Namespaces
namespace	grid

Classes
struct	GridSettings
	Extended settings for Becke grid generation. More...
struct	PartitionMethod
struct	RadialGrid
	Structure to hold radial grid data. More...
struct	AtomGrid
	Structure to hold atom-centered grid data. More...
class	BeckeWeightDerivative
	Per-point derivatives of Becke-partitioned weights. More...
struct	OrcaCOSXAtomicGrid
	One ORCA 6.1 COSX atomic grid: number of radial shells, the M3 scaling xi (bohr), and the angular grid as runs of (Lebedev order, shell count) from the nucleus outwards. More...
class	MolecularGrid
	Class for molecular integration grids in DFT. More...
class	MolecularGridPoints
	Class for storing and managing molecular grid points. More...
struct	GridBoundingSphere
	Bounding sphere for a group of grid points. More...
struct	GridBatchLeaf
	Leaf node containing grid point batch. More...
struct	SpatialHierarchySettings
	Configuration for hierarchy construction. More...
class	SpatialGridHierarchy
	Spatial hierarchy for molecular grid points using Morton ordering. More...
class	AOBasis

Enumerations
enum class	PruningScheme { None , NWChem , NumGrid }
	Enumeration of available angular pruning schemes. More...
enum class	GridQuality { Coarse , Standard , Fine , VeryFine }
	Standard grid quality levels (similar to Q-Chem's SG system). More...
enum class	COSXGridLevel { Grid1 , Grid2 , Grid3 }
	COSX/SGX grid levels matching ORCA's GRIDX scheme. More...
enum class	PartitionFunction { Becke , StratmannScuseria }
	Enumeration of available partition functions for atomic weight calculation. More...
enum class	RadialGridMethod { LMG , TreutlerAlrichs , MuraKnowles , Becke , GaussChebyshev , EulerMaclaurin }
	Enumeration of available radial grid generation methods. More...

