Class that calculates and manages the radial integrals for given B-spline bases. More...

#include <radial.h>

Public Member Functions
	RadialIntegrals (Bspline const &bspline_x, Bspline const &bspline_y, int Nlambdas)
	Constructor. More...

void	setupOneElectronIntegrals (bool sharedscratch=false, bool iammaster=true)
	Calculate one-electron integral matrices. More...

void	setupOneElectronIntegrals (Parallel const &par, CommandLine const &cmd)
	Calculate one-electron integral matrices. More...

void	setupTwoElectronIntegrals (Parallel const &par, CommandLine const &cmd)
	Calculate the two-electron integral matrix. More...

void	verbose (bool v)
	Verbosity control. More...

int	maxlambda () const
	Maximal multipole moment. More...

Complex	computeOverlapMatrixElement_iknot (Bspline const &bspline, GaussLegendre const &g, int i, int j, CCFunction func, int iknot) const

Complex	computeOverlapMatrixElement (Bspline const &bspline, GaussLegendre const &g, int i, int j, CCFunction func) const

SymBandMatrix< Complex >	computeOverlapMatrix (Bspline const &bspline, CCFunction func) const

Complex	computeD_iknot (Bspline const &bspline, GaussLegendre const &g, int i, int j, int iknot) const
	Partial derivative overlap. More...

Complex	computeD (Bspline const &bspline, GaussLegendre const &g, int i, int j) const
	Derivative overlap. More...

Complex	computeM_iknot (Bspline const &bspline, GaussLegendre const &g, int a, int i, int j, int iknot, Real rmin, Real rmax, Real scale) const
	Partial integral moment. More...

Complex	computeM (Bspline const &bspline, GaussLegendre const &g, int a, int i, int j, Real rmin, Real rmax, bool scale=false) const
	Integral moments. More...

Complex	computeM (Bspline const &bspline, GaussLegendre const &g, int a, int i, int j, int begin_knot, int end_knot, Real rmin, Real rmax, bool scale=false) const
	Integral moments. More...

cArray	computeMi (Bspline const &bspline, GaussLegendre const &g, int a, Real rmin, Real rmax) const
	Partial integral moments. More...

Complex	computeR (int lambda, int i, int j, int k, int l) const
	Two-electron integral for multipole \( lambda \). More...

void	diagonalR (int lambda)

void	coupledR (int lambda)

Complex	computeRtri (int lambda, Bspline const &xspline, GaussLegendre const &xg, int k, int l, int iknotx, Real tx1, Real tx2, Bspline const &yspline, GaussLegendre const &yg, int m, int n, int iknoty, Real ty1, Real ty2) const
	Triangular R-integral. More...

SymBandMatrix< Complex >	calc_R_tr_dia_block (unsigned lambda, int i, int k) const
	Calculate particular sub-matrix of the radial integrals matrix. More...

void	apply_R_matrix (unsigned int lambda, Complex a, const cArrayView src, Complex b, cArrayView dst, int x_row_limit=-1, int y_row_limit=-1) const
	Multiply vector by matrix of two-electron integrals. More...

cArray	overlap (Bspline const &bspline, GaussLegendre const &g, CCFunction funct) const
	Compute B-spline overlaps of arbitrary one-dimensional function. More...

cArray	overlap (Bspline const &bsplinex, GaussLegendre const &gx, Bspline const &bspliney, GaussLegendre const &gy, C2CFunction funct) const
	Compute B-spline overlaps of arbitrary two-dimensional function. More...

cArray	overlapj (Bspline const &bspline, GaussLegendre const &g, int maxell, const rArrayView vk, bool fast_bessel=false) const
	Compute j-overlaps. More...

Bspline const &	bspline () const

Bspline const &	bspline_x () const

Bspline const &	bspline_y () const

GaussLegendre const &	gaussleg () const

GaussLegendre const &	gaussleg_x () const

GaussLegendre const &	gaussleg_y () const

	OneElectronMatrixAccessors (D) OneElectronMatrixAccessors(S) OneElectronMatrixAccessors(Mm1) OneElectronMatrixAccessors(Mm2) OneElectronMatrixArrayAccessors(Mtr_L) OneElectronMatrixArrayAccessors(Mtr_mLm1) OneElectronPartialMatrixAccessors(Mitr_L) OneElectronPartialMatrixAccessors(Mitr_mLm1) BlockSymBandMatrix< Complex > const &R_tr_dia(unsigned i) const

CsrMatrix< LU_int_t, Complex > const &	R_coupled (unsigned i) const

Static Public Member Functions
static CsrMatrix< LU_int_t, Complex >	computeOverlapMatrix (Bspline const &bspline1, Bspline const &bspline2, Real R_left, Real R_right)
	Compute overlap matrix of two B-spline bases. More...

static cArray	overlapP (Bspline const &bspline, GaussLegendre const &g, Real Z, int n, int l)
	Compute P-overlaps. More...

