Table of Gaunt coefficients of real spherical harmonics. More...

#include <real_gaunt_table.h>

Collaboration diagram for RealGauntTable:

Public Member Functions
	RealGauntTable (RealGauntTable const &)=delete

RealGauntTable &	operator= (RealGauntTable const &)=delete

	~RealGauntTable ()

void	build (const int lmax)
	Builds the Gaunt table of real spherical harmonics for two-center integrals.

const double &	operator() (const int l1, const int l2, const int l3, const int m1, const int m2, const int m3) const
	gets the tabulated real Gaunt coefficient

int	lmax () const
	returns the maximum l (for the first two dimensions; the third dimension is 2*lmax)

size_t	memory () const
	Returns the amount of heap memory used by this class (in bytes).

double	gaunt (const int l1, const int l2, const int l3, const int m1, const int m2, const int m3) const
	Computes the standard Gaunt coefficients.

Static Public Member Functions
static RealGauntTable &	instance ()

Private Member Functions
	RealGauntTable ()

bool	gaunt_select_l (const int l1, const int l2, const int l3) const
	selection rule of standard & real Gaunt coefficients regarding l1, l2, l3

bool	gaunt_select_m (const int m1, const int m2, const int m3) const
	selection rule of standard Gaunt coefficients regarding m1, m2, m3

bool	real_gaunt_select_m (const int m1, const int m2, const int m3) const
	selection rule of real Gaunt coefficients regarding m1, m2, m3

bool	is_valid_lm (const int l1, const int l2, const int l3, const int m1, const int m2, const int m3) const
	Returns whether the given l & m are valid quantum numbers.

double	gaunt_lookup (const int l1, const int l2, const int l3, const int m1, const int m2, const int m3) const
	Get a Gaunt coefficient by looking up the table.

double	real_gaunt_lookup (const int l1, const int l2, const int l3, const int m1, const int m2, const int m3) const
	Get a real Gaunt coefficient from the stored Gaunt coefficients.

std::array< int, 6 >	gaunt_key (const int l1, const int l2, const int l3, const int m1, const int m2, const int m3) const
	Symmetry-adapted key for gaunt_table_.

void	arrange (int &l1, int &l2, int &m1, int &m2) const
	swap (l1,m1) <--> (l2,m2) if l1 < l2; do nothing otherwise

double	factorial (const int n) const
	returns n! as a double

int	index_map (int l, int m) const
	Returns the linearized index of Y(l,m).

int	minus_1_pow (int m) const
	returns pow(-1, m)

Private Attributes
int	lmax_ = -1
	maximum angular momentum of the table (for the first two dimensions)

std::map< std::array< int, 6 >, double >	gaunt_table_
	Table of standard Gaunt coefficients.

container::Tensor	real_gaunt_table_ {container::DataType::DT_DOUBLE, container::TensorShape({0})}
	table of real Gaunt coefficients

Detailed Description

Table of Gaunt coefficients of real spherical harmonics.

This class computes and tabulates the Gaunt coefficients of real spherical harmonics used in two-center integrals. The implementation follows the singleton pattern.

Usage:

 int lmax = 5;
 RealGauntTable::instance().build(lmax);

 // get the real Gaunt coefficient
 double G = RealGauntTable::instance()(l1, l2, l3, m1, m2, m3);

Constructor & Destructor Documentation

◆ RealGauntTable() [1/2]

RealGauntTable::RealGauntTable ( RealGauntTable const & )

delete

◆ ~RealGauntTable()

RealGauntTable::~RealGauntTable ( )

inline

◆ RealGauntTable() [2/2]

RealGauntTable::RealGauntTable ( )

inlineprivate

Member Function Documentation

◆ arrange()

void RealGauntTable::arrange	(	int &	l1,
		int &	l2,
		int &	m1,
		int &	m2
	)		const

private

swap (l1,m1) <--> (l2,m2) if l1 < l2; do nothing otherwise

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◆ build()

void RealGauntTable::build ( const int lmax )

Builds the Gaunt table of real spherical harmonics for two-center integrals.

This function tabulates the Gaunt coefficients of real spherical harmonics

                        /
 G(l1,l2,l3,m1,m2,m3) = | Z(l1,m1) Z(l2,m2) Z(l3,m3) d Omega
                        /

for l1,l2 <= lmax and l3 <= 2*lmax. Here Z is the real spherical harmonics defined as

          / sqrt(2) * Re[Y(l,|m|)]   m > 0
          |
 Z(l,m) = | Y(l,0)                   m = 0
          |
          \ sqrt(2) * Im[Y(l,|m|)]   m < 0

Note

In some literature an extra pow(-1, m) is introduced to yield a signless Cartesian expression. The definition here does not adopt this convention and is consistent with ModuleBase::Ylm::sph_harm. Consequently, Cartesian expression of Z may involve some minus signs, for example,

   Z(1,-1) = -c * y / r
   Z(1, 0) = +c * z / r
   Z(1, 1) = -c * x / r

where c = sqrt(3/4/pi) and r = sqrt(x^2 + y^2 + z^2).

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◆ factorial()

double RealGauntTable::factorial ( const int n ) const

private

returns n! as a double

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◆ gaunt()

double RealGauntTable::gaunt	(	const int	l1,
		const int	l2,
		const int	l3,
		const int	m1,
		const int	m2,
		const int	m3
	)		const

Computes the standard Gaunt coefficients.

This function computes the standard Gaunt coefficients

                        /
 G(l1,l2,l3,m1,m2,m3) = | Y(l1,m1) Y(l2,m2) Y(l3,m3) d Omega
                        /

where Y is the (standard) spherical harmonics and Omega is the solid angle element.