Functions
GridSettings	get_grid_settings (GridQuality quality)
	Get grid settings for a specific quality level.
std::string	grid_quality_to_string (GridQuality quality)
	Convert GridQuality enum to string.
GridQuality	grid_quality_from_string (const std::string &str)
	Convert string to GridQuality enum.
std::string	cosx_grid_level_to_string (COSXGridLevel level)
	Convert COSXGridLevel enum to string.
double	cosx_grid_int_acc (COSXGridLevel level)
	Get IntAcc value for a COSX grid level.
size_t	calculate_radial_points_orca (double int_acc, size_t atomic_number, double b=5.0)
	Calculate number of radial points using ORCA's formula.
void	print_cosx_grid_summary (const GridSettings &settings, COSXGridLevel level, size_t num_points, size_t num_batches, size_t num_atoms)
	Print ORCA-style grid generation summary.
uint_fast16_t	nearest_grid_level_at_or_above (uint_fast16_t n)
	Find the nearest Lebedev grid level at or above the requested number of points.
uint_fast16_t	nearest_grid_level_below (uint_fast16_t n)
	Find the nearest Lebedev grid level below the requested number of points.
size_t	angular_point_count (size_t level)
	Get the number of points in a Lebedev grid.
double	get_atomic_radius (size_t atomic_number)
	Get the atomic radius for a given atomic number.
Vec	becke_partition (const Vec &w)
	Apply the Becke partition function.
Vec	stratmann_scuseria_partition (const Vec &w)
	Apply the Stratmann-Scuseria partition function.
Mat	calculate_interatomic_distances (const Mat3N &positions)
	Calculate interatomic distances.
Mat3N	becke_weight_gradient (const Mat3N &points, const Vec &weights, const std::vector< std::pair< size_t, size_t > > &atom_blocks, const Vec &e, const Mat3N &positions, const Vec &radii, bool treutler_ahlrichs)
	Grid-weight response: sum_g e_g dw_g/dR_B for every atom B.
Mat	becke_weight_hessian (const Mat3N &points, const Vec &weights, const std::vector< std::pair< size_t, size_t > > &atom_blocks, const Vec &e, const Mat3N &positions, const Vec &radii, bool treutler_ahlrichs)
	Second-order grid-weight response: sum_g e_g d2 w_g / dR_X dR_Y (3N x 3N), points moving with their parent atoms, e_g held fixed.
Mat	calculate_atomic_grid_weights (PartitionMethod method, const Mat &grid_points, const Mat &atomic_positions, const Vec &radii, const Mat &interatomic_distances)
	Calculate Becke atomic weights for grid points.
IVec	prune_nwchem_scheme (size_t nuclear_charge, size_t max_angular, size_t num_radial, const Vec &radii)
	Apply the NWChem pruning scheme to a radial grid.
IVec	prune_numgrid_scheme (size_t atomic_number, size_t max_angular, size_t min_angular, const Vec &radii)
	Apply the NumGrid pruning scheme to a radial grid.
IVec	prune_orca_scheme (size_t atomic_number, const std::array< size_t, 5 > &region_grids, const Vec &radii)
	Apply ORCA-style 5-region angular pruning.
RadialGrid	generate_becke_radial_grid (size_t num_points, double rm=1.0)
	Generate a Becke radial grid.
RadialGrid	generate_mura_knowles_radial_grid (size_t num_points, size_t charge)
	Generate a Mura-Knowles radial grid.
RadialGrid	generate_treutler_alrichs_radial_grid (size_t num_points)
	Generate a Treutler-Alrichs radial grid.
RadialGrid	generate_perez_jorda_m3_radial_grid (size_t num_points, double xi)
	Perez-Jorda Gauss-Chebyshev (second kind) quadrature with the Treutler-Ahlrichs M3 mapping r = xi / ln 2 * ln(2 / (1 - x)).
OrcaCOSXAtomicGrid	orca_cosx_atomic_grid (int atomic_number, int level)
	ORCA 6.1 COSX atomic grid for an element and GRIDX level (1-3).
RadialGrid	generate_gauss_chebyshev_radial_grid (size_t num_points)
	Generate a Gauss-Chebyshev radial grid.
RadialGrid	generate_gauss_chebyshev_m3_radial_grid (size_t num_points, size_t atomic_number)
	Generate a Gauss-Chebyshev radial grid with M3 mapping.
RadialGrid	generate_euler_maclaurin_radial_grid (size_t num_points, double alpha)
	Generate a Euler-Maclaurin radial grid.
RadialGrid	generate_lmg_radial_grid (size_t atomic_number, double radial_precision, double alpha_max, int l_max, const occ::Vec &alpha_min)
	Generate a Lindh-Malmqvist-Gagliardi (LMG) radial grid.
double	lmg_inner (const double max_error, const double alpha_inner)
	Helper function to calculate LMG inner radial boundary.
double	lmg_outer (const double max_error, const double alpha_outer, const int l, const double guess)
	Helper function to calculate LMG outer radial boundary.
double	lmg_h (const double max_error, const int l, const double guess)
	Helper function to calculate LMG grid spacing parameter.
AtomGrid	generate_atom_grid (size_t atomic_number, const GridSettings &settings, RadialGridMethod method=RadialGridMethod::LMG, double alpha_max=0.0, int l_max=0, const Vec &alpha_min=Vec())
	Generate an atom-centered grid using the specified method.

Variables
const std::array< uint_fast16_t, 33 >	lebedev_grid_levels
	Array of available Lebedev grid levels.

Enumeration Type Documentation

◆ COSXGridLevel

enum class occ::numint::COSXGridLevel

strong

COSX/SGX grid levels matching ORCA's GRIDX scheme.

ORCA uses 3 grid levels for COSX with specific IntAcc values and 5-region angular pruning. These correspond to GRIDX 1, 2, 3 in ORCA.

Enumerator
Grid1	GRIDX 1: IntAcc=3.816, max 50 Lebedev points.
Grid2	GRIDX 2: IntAcc=4.020, max 110 Lebedev points.
Grid3	GRIDX 3: IntAcc=4.338, max 194 Lebedev points.