Detailed Description

This class contains important routines for calculation of overlap matrices and matrix elements of various functions for given B-spline bases. The calculation of the overlaps uses Gauss-Legendre quadrature, that is to be specified in a form of a preallocated GaussLegendre object.

The overlaps considered are one-electron integrals of various one-dimensional functions (or given arbitrary function) and two-electron overlaps of the multipole elements (or given adbitrary function).

There are two variants of the B-spline basis specified for every axis: the inner basis and the full basis. These bases can be identical, when no channel reduction is used in the solution of the scattering equations (typically for above-ionization energies). However, the inner and outer bases can be different, when the channel reduction will take place. The inner basis then contains real B-spline knots only and will be used for the full-channel calculation of the inner problem. The full basis contains the inner basis knots but is generally much longer and terminated by the ECS complex grid section. This basis is used for solution of energy-allowed channels. The energy-forbidden channels are thus solved only in the inner basis, which is all right, because the energy-forbidden channel functions exponentially decrease with the radial distance.

Constructor & Destructor Documentation

◆ RadialIntegrals()

RadialIntegrals::RadialIntegrals	(	Bspline const &	bspline_x,
		Bspline const &	bspline_y,
		int	Nlambdas
	)

This constructor will initialize the RadialIntegrals object with different B-spline bases for every axis.

Member Function Documentation

◆ apply_R_matrix()

void RadialIntegrals::apply_R_matrix	(	unsigned int	lambda,
		Complex	a,
		const cArrayView	src,
		Complex	b,
		cArrayView	dst,
		int	x_row_limit = `-1`,
		int	y_row_limit = `-1`
	)		const

This routine will multiply several source vectors by the matrix of two-electron integrals for given multipole 'lambda'. The matrix elements are computed anew and applied directly to the vectors to minimize memory requirements. This method - instead of caching the whole integral matrix in memory or on disk - is used in the 'lightweight' mode, which can be requested by the command line option –lightweight.

◆ bspline()

Bspline const& RadialIntegrals::bspline ( ) const

inline

◆ bspline_x()

Bspline const& RadialIntegrals::bspline_x ( ) const

inline

◆ bspline_y()

Bspline const& RadialIntegrals::bspline_y ( ) const

inline

◆ calc_R_tr_dia_block()

SymBandMatrix< Complex > RadialIntegrals::calc_R_tr_dia_block	(	unsigned	lambda,
		int	i,
		int	k
	)		const

Calculate particular sub-matrix of the radial integrals matrix (with block indices "i" and "k") and return it in a form of a dense array (copying structure of the overlap matrix).

◆ computeD()

Complex RadialIntegrals::computeD	(	Bspline const &	bspline,
		GaussLegendre const &	g,
		int	i,
		int	j
	)		const

Compute derivative overlap for B-splines \( B_i \) and \( B_j \).

Parameters

i	B-spline index.
j	B-spline index.

◆ computeD_iknot()

Complex RadialIntegrals::computeD_iknot	(	Bspline const &	bspline,
		GaussLegendre const &	g,
		int	i,
		int	j,
		int	iknot
	)		const

Compute derivative overlap of B-splines \( B_i \) and \( B_j \) over the knot "iknot", using Gauss-Legendre integration.

Parameters

i	B-spline index.
j	B-spline index.
iknot	Interval index.

◆ computeM() [1/2]

Complex RadialIntegrals::computeM	(	Bspline const &	bspline,
		GaussLegendre const &	g,
		int	a,
		int	i,
		int	j,
		Real	rmin,
		Real	rmax,
		bool	scale = `false`
	)		const

Compute integral moment of coordinate power between the B-splines \( B_i \) and \( B_j \)

Parameters

a	Exponent.
i	B-spline index.
j	B-spline index.

◆ computeM() [2/2]

Complex RadialIntegrals::computeM	(	Bspline const &	bspline,
		GaussLegendre const &	g,
		int	a,
		int	i,
		int	j,
		int	begin_knot,
		int	end_knot,
		Real	rmin,
		Real	rmax,
		bool	scale = `false`
	)		const

Compute integral moment of coordinate power between the B-splines \( B_i \) and \( B_j \)

Parameters

a	Exponent.
i	B-spline index.
j	B-spline index.

◆ computeM_iknot()

Complex RadialIntegrals::computeM_iknot	(	Bspline const &	bspline,
		GaussLegendre const &	g,
		int	a,
		int	i,
		int	j,
		int	iknot,
		Real	rmin,
		Real	rmax,
		Real	scale
	)		const

Compute integral moment of coordinate power between the B-splines \( B_i \) and \( B_j \) over the knot "iknot", using Gauss-Legendre integration.

Parameters

a	Exponent.
i	B-spline index.
j	B-spline index.
iknot	Interval index.
R	Potential truncation point.

◆ computeMi()

cArray RadialIntegrals::computeMi	(	Bspline const &	bspline,
		GaussLegendre const &	g,
		int	a,
		Real	rmin,
		Real	rmax
	)		const

Compute logarithms of integral moment of degree "a" for every B-spline pair and every interknot sub-interval. Store in 1-D array of shape

[ Nspline × Nspline × Nintval ]

Parameters

a	Moment degree.

◆ computeOverlapMatrix() [1/2]

SymBandMatrix< Complex > RadialIntegrals::computeOverlapMatrix	(	Bspline const &	bspline,
		CCFunction	func
	)		const

◆ computeOverlapMatrix() [2/2]

CsrMatrix< LU_int_t, Complex > RadialIntegrals::computeOverlapMatrix	(	Bspline const &	bspline1,
		Bspline const &	bspline2,
		Real	R_left,
		Real	R_right
	)

static

This function returns a matrix of B-spline overlaps across two bases. The overlap integrals are calculated only between the given bounds R_left and R_right (these are the real parts of the potentially complex coordinates). Both bounds must be aligned to some B-spline knot.

◆ computeOverlapMatrixElement()

Complex RadialIntegrals::computeOverlapMatrixElement	(	Bspline const &	bspline,
		GaussLegendre const &	g,
		int	i,
		int	j,
		CCFunction	func
	)		const

◆ computeOverlapMatrixElement_iknot()

Complex RadialIntegrals::computeOverlapMatrixElement_iknot	(	Bspline const &	bspline,
		GaussLegendre const &	g,
		int	i,
		int	j,
		CCFunction	func,
		int	iknot
	)		const

◆ computeR()

Complex RadialIntegrals::computeR	(	int	lambda,
		int	i,
		int	j,
		int	k,
		int	l
	)		const

Compute the two-electron (Slater-type) four-B-spline integral. Uses the "diagonal" contribution from diagonalR and the off-diagonal contributions calculated from products of the partial moments.

◆ computeRtri()

Complex RadialIntegrals::computeRtri	(	int	lambda,
		Bspline const &	xspline,
		GaussLegendre const &	xg,
		int	k,
		int	l,
		int	iknotx,
		Real	tx1,
		Real	tx2,
		Bspline const &	yspline,
		GaussLegendre const &	yg,
		int	m,
		int	n,
		int	iknoty,
		Real	ty1,
		Real	ty2
	)		const

Calculates integral

\[ R_{klmn}^{\lambda\triangle} = \int\limits_{t_i}^{t_{i+1}} B_k(r_1) B_l(r_1) \frac{1}{r_1} \left( \int\limits_{t_i}^{r_1} \left(\frac{r_2}{r_1}\right)^\lambda B_m(r_2) B_n(r_2) \mathrm{d}r_2 \right) \mathrm{d}r_1 \,, \]

which is used to calculate a diagonal contribution to the full R-integral, see computeRdiag. Inner integrand is a polynomial, so a simple Gauss-Legendre quadrature is used (with a sufficient order). The outer integral is polynomial only when the integration starts from zero ( \( t_i = 0 \)). In such case it has degree equal to the combined order of the B-splines. It is assumed that also elsewhere the integrand can be well approximated by a polynomial of the same degree.

Uses functions R_outer_integrand and R_inner_integrand.

Parameters

lambda	The multipole ( \( \lambda \)).
k	First B-spline index.
l	Second B-spline index.
iknotx	Integration interval (tx[iknotx] ... tx[iknotx+1]).
tx1	Left integration bound.
tx2	Right integration bound.
m	Third B-spline index.
n	Fourth B-spline index.
iknoty	Integration interval (ty[iknoty] ... ty[iknoty+1]).
ty1	Left integration bound.
ty2	Right integration bound.

◆ coupledR()

void RadialIntegrals::coupledR ( int lambda )

◆ diagonalR()

void RadialIntegrals::diagonalR ( int lambda )

◆ gaussleg()

GaussLegendre const& RadialIntegrals::gaussleg ( ) const

inline

◆ gaussleg_x()

GaussLegendre const& RadialIntegrals::gaussleg_x ( ) const

inline

◆ gaussleg_y()

GaussLegendre const& RadialIntegrals::gaussleg_y ( ) const

inline

◆ maxlambda()

int RadialIntegrals::maxlambda ( ) const

inline

Returns maximal multipole (also known as "angular momentum transfer"), for which there are precomputed two-electron integrals.

◆ OneElectronMatrixAccessors()

RadialIntegrals::OneElectronMatrixAccessors ( D ) const &

inline

◆ overlap() [1/2]

cArray RadialIntegrals::overlap	(	Bspline const &	bspline,
		GaussLegendre const &	g,
		CCFunction	funct
	)		const

Parameters

bspline	B-spline basis.
g	Gauss-Legendre integrator adapted to the B-spline basis.
funct	Some one-dimensional function (Complex -> Complex).

◆ overlap() [2/2]

cArray RadialIntegrals::overlap	(	Bspline const &	bsplinex,
		GaussLegendre const &	gx,
		Bspline const &	bspliney,
		GaussLegendre const &	gy,
		C2CFunction	funct
	)		const

Parameters

bsplinex	B-spline basis for x-axis.
gx	Gauss-Legendre integrator adapted to the B-spline basis for x-axis.
bspliney	B-spline basis for y-axis.
gy	Gauss-Legendre integrator adapted to the B-spline basis for y-axis.
funct	Some one-dimensional function (Complex x Complex -> Complex).

◆ overlapj()

cArray RadialIntegrals::overlapj	(	Bspline const &	bspline,
		GaussLegendre const &	g,
		int	maxell,
		const rArrayView	vk,
		bool	fast_bessel = `false`
	)		const

Compute B-spline overlap integrals for Riccati-Bessel function.

Parameters

maxell	Maximal degree of the Riccati-Bessel function.
vk	Array containing linear momenta.

Returns: Array of shape [vk.size() × (maxell + 1) × Nspline] in column-major format.

◆ overlapP()

cArray RadialIntegrals::overlapP	(	Bspline const &	bspline,
		GaussLegendre const &	g,
		Real	Z,
		int	n,
		int	l
	)

static

Compute overlap vector of B-splines vs. hydrogen Pnl function.

Parameters

bspline	B-spline basis.
g	Gauss-Legendre integrator adapted to the B-spline basis.
n	Principal quantum number.
l	Orbital quantum number.

◆ R_coupled()

CsrMatrix<LU_int_t,Complex> const& RadialIntegrals::R_coupled ( unsigned i ) const

inline

◆ setupOneElectronIntegrals() [1/2]

void RadialIntegrals::setupOneElectronIntegrals	(	bool	sharedscratch = `false`,
		bool	iammaster = `true`
	)

This function will calculate the common one-electron overlap matrices S, D, Mm1, Mm2. The parameters sharedscratch and iammaster are used when writing the results to disk to avoid multiple processes writing the same file.

◆ setupOneElectronIntegrals() [2/2]

void RadialIntegrals::setupOneElectronIntegrals	(	Parallel const &	par,
		CommandLine const &	cmd
	)

This function calls the other setupOneElectronIntegrals with the appropriate parameters.

◆ setupTwoElectronIntegrals()

void RadialIntegrals::setupTwoElectronIntegrals	(	Parallel const &	par,
		CommandLine const &	cmd
	)

This function will calculate the two-electron multipole overlap matrices.

◆ verbose()

void RadialIntegrals::verbose ( bool v )

inline

Setting this to false will inhibit standard output messages from this class.

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

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ RadialIntegrals()

Member Function Documentation

◆ apply_R_matrix()

◆ bspline()

◆ bspline_x()

◆ bspline_y()

◆ calc_R_tr_dia_block()

◆ computeD()

◆ computeD_iknot()

◆ computeM() [1/2]

◆ computeM() [2/2]

◆ computeM_iknot()

◆ computeMi()

◆ computeOverlapMatrix() [1/2]

◆ computeOverlapMatrix() [2/2]

◆ computeOverlapMatrixElement()

◆ computeOverlapMatrixElement_iknot()

◆ computeR()

◆ computeRtri()

◆ coupledR()

◆ diagonalR()

◆ gaussleg()

◆ gaussleg_x()

◆ gaussleg_y()

◆ maxlambda()

◆ OneElectronMatrixAccessors()

◆ overlap() [1/2]

◆ overlap() [2/2]

◆ overlapj()

◆ overlapP()

◆ R_coupled()

◆ setupOneElectronIntegrals() [1/2]

◆ setupOneElectronIntegrals() [2/2]

◆ setupTwoElectronIntegrals()

◆ verbose()