theta, \phi) = \sum_{m'} a^{\ell}_{m' m}Y_{\ell m'}(\theta, \phi) \]

where

\[ a^{\ell}_{m' m} = \langle Y_{\ell m'} | R_{\ell m} \rangle \]

which gives the name to this function.

Transformation from real to complex spherical harmonics is conjugate transpose:

\[ Y_{\ell m}(\theta, \phi) = \sum_{m'} a^{\ell*}_{m m'}R_{\ell m'}(\theta, \phi) \]

Mathematica code:

b[m1_, m2_] :=
 If[m1 == 0, 1,
 If[m1 < 0 && m2 < 0, -I/Sqrt[2],
 If[m1 > 0 && m2 < 0, (-1)^m1*I/Sqrt[2],
 If[m1 < 0 && m2 > 0, (-1)^m2/Sqrt[2],
 If[m1 > 0 && m2 > 0, 1/Sqrt[2]]]]]]

a[m1_, m2_] := If[Abs[m1] == Abs[m2], b[m1, m2], 0]

Rlm[l_, m_, t_, p_] := Sum[a[m1, m]*SphericalHarmonicY[l, m1, t, p], {m1, -l, l}]

Definition at line 363 of file sht.hpp.

◆ rlm_dot_ylm()

static std::complex< double > sirius::SHT::rlm_dot_ylm	(	int	l,
		int	m1,
		int	m2
	)

inlinestatic

Definition at line 392 of file sht.hpp.

◆ gaunt_yyy()

static double sirius::SHT::gaunt_yyy	(	int	l1,
		int	l2,
		int	l3,
		int	m1,
		int	m2,
		int	m3
	)

inlinestatic

Gaunt coefficent of three complex spherical harmonics.

\[ \langle Y_{\ell_1 m_1} | Y_{\ell_2 m_2} | Y_{\ell_3 m_3} \rangle \]

Definition at line 403 of file sht.hpp.

◆ gaunt_rrr()

static double sirius::SHT::gaunt_rrr	(	int	l1,
		int	l2,
		int	l3,
		int	m1,
		int	m2,
		int	m3
	)

inlinestatic

Gaunt coefficent of three real spherical harmonics.

\[ \langle R_{\ell_1 m_1} | R_{\ell_2 m_2} | R_{\ell_3 m_3} \rangle \]

Definition at line 423 of file sht.hpp.

◆ gaunt_rlm_ylm_rlm()

static double sirius::SHT::gaunt_rlm_ylm_rlm	(	int	l1,
		int	l2,
		int	l3,
		int	m1,
		int	m2,
		int	m3
	)

inlinestatic

Gaunt coefficent of two real spherical harmonics with a complex one.

\[ \langle R_{\ell_1 m_1} | Y_{\ell_2 m_2} | R_{\ell_3 m_3} \rangle \]

Definition at line 452 of file sht.hpp.

◆ gaunt_hybrid()

static std::complex< double > sirius::SHT::gaunt_hybrid	(	int	l1,
		int	l2,
		int	l3,
		int	m1,
		int	m2,
		int	m3
	)

inlinestatic

Gaunt coefficent of two complex and one real spherical harmonics.

\[ \langle Y_{\ell_1 m_1} | R_{\ell_2 m_2} | Y_{\ell_3 m_3} \rangle \]

Definition at line 478 of file sht.hpp.

◆ uniform_coverage()

void sirius::SHT::uniform_coverage ( )

inline

Definition at line 495 of file sht.hpp.

◆ clebsch_gordan()

static double sirius::SHT::clebsch_gordan	(	int	l1,
		int	l2,
		int	l3,
		int	m1,
		int	m2,
		int	m3
	)

inlinestatic

Return Clebsch-Gordan coefficient.

Clebsch-Gordan coefficients arise when two angular momenta are combined into a total angular momentum.

Definition at line 515 of file sht.hpp.

◆ ylm_backward()

auto sirius::SHT::ylm_backward	(	int	lm,
		int	itp
	)		const

inline

Definition at line 528 of file sht.hpp.

◆ rlm_backward()

auto sirius::SHT::rlm_backward	(	int	lm,
		int	itp
	)		const

inline

Definition at line 533 of file sht.hpp.

◆ coord() [1/2]

auto sirius::SHT::coord	(	int	x,
		int	itp
	)		const

inline

Definition at line 538 of file sht.hpp.

◆ coord() [2/2]

auto sirius::SHT::coord ( int idx__ ) const

inline

Definition at line 543 of file sht.hpp.