◆ GridQuality

enum class occ::numint::GridQuality

strong

Standard grid quality levels (similar to Q-Chem's SG system).

Enumerator
Coarse	Like SG-0: ~23 radial, ~170 angular points - fastest, basic accuracy.
Standard	Like SG-1: ~50 radial, ~194 angular points - standard for LDA/GGA.
Fine	Like SG-2: ~75 radial, ~302 angular points - meta-GGAs, B95/B97.
VeryFine	Like SG-3: ~99 radial, ~590 angular points - Minnesota functionals, highest accuracy.

◆ PartitionFunction

enum class occ::numint::PartitionFunction

strong

Enumeration of available partition functions for atomic weight calculation.

Enumerator
Becke	Becke's original partition function.
StratmannScuseria	Stratmann-Scuseria partition function.

◆ PruningScheme

enum class occ::numint::PruningScheme

strong

Enumeration of available angular pruning schemes.

Enumerator
None	No pruning (uniform angular points).
NWChem	NWChem pruning scheme.
NumGrid	NumGrid pruning scheme.

◆ RadialGridMethod

enum class occ::numint::RadialGridMethod

strong

Enumeration of available radial grid generation methods.

Enumerator
LMG	Lindh-Malmqvist-Gagliardi method.
TreutlerAlrichs	Treutler-Alrichs method.
MuraKnowles	Mura-Knowles method.
Becke	Becke's transformation method.
GaussChebyshev	Gauss-Chebyshev method.
EulerMaclaurin	Gauss-Chebyshev method.

Function Documentation

◆ angular_point_count()

size_t occ::numint::angular_point_count ( size_t level )

Get the number of points in a Lebedev grid.

Parameters

level The Lebedev grid level

Returns: size_t The number of points in the grid

◆ becke_partition()

Vec occ::numint::becke_partition ( const Vec & w )

Apply the Becke partition function.

Parameters

w	Input values to partition

Returns: Vec Partitioned values

◆ becke_weight_gradient()

Mat3N occ::numint::becke_weight_gradient	(	const Mat3N &	points,
		const Vec &	weights,
		const std::vector< std::pair< size_t, size_t > > &	atom_blocks,
		const Vec &	e,
		const Mat3N &	positions,
		const Vec &	radii,
		bool	treutler_ahlrichs )

Grid-weight response: sum_g e_g dw_g/dR_B for every atom B.

w_g = w0_g P_A(r_g) / sum_C P_C(r_g) is the Becke-partitioned weight of a point attached to its parent atom A (the point moves with A). This returns the derivative of sum_g e_g w_g with e_g held fixed, 3 x natoms. Only the Becke partition (with Becke or Treutler-Ahlrichs size adjustment, exactly as calculate_atomic_grid_weights builds it) is supported.

Parameters

points	all grid points (3 x N), grouped by parent atom
weights	the partitioned weights w_g
atom_blocks	(offset, count) of each atom's points
e	per-point quantity e_g (e.g. the XC energy density)
positions	atomic positions (3 x natoms)
radii	atomic radii used by the size adjustment
treutler_ahlrichs	whether the Treutler-Ahlrichs adjustment was used

◆ becke_weight_hessian()

Mat occ::numint::becke_weight_hessian	(	const Mat3N &	points,
		const Vec &	weights,
		const std::vector< std::pair< size_t, size_t > > &	atom_blocks,
		const Vec &	e,
		const Mat3N &	positions,
		const Vec &	radii,
		bool	treutler_ahlrichs )

Second-order grid-weight response: sum_g e_g d2 w_g / dR_X dR_Y (3N x 3N), points moving with their parent atoms, e_g held fixed.

Same partition as becke_weight_gradient.

◆ calculate_atomic_grid_weights()

Mat occ::numint::calculate_atomic_grid_weights	(	PartitionMethod	method,
		const Mat &	grid_points,
		const Mat &	atomic_positions,
		const Vec &	radii,
		const Mat &	interatomic_distances )

Calculate Becke atomic weights for grid points.

Parameters

grid_points	Grid points (3 x N)
atomic_positions	Atomic positions (3 x M)
radii	atomic radii (M)
interatomic_distances	Matrix of interatomic distances (M x M)

Returns: Mat Matrix of weights (N x M)

◆ calculate_interatomic_distances()

Mat occ::numint::calculate_interatomic_distances ( const Mat3N & positions )

Calculate interatomic distances.

Parameters

positions Atomic positions (3 x N)

Returns: Mat Matrix of interatomic distances

◆ calculate_radial_points_orca()

size_t occ::numint::calculate_radial_points_orca	(	double	int_acc,
		size_t	atomic_number,
		double	b = 5.0 )

Calculate number of radial points using ORCA's formula.

Uses: nr = (15 * IntAcc - 40) + b * ROW where ROW is the periodic table row of the element

Parameters

int_acc	IntAcc parameter
atomic_number	Atomic number of the element
b	Row multiplier (default 5.0)

Returns: size_t Number of radial points

◆ cosx_grid_int_acc()

double occ::numint::cosx_grid_int_acc ( COSXGridLevel level )

Get IntAcc value for a COSX grid level.

Returns the ORCA IntAcc parameter for the given COSX grid level

Parameters

level COSX grid level

Returns: double IntAcc value

◆ cosx_grid_level_to_string()

std::string occ::numint::cosx_grid_level_to_string ( COSXGridLevel level )

Convert COSXGridLevel enum to string.

Parameters

level COSX grid level

Returns: std::string String representation (e.g., "GRIDX 1")

◆ generate_atom_grid()

AtomGrid occ::numint::generate_atom_grid	(	size_t	atomic_number,
		const GridSettings &	settings,
		RadialGridMethod	method = RadialGridMethod::LMG,
		double	alpha_max = 0.0,
		int	l_max = 0,
		const Vec &	alpha_min = Vec() )

Generate an atom-centered grid using the specified method.

Parameters

atomic_number	Atomic number of the center atom
settings	Grid generation settings
method	The radial grid method to use
alpha_max	Maximum exponent from basis set (optional)
l_max	Maximum angular momentum from basis set (optional)
alpha_min	Minimum exponents for each angular momentum (optional)

Returns: AtomGrid The generated atom-centered grid

◆ generate_becke_radial_grid()

RadialGrid occ::numint::generate_becke_radial_grid	(	size_t	num_points,
		double	rm = 1.0 )

Generate a Becke radial grid.

Reference: A. D. Becke, J. Chem. Phys. 88, 2547 (1988)

Parameters

num_points	Number of radial points
rm	Scaling parameter

Returns: RadialGrid The generated radial grid

◆ generate_euler_maclaurin_radial_grid()

RadialGrid occ::numint::generate_euler_maclaurin_radial_grid	(	size_t	num_points,
		double	alpha )

Generate a Euler-Maclaurin radial grid.

Parameters

num_points	Number of radial points
alpha

Returns: RadialGrid The generated radial grid

◆ generate_gauss_chebyshev_m3_radial_grid()

RadialGrid occ::numint::generate_gauss_chebyshev_m3_radial_grid	(	size_t	num_points,
		size_t	atomic_number )

Generate a Gauss-Chebyshev radial grid with M3 mapping.

Uses the Treutler-Alrichs M3 mapping for transforming the Gauss-Chebyshev quadrature to an infinite interval suitable for atomic integrals: r = (xi / ln2) * ln(2 / (1-x)) where xi is a scaling parameter based on the atomic radius.

Reference: O. Treutler and R. Ahlrichs, J. Chem. Phys. 102, 346 (1995)

Parameters

num_points	Number of radial points
atomic_number	Atomic number (used for xi scaling)

Returns: RadialGrid The generated radial grid

◆ generate_gauss_chebyshev_radial_grid()

RadialGrid occ::numint::generate_gauss_chebyshev_radial_grid ( size_t num_points )

Generate a Gauss-Chebyshev radial grid.

