#include <gfit.h>

Public Member Functions
	GFit ()=default
	default ctor does nothing
	GFit (const GFit &other)=default
	copy constructor
	GFit (OperatorInfo info)
Tensor< T >	coeffs () const
	return the coefficients of the fit
Tensor< T >	exponents () const
	return the exponents of the fit
GFit &	operator= (const GFit &other)
	assignment operator

Static Public Member Functions
static GFit	BSHFit (double mu, double lo, double hi, double eps, bool prnt=false)
	return a fit for the bound-state Helmholtz function
static GFit	CoulombFit (double lo, double hi, double eps, bool prnt=false)
	return a fit for the Coulomb function
static GFit	F12Fit (double gamma, double lo, double hi, double eps, bool prnt=false)
	return a fit for the F12 correlation factor
static GFit	F12sqFit (double gamma, double lo, double hi, double eps, bool prnt=false)
	return a fit for the F12^2 correlation factor
static GFit	F2GFit (double gamma, double lo, double hi, double eps, bool prnt=false)
	return a fit for the F2G function
static GFit	FGFit (double gamma, double lo, double hi, double eps, bool prnt=false)
	return a fit for the FG function
static GFit	GaussFit (double gamma, double lo, double hi, double eps, bool prnt=false)
	return a (trivial) fit for a single Gauss function
static GFit	GeneralFit ()
	return a fit for a general isotropic function
static GFit	InverseRSqFit (double lo, double hi, double eps, bool prnt=false)
	return a fit for the inverse-square function 1/r^2
static void	prune_small_coefficients (const double eps, const double lo, const double hi, Tensor< double > &coeff, Tensor< double > &expnt, const long first_prunable=1)
static GFit	SlaterFit (double gamma, double lo, double hi, double eps, bool prnt=false)
	return a fit for the Slater function
static void	truncate_mixed_expansion (Tensor< double > &c, Tensor< double > &e, const std::array< LatticeRange, NDIM > &lattice_ranges, const Tensor< double > &cell_width, double lo, double hi_fin, double eps)
	Truncate the fit of a kernel that is lattice-summed along some axes.

Private Member Functions
template<typename funcT>
	GFit (const funcT &f)
	ctor taking an isotropic function
template<typename opT>
void	print_accuracy (opT op, const double lo, const double hi) const
	print coefficients and exponents, and values and errors

Static Private Member Functions
static void	bsh_fit (double mu, double lo, double hi, double eps, Tensor< double > &pcoeff, Tensor< double > &pexpnt, bool prnt, bool fix_interval)
	fit the function exp(-mu r)/r
static void	bsh_fit_ndim (int ndim, double mu, double lo, double hi, double eps, Tensor< double > &pcoeff, Tensor< double > &pexpnt, bool prnt)
static void	bsh_spherical_moments (double mu, double R, Tensor< double > &q)
static void	f12_fit (double gamma, double lo, double hi, double eps, Tensor< double > &pcoeff, Tensor< double > &pexpnt, bool prnt)
	fit a correlation factor (1- exp(-mu r))
static void	f12sq_fit (double gamma, double lo, double hi, double eps, Tensor< double > &pcoeff, Tensor< double > &pexpnt, bool prnt)
	fit a correlation factor f12^2 = (1- exp(-mu r))^2 = 1 - 2 exp(-mu r) + exp(-2 mu r)
static void	gaussian_spherical_moments (double alpha, double R, Tensor< double > &q)
static void	slater_fit (double gamma, double lo, double hi, double eps, Tensor< double > &pcoeff, Tensor< double > &pexpnt, bool prnt)
	fit a Slater function using a sum of Gaussians

Private Attributes
Tensor< T >	coeffs_
	the coefficients of the expansion f(x) = \sum_m coeffs[m] exp(-exponents[m] * x^2)
Tensor< T >	exponents_
	the exponents of the expansion f(x) = \sum_m coeffs[m] exp(-exponents[m] * x^2)

Constructor & Destructor Documentation

◆ GFit() [1/4]

template<typename T, std::size_t NDIM>

madness::GFit< T, NDIM >::GFit ( )

default

default ctor does nothing

Referenced by GFit(), BSHFit(), CoulombFit(), F12Fit(), F12sqFit(), F2GFit(), FGFit(), GaussFit(), GeneralFit(), InverseRSqFit(), operator=(), and SlaterFit().