◆ theta()

auto sirius::SHT::theta ( int idx__ ) const

inline

Definition at line 548 of file sht.hpp.

◆ phi()

auto sirius::SHT::phi ( int idx__ ) const

inline

Definition at line 553 of file sht.hpp.

◆ weight()

auto sirius::SHT::weight ( int idx__ ) const

inline

Definition at line 558 of file sht.hpp.

◆ num_points()

auto sirius::SHT::num_points ( ) const

inline

Definition at line 563 of file sht.hpp.

◆ lmax()

auto sirius::SHT::lmax ( ) const

inline

Definition at line 568 of file sht.hpp.

◆ lmmax()

auto sirius::SHT::lmmax ( ) const

inline

Definition at line 573 of file sht.hpp.

◆ backward_transform() [2/2]

template<>

void sirius::SHT::backward_transform	(	int	ld,
		double const *	flm,
		int	nr,
		int	lmmax,
		double *	ftp
	)		const

Definition at line 245 of file sht.cpp.

◆ forward_transform() [2/2]

template<>

void sirius::SHT::forward_transform	(	double const *	ftp,
		int	nr,
		int	lmmax,
		int	ld,
		double *	flm
	)		const

Definition at line 265 of file sht.cpp.

Member Data Documentation

◆ pu_

sddk::device_t sirius::SHT::pu_

private

Type of processing unit.

Definition at line 84 of file sht.hpp.

◆ lmax_

int sirius::SHT::lmax_

private

Maximum \( \ell \) of spherical harmonics.

Definition at line 87 of file sht.hpp.

◆ lmmax_

int sirius::SHT::lmmax_

private

Maximum number of \( \ell, m \) components.

Definition at line 90 of file sht.hpp.

◆ num_points_

int sirius::SHT::num_points_

private

Number of real-space \( (\theta, \phi) \) points on the sphere.

Definition at line 93 of file sht.hpp.

◆ coord_

sddk::mdarray<double, 2> sirius::SHT::coord_

private

Cartesian coordinates of points (normalized to 1).

Definition at line 96 of file sht.hpp.

◆ tp_

sddk::mdarray<double, 2> sirius::SHT::tp_

private

\( (\theta, \phi) \) angles of points.

Definition at line 99 of file sht.hpp.

◆ w_

std::vector<double> sirius::SHT::w_

private

Point weights.

Definition at line 102 of file sht.hpp.

◆ ylm_backward_

sddk::mdarray<std::complex<double>, 2> sirius::SHT::ylm_backward_

private

Backward transformation from Ylm to spherical coordinates.

Definition at line 105 of file sht.hpp.

◆ ylm_forward_

sddk::mdarray<std::complex<double>, 2> sirius::SHT::ylm_forward_

private

Forward transformation from spherical coordinates to Ylm.

Definition at line 108 of file sht.hpp.

◆ rlm_backward_

sddk::mdarray<double, 2> sirius::SHT::rlm_backward_

private

Backward transformation from Rlm to spherical coordinates.

Definition at line 111 of file sht.hpp.

◆ rlm_forward_

sddk::mdarray<double, 2> sirius::SHT::rlm_forward_

private

Forward transformation from spherical coordinates to Rlm.

Definition at line 114 of file sht.hpp.

◆ mesh_type_

int sirius::SHT::mesh_type_ {0}

private

Type of spherical grid (0: Lebedev-Laikov, 1: uniform).

Definition at line 117 of file sht.hpp.

The documentation for this class was generated from the following files:

Public Member Functions

Static Public Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

◆ SHT()

Member Function Documentation

◆ check()

◆ backward_transform() [1/2]

◆ forward_transform() [1/2]

◆ convert() [1/2]

◆ convert() [2/2]

◆ ylm_dot_rlm()

◆ rlm_dot_ylm()

◆ gaunt_yyy()

◆ gaunt_rrr()

◆ gaunt_rlm_ylm_rlm()

◆ gaunt_hybrid()

◆ uniform_coverage()

◆ clebsch_gordan()

◆ ylm_backward()

◆ rlm_backward()

◆ coord() [1/2]

◆ coord() [2/2]

◆ theta()

◆ phi()

◆ weight()

◆ num_points()

◆ lmax()

◆ lmmax()

◆ backward_transform() [2/2]

◆ forward_transform() [2/2]

Member Data Documentation

◆ pu_

◆ lmax_

◆ lmmax_

◆ num_points_

◆ coord_

◆ tp_

◆ w_

◆ ylm_backward_

◆ ylm_forward_

◆ rlm_backward_

◆ rlm_forward_

◆ mesh_type_