Note: This function computes the standard Gaunt coefficients, which is different from Gaunt coefficients of real spherical harmonics.; Currently the algorithm computes the Gaunt coefficients with the Wigner-3j symbols, which in turn is evaluated with the Racah formula. This might have some numerical issue for large l and is yet to be studied later.

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◆ gaunt_key()

std::array< int, 6 > RealGauntTable::gaunt_key	(	const int	l1,
		const int	l2,
		const int	l3,
		const int	m1,
		const int	m2,
		const int	m3
	)		const

private

Symmetry-adapted key for gaunt_table_.

Standard Gaunt coefficients have the following symmetries:

 Gaunt(l1,l2,l3,m1,m2,m3) = Gaunt(l1,l2,l3,-m1,-m2,-m3)

 Gaunt(1,2,3) = Gaunt(2,3,1) = Gaunt(3,1,2) =
 Gaunt(2,1,3) = Gaunt(1,3,2) = Gaunt(3,2,1)

These symmetries enable us to store merely a small portion of the Gaunt coefficients. This function permutes 1/2/3 and flips the signs of m1/m2/m3 if necessary so that the returned key {l1,l2,l3,m1,m2,m3} satisfies l1 >= l2 >= l3 and m3 >= 0.

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◆ gaunt_lookup()

double RealGauntTable::gaunt_lookup	(	const int	l1,
		const int	l2,
		const int	l3,
		const int	m1,
		const int	m2,
		const int	m3
	)		const

private

Get a Gaunt coefficient by looking up the table.

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◆ gaunt_select_l()

bool RealGauntTable::gaunt_select_l	(	const int	l1,
		const int	l2,
		const int	l3
	)		const

private

selection rule of standard & real Gaunt coefficients regarding l1, l2, l3

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◆ gaunt_select_m()

bool RealGauntTable::gaunt_select_m	(	const int	m1,
		const int	m2,
		const int	m3
	)		const

inlineprivate

selection rule of standard Gaunt coefficients regarding m1, m2, m3

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◆ index_map()

int RealGauntTable::index_map	(	int	l,
		int	m
	)		const

private

Returns the linearized index of Y(l,m).

 l       0   1   1   1   2   2   2   2   2   3 ...
 m       0  -1   0   1  -2  -1   0   1   2  -3 ...

index 0 1 2 3 4 5 6 7 8 9 ...

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◆ instance()

static RealGauntTable & RealGauntTable::instance ( )

inlinestatic

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◆ is_valid_lm()

bool RealGauntTable::is_valid_lm	(	const int	l1,
		const int	l2,
		const int	l3,
		const int	m1,
		const int	m2,
		const int	m3
	)		const

private

Returns whether the given l & m are valid quantum numbers.

This function checks whether abs(mi) <= li (i=1,2,3) is satisfied. This implies li >= 0.

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◆ lmax()

int RealGauntTable::lmax ( ) const

inline

returns the maximum l (for the first two dimensions; the third dimension is 2*lmax)

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◆ memory()

size_t RealGauntTable::memory ( ) const

inline

Returns the amount of heap memory used by this class (in bytes).

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◆ minus_1_pow()

int RealGauntTable::minus_1_pow ( int m ) const

inlineprivate

returns pow(-1, m)

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◆ operator()()

const double & RealGauntTable::operator()	(	const int	l1,
		const int	l2,
		const int	l3,
		const int	m1,
		const int	m2,
		const int	m3
	)		const

gets the tabulated real Gaunt coefficient

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◆ operator=()

RealGauntTable & RealGauntTable::operator= ( RealGauntTable const & )

delete

◆ real_gaunt_lookup()

double RealGauntTable::real_gaunt_lookup	(	const int	l1,
		const int	l2,
		const int	l3,
		const int	m1,
		const int	m2,
		const int	m3
	)		const

private

Get a real Gaunt coefficient from the stored Gaunt coefficients.

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◆ real_gaunt_select_m()

bool RealGauntTable::real_gaunt_select_m	(	const int	m1,
		const int	m2,
		const int	m3
	)		const

private

selection rule of real Gaunt coefficients regarding m1, m2, m3

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Member Data Documentation

◆ gaunt_table_

std::map<std::array<int, 6>, double> RealGauntTable::gaunt_table_

private

Table of standard Gaunt coefficients.

This table maps (l1,l2,l3,m1,m2,m3) to a standard Gaunt coefficient. Due to the selection rule and symmetry, only those which survive the selection rule and satisfy l1 >= l2 >= l3 && m3 >= 0 are stored.

◆ lmax_

int RealGauntTable::lmax_ = -1

private

maximum angular momentum of the table (for the first two dimensions)

◆ real_gaunt_table_

container::Tensor RealGauntTable::real_gaunt_table_ {container::DataType::DT_DOUBLE, container::TensorShape({0})}

private

table of real Gaunt coefficients

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

/home/runner/work/abacus-develop/abacus-develop/source/source_basis/module_nao/real_gaunt_table.h
/home/runner/work/abacus-develop/abacus-develop/source/source_basis/module_nao/real_gaunt_table.cpp

Public Member Functions

Static Public Member Functions

Private Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

◆ RealGauntTable() [1/2]

◆ ~RealGauntTable()

◆ RealGauntTable() [2/2]

Member Function Documentation

◆ arrange()

◆ build()

◆ factorial()

◆ gaunt()

◆ gaunt_key()

◆ gaunt_lookup()

◆ gaunt_select_l()

◆ gaunt_select_m()

◆ index_map()

◆ instance()

◆ is_valid_lm()

◆ lmax()

◆ memory()

◆ minus_1_pow()

◆ operator()()

◆ operator=()

◆ real_gaunt_lookup()

◆ real_gaunt_select_m()

Member Data Documentation

◆ gaunt_table_

◆ lmax_

◆ real_gaunt_table_