◆ GFit() [2/4]

template<typename T, std::size_t NDIM>

madness::GFit< T, NDIM >::GFit ( OperatorInfo info )

inline

◆ GFit() [3/4]

template<typename T, std::size_t NDIM>

madness::GFit< T, NDIM >::GFit ( const GFit< T, NDIM > & other )

default

copy constructor

References GFit().

◆ GFit() [4/4]

template<typename T, std::size_t NDIM>

template<typename funcT>

madness::GFit< T, NDIM >::GFit ( const funcT & f )

inlineprivate

ctor taking an isotropic function

the function will be represented with a uniform error on a uniform grid

Parameters

[in] f a 1d-function that implements T operator()

References madness::f.

Member Function Documentation

◆ bsh_fit()

template<typename T, std::size_t NDIM>

void madness::GFit< T, NDIM >::bsh_fit	(	double	mu,
		double	lo,
		double	hi,
		double	eps,
		Tensor< double > &	pcoeff,
		Tensor< double > &	pexpnt,
		bool	prnt,
		bool	fix_interval )

inlinestaticprivate

fit the function exp(-mu r)/r

formulas taken from G. Beylkin and L. Monzon, On approximation of functions by exponential sums, Appl Comput Harmon A, vol. 19, no. 1, pp. 17-48, Jul. 2005. and R. J. Harrison, G. I. Fann, T. Yanai, and G. Beylkin, Multiresolution Quantum Chemistry in Multiwavelet Bases, Lecture Notes in Computer Science, vol. 2660, p. 103, 2003.

References bsh_spherical_moments(), e(), gaussian_spherical_moments(), madness::gesv(), h(), lo, mu, madness::constants::pi, madness::print(), prune_small_coefficients(), and q().

Referenced by BSHFit(), F2GFit(), and FGFit().

◆ bsh_fit_ndim()

template<typename T, std::size_t NDIM>

void madness::GFit< T, NDIM >::bsh_fit_ndim	(	int	ndim,
		double	mu,
		double	lo,
		double	hi,
		double	eps,
		Tensor< double > &	pcoeff,
		Tensor< double > &	pexpnt,
		bool	prnt )

inlinestaticprivate

References std::abs(), c, h(), lo, mu, p(), madness::constants::pi, pow(), madness::print(), and test().

Referenced by BSHFit().

◆ bsh_spherical_moments()

template<typename T, std::size_t NDIM>

void madness::GFit< T, NDIM >::bsh_spherical_moments	(	double	mu,
		double	R,
		Tensor< double > &	q )

inlinestaticprivate

References mu, madness::constants::pi, pow(), q(), and R.

Referenced by bsh_fit().

◆ BSHFit()

template<typename T, std::size_t NDIM>

GFit madness::GFit< T, NDIM >::BSHFit	(	double	mu,
		double	lo,
		double	hi,
		double	eps,
		bool	prnt = false )

inlinestatic

return a fit for the bound-state Helmholtz function

the BSH function is defined by f(r) = 1/ (4 pi) exp(-\mu r)/r

Parameters

[in]	mu	the exponent of the BSH
[in]	lo	the smallest length scale that needs to be precisely represented
[in]	hi	the largest length scale that needs to be precisely represented
[in]	eps	the precision threshold @parma[in] prnt print level

References GFit(), bsh_fit(), bsh_fit_ndim(), exact, fit(), lo, mu, NDIM, madness::constants::pi, and madness::print().

Referenced by GFit(), madness::BSHOperatorPtr3D(), madness::DiagnosticMatrix< T, NDIM >::build_Gab_bra(), CoulombFit(), madness::BSHFunctionInterface::fit(), madness::GradBSHOperator(), and GradBSHOperator_Joel().

◆ coeffs()

template<typename T, std::size_t NDIM>

Tensor< T > madness::GFit< T, NDIM >::coeffs ( ) const

inline

return the coefficients of the fit

References coeffs_.

Referenced by F2GFit(), and FGFit().

◆ CoulombFit()

template<typename T, std::size_t NDIM>

GFit madness::GFit< T, NDIM >::CoulombFit	(	double	lo,
		double	hi,
		double	eps,
		bool	prnt = false )

inlinestatic

return a fit for the Coulomb function

f(r) = 1/r

References GFit(), BSHFit(), fit(), lo, and madness::constants::pi.

Referenced by GFit(), madness::ExchangeCommutator::apply_KffK_lowrank_three_range(), madness::ExchangeCommutator::compute_lrf_exchange_operator(), madness::ElectronRepulsionInterface< T, NDIM >::fit(), madness::GradCoulombOperator(), and test_coulomb_gfit_discard_large_alpha().

◆ exponents()

template<typename T, std::size_t NDIM>

Tensor< T > madness::GFit< T, NDIM >::exponents ( ) const

inline

return the exponents of the fit

References exponents_.

Referenced by F2GFit(), and FGFit().

◆ f12_fit()

template<typename T, std::size_t NDIM>

void madness::GFit< T, NDIM >::f12_fit	(	double	gamma,
		double	lo,
		double	hi,
		double	eps,
		Tensor< double > &	pcoeff,
		Tensor< double > &	pexpnt,
		bool	prnt )

inlinestaticprivate

fit a correlation factor (1- exp(-mu r))

use the Slater fit with an additional term: 1*exp(-0 r^2)

References madness::_(), lo, madness::BaseTensor::size(), and slater_fit().

Referenced by F12Fit().

◆ F12Fit()

template<typename T, std::size_t NDIM>

GFit madness::GFit< T, NDIM >::F12Fit	(	double	gamma,
		double	lo,
		double	hi,
		double	eps,
		bool	prnt = false )

inlinestatic

return a fit for the F12 correlation factor

the Slater function is defined by f(r) = 1/(2 gamma) * (1 - exp(-\gamma r))

Parameters

[in]	gamma	the exponent of the Slater function
[in]	lo	the smallest length scale that needs to be precisely represented
[in]	hi	the largest length scale that needs to be precisely represented
[in]	eps	the precision threshold @parma[in] prnt print level

References GFit(), exact, f12_fit(), fit(), lo, and madness::print().

Referenced by GFit(), and madness::SlaterF12Interface::fit().

◆ f12sq_fit()

template<typename T, std::size_t NDIM>

void madness::GFit< T, NDIM >::f12sq_fit	(	double	gamma,
		double	lo,
		double	hi,
		double	eps,
		Tensor< double > &	pcoeff,
		Tensor< double > &	pexpnt,
		bool	prnt )

inlinestaticprivate

fit a correlation factor f12^2 = (1- exp(-mu r))^2 = 1 - 2 exp(-mu r) + exp(-2 mu r)

no factor 1/(2 mu) or square of it included! use the Slater fit with an additional term: 1*exp(-0 r^2)

References madness::_(), lo, MADNESS_CHECK, madness::BaseTensor::size(), and slater_fit().

Referenced by F12sqFit().

◆ F12sqFit()

template<typename T, std::size_t NDIM>

GFit madness::GFit< T, NDIM >::F12sqFit	(	double	gamma,
		double	lo,
		double	hi,
		double	eps,
		bool	prnt = false )

inlinestatic

return a fit for the F12^2 correlation factor

the Slater function square is defined by f(r) = [ 1/(2 gamma) * (1 - exp(-\gamma r)) ] ^2

Parameters

[in]	gamma	the exponent of the Slater function
[in]	lo	the smallest length scale that needs to be precisely represented
[in]	hi	the largest length scale that needs to be precisely represented
[in]	eps	the precision threshold @parma[in] prnt print level

References GFit(), exact, f12sq_fit(), fit(), lo, and madness::print().

Referenced by GFit().

◆ F2GFit()

template<typename T, std::size_t NDIM>

GFit madness::GFit< T, NDIM >::F2GFit	(	double	gamma,
		double	lo,
		double	hi,
		double	eps,
		bool	prnt = false )

inlinestatic

return a fit for the F2G function

f2g = (1/(2 mu) * (1 - exp(-gamma r12)))^2 / r12 = 1/(4 mu^2) * [ 1/r12 - 2 exp(-gamma r12)/r12) + exp(-2 gamma r12)/r12 ]

Parameters

[in]	gamma	the exponent of the Slater function
[in]	lo	the smallest length scale that needs to be precisely represented
[in]	hi	the largest length scale that needs to be precisely represented
[in]	eps	the precision threshold @parma[in] prnt print level

References GFit(), bsh_fit(), coeffs(), coeffs_, exact, exponents(), exponents_, lo, MADNESS_CHECK, madness::constants::pi, madness::print(), print_accuracy(), and prune_small_coefficients().

Referenced by GFit().

◆ FGFit()

template<typename T, std::size_t NDIM>

GFit madness::GFit< T, NDIM >::FGFit	(	double	gamma,
		double	lo,
		double	hi,
		double	eps,
		bool	prnt = false )

inlinestatic

return a fit for the FG function

fg = 1/(2 mu) * (1 - exp(-gamma r12)) / r12 = 1/(2 mu) *( 1/r12 - exp(-gamma r12)/r12) = 1/(2 mu) * (coulomb - bsh)

Parameters

[in]	gamma	the exponent of the Slater function
[in]	lo	the smallest length scale that needs to be precisely represented
[in]	hi	the largest length scale that needs to be precisely represented
[in]	eps	the precision threshold @parma[in] prnt print level

References GFit(), bsh_fit(), coeffs(), coeffs_, exact, exponents(), exponents_, lo, MADNESS_CHECK, madness::constants::pi, madness::print(), print_accuracy(), and prune_small_coefficients().

Referenced by GFit().

◆ GaussFit()

template<typename T, std::size_t NDIM>

GFit madness::GFit< T, NDIM >::GaussFit	(	double	gamma,
		double	lo,
		double	hi,
		double	eps,
		bool	prnt = false )

inlinestatic

return a (trivial) fit for a single Gauss function

the Gauss function is defined by f(r) = exp(-\gamma r^2)

Parameters

[in]	gamma	the exponent of the Gauss function
[in]	lo	the smallest length scale that needs to be precisely represented
[in]	hi	the largest length scale that needs to be precisely represented
[in]	eps	the precision threshold @parma[in] prnt print level

References GFit(), exact, fit(), lo, and madness::print().

Referenced by GFit().

◆ gaussian_spherical_moments()

template<typename T, std::size_t NDIM>

void madness::GFit< T, NDIM >::gaussian_spherical_moments	(	double	alpha,
		double	R,
		Tensor< double > &	q )

inlinestaticprivate

References alpha, madness::constants::pi, pow(), q(), and R.

Referenced by bsh_fit().

◆ GeneralFit()

template<typename T, std::size_t NDIM>

GFit madness::GFit< T, NDIM >::GeneralFit ( )

inlinestatic

return a fit for a general isotropic function

note that the error is controlled over a uniform grid, the boundaries will be poorly represented in general. Following Beylkin 2005

References GFit(), and MADNESS_EXCEPTION.

◆ InverseRSqFit()

template<typename T, std::size_t NDIM>

GFit madness::GFit< T, NDIM >::InverseRSqFit	(	double	lo,
		double	hi,
		double	eps,
		bool	prnt = false )

inlinestatic

return a fit for the inverse-square function 1/r^2

Built from the Beylkin-Monzon integral representation 1/r^2 = ∫_0^∞ exp(-t r^2) dt After the substitution t = exp(2s), this is trapezoidally discretized to give 1/r^2 ≈ Σ_k c_k exp(-α_k r^2) with c_k = 2 h exp(2 s_k), α_k = exp(2 s_k).

Implementation is a thin wrapper around bsh_fit_ndim(4, 0, ...), which fits the 4-D Poisson Green's function (proportional to 1/r^2 with a factor 0.5/π^2). The coefficients are rescaled by 4π^2 so that the returned fit is the pointwise approximation of 1/r^2.

Parameters

[in]	lo	the smallest length scale that needs to be precisely represented
[in]	hi	the largest length scale that needs to be precisely represented
[in]	eps	the precision threshold
[in]	prnt	print level

References GFit(), alpha, e(), exact, fit(), h(), lo, and madness::print().

Referenced by GFit(), and test_invrsq_operator().

◆ operator=()

template<typename T, std::size_t NDIM>

GFit & madness::GFit< T, NDIM >::operator= ( const GFit< T, NDIM > & other )

inline

assignment operator

References GFit(), coeffs_, and exponents_.

◆ print_accuracy()

template<typename T, std::size_t NDIM>

template<typename opT>

void madness::GFit< T, NDIM >::print_accuracy	(	opT	op,
		const double	lo,
		const double	hi ) const

inlineprivate

print coefficients and exponents, and values and errors

Parameters

[in]	op	the exact function, e.g. 1/r, exp(-mu r), etc
[in]	lo	lower bound for the range r
[in]	hi	higher bound for the range r

References coeffs_, exact, exponents_, lo, op(), pow(), and test().

Referenced by F2GFit(), and FGFit().

◆ prune_small_coefficients()

template<typename T, std::size_t NDIM>

void madness::GFit< T, NDIM >::prune_small_coefficients	(	const double	eps,
		const double	lo,
		const double	hi,
		Tensor< double > &	coeff,
		Tensor< double > &	expnt,
		const long	first_prunable = 1 )

inlinestatic

Fold the most diffuse Gaussians of a fit into their neighbours while the fit stays accurate to eps on [lo, hi]. Works from the tail inwards and stops at the first term that cannot be merged; terms before first_prunable are never merged into (so first_prunable - 1 and everything before it keep their coefficients).

References std::abs(), lo, and madness::BaseTensor::size().

Referenced by bsh_fit(), F2GFit(), FGFit(), and truncate_mixed_expansion().

◆ slater_fit()

template<typename T, std::size_t NDIM>

void madness::GFit< T, NDIM >::slater_fit	(	double	gamma,
		double	lo,
		double	hi,
		double	eps,
		Tensor< double > &	pcoeff,
		Tensor< double > &	pexpnt,
		bool	prnt )

inlinestaticprivate

fit a Slater function using a sum of Gaussians

formula inspired by the BSH fit, with the roles of r and mu exchanged see also Eq. (A3) in S. Ten-no, Initiation of explicitly correlated Slater-type geminal theory, Chem. Phys. Lett., vol. 398, no. 1, pp. 56-61, 2004.

References e(), exact, h(), lo, MADNESS_CHECK, madness::constants::pi, pow(), and test().

Referenced by f12_fit(), f12sq_fit(), and SlaterFit().

◆ SlaterFit()

template<typename T, std::size_t NDIM>

GFit madness::GFit< T, NDIM >::SlaterFit	(	double	gamma,
		double	lo,
		double	hi,
		double	eps,
		bool	prnt = false )

inlinestatic

return a fit for the Slater function

the Slater function is defined by f(r) = exp(-\gamma r)

Parameters

[in]	gamma	the exponent of the Slater function
[in]	lo	the smallest length scale that needs to be precisely represented
[in]	hi	the largest length scale that needs to be precisely represented
[in]	eps	the precision threshold @parma[in] prnt print level

References GFit(), exact, fit(), lo, madness::print(), and slater_fit().

Referenced by GFit(), madness::SlaterFunctionInterface::fit(), madness::GradSlaterOperator(), and test_combine_generic().

◆ truncate_mixed_expansion()

template<typename T, std::size_t NDIM>

void madness::GFit< T, NDIM >::truncate_mixed_expansion	(	Tensor< double > &	c,
		Tensor< double > &	e,
		const std::array< LatticeRange, NDIM > &	lattice_ranges,
		const Tensor< double > &	cell_width,
		double	lo,
		double	hi_fin,
		double	eps )

inlinestatic

Truncate the fit of a kernel that is lattice-summed along some axes.

Along an axis with an infinite lattice sum, a Gaussian with exponent below tcut = 0.25/L^2 (L the largest such cell width) has a lattice sum that is flat to exp(-4 pi^2) over the cell: a gauge constant. If every axis is summed, those terms are dropped (all but the first, as before). If some axes are not summed, the same terms are not flat along them and carry real potential there – so they are kept as far as the finite axes need them: the tail is folded into its neighbours while the fit stays accurate to eps on [lo, hi_fin], where hi_fin is the largest distance the finite axes can reach. The first term below tcut is never merged into a term above it, whose lattice sum is not flat.

Parameters

[in,out]	c	coefficients of the fit; truncated (and possibly rescaled) on return
[in,out]	e	exponents of the fit, in decreasing order
[in]	lattice_ranges	lattice range of each axis
[in]	cell_width	width of each axis of the cell
[in]	lo	smallest distance the fit must represent accurately
[in]	hi_fin	largest distance the finite (non-infinite) axes must represent accurately
[in]	eps	accuracy of the fit on [lo, hi_fin]

References b, c, d, e(), madness::FunctionDefaults< NDIM >::cell_width, lo, NDIM, and prune_small_coefficients().

Member Data Documentation

◆ coeffs_

template<typename T, std::size_t NDIM>

Tensor<T> madness::GFit< T, NDIM >::coeffs_

private

the coefficients of the expansion f(x) = \sum_m coeffs[m] exp(-exponents[m] * x^2)

Referenced by coeffs(), F2GFit(), FGFit(), operator=(), and print_accuracy().

◆ exponents_

template<typename T, std::size_t NDIM>

Tensor<T> madness::GFit< T, NDIM >::exponents_

private

the exponents of the expansion f(x) = \sum_m coeffs[m] exp(-exponents[m] * x^2)

Referenced by exponents(), F2GFit(), FGFit(), operator=(), and print_accuracy().

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

gfit.h

Public Member Functions

Static Public Member Functions

Private Member Functions

Static Private Member Functions

Private Attributes

Constructor & Destructor Documentation

◆ GFit() [1/4]

◆ GFit() [2/4]

◆ GFit() [3/4]

◆ GFit() [4/4]

Member Function Documentation

◆ bsh_fit()

◆ bsh_fit_ndim()

◆ bsh_spherical_moments()

◆ BSHFit()

◆ coeffs()

◆ CoulombFit()

◆ exponents()

◆ f12_fit()

◆ F12Fit()

◆ f12sq_fit()

◆ F12sqFit()

◆ F2GFit()

◆ FGFit()

◆ GaussFit()

◆ gaussian_spherical_moments()

◆ GeneralFit()

◆ InverseRSqFit()

◆ operator=()

◆ print_accuracy()

◆ prune_small_coefficients()

◆ slater_fit()

◆ SlaterFit()

◆ truncate_mixed_expansion()

Member Data Documentation

◆ coeffs_

◆ exponents_