Parameters

num_points Number of radial points

Returns: RadialGrid The generated radial grid

◆ generate_lmg_radial_grid()

RadialGrid occ::numint::generate_lmg_radial_grid	(	size_t	atomic_number,
		double	radial_precision,
		double	alpha_max,
		int	l_max,
		const occ::Vec &	alpha_min )

Generate a Lindh-Malmqvist-Gagliardi (LMG) radial grid.

Reference: T. Helgaker, P. Jørgensen, J. Olsen, "Molecular Electronic Structure Theory", John Wiley & Sons, 2000

Parameters

atomic_number	Atomic number
radial_precision	Precision parameter for the radial grid
alpha_max	Maximum exponent
l_max	Maximum angular momentum
alpha_min	Minimum exponents for each angular momentum

Returns: RadialGrid The generated radial grid

◆ generate_mura_knowles_radial_grid()

RadialGrid occ::numint::generate_mura_knowles_radial_grid	(	size_t	num_points,
		size_t	charge )

Generate a Mura-Knowles radial grid.

Reference: M. E. Mura and P. J. Knowles, J. Chem. Phys. 104, 9848 (1996)

Parameters

num_points	Number of radial points
charge	Atomic number

Returns: RadialGrid The generated radial grid

◆ generate_perez_jorda_m3_radial_grid()

RadialGrid occ::numint::generate_perez_jorda_m3_radial_grid	(	size_t	num_points,
		double	xi )

Perez-Jorda Gauss-Chebyshev (second kind) quadrature with the Treutler-Ahlrichs M3 mapping r = xi / ln 2 * ln(2 / (1 - x)).

Nodes x_i = (n+1-2i)/(n+1) + 2/pi (1 + 2/3 sin^2 t_i) cos t_i sin t_i, weights 16/(3(n+1)) sin^4 t_i, t_i = i pi / (n+1) [Perez-Jorda, San-Fabian, Moscardo, Comput. Phys. Commun. 70, 271 (1992)]. This is ORCA's "OptM3 with GC (2021)" radial grid. Weights include r^2 (not 4 pi); points ascending.

◆ generate_treutler_alrichs_radial_grid()

RadialGrid occ::numint::generate_treutler_alrichs_radial_grid ( size_t num_points )

Generate a Treutler-Alrichs radial grid.

Reference: O. Treutler and R. Ahlrichs, J. Chem. Phys. 102, 346 (1995)

Parameters

num_points Number of radial points

Returns: RadialGrid The generated radial grid

◆ get_atomic_radius()

double occ::numint::get_atomic_radius ( size_t atomic_number )

Get the atomic radius for a given atomic number.

Parameters

atomic_number The atomic number

Returns: double The atomic radius in Bohr

◆ get_grid_settings()

GridSettings occ::numint::get_grid_settings ( GridQuality quality )

Get grid settings for a specific quality level.

Parameters

quality Grid quality level

Returns: GridSettings configured for the specified quality

◆ grid_quality_from_string()

GridQuality occ::numint::grid_quality_from_string ( const std::string & str )

Convert string to GridQuality enum.

Parameters

str	String representation (case-insensitive)

Returns: GridQuality Grid quality level

Exceptions

std::invalid_argument if string is not recognized

◆ grid_quality_to_string()

std::string occ::numint::grid_quality_to_string ( GridQuality quality )

Convert GridQuality enum to string.

Parameters

quality Grid quality level

Returns: std::string String representation of the quality level

◆ lmg_h()

double occ::numint::lmg_h	(	const double	max_error,
		const int	l,
		const double	guess )

Helper function to calculate LMG grid spacing parameter.

Parameters

max_error	Maximum error parameter
l	Angular momentum
guess	Initial guess for the spacing

Returns: double Grid spacing parameter

◆ lmg_inner()

double occ::numint::lmg_inner	(	const double	max_error,
		const double	alpha_inner )

Helper function to calculate LMG inner radial boundary.

Parameters

max_error	Maximum error parameter
alpha_inner	Inner alpha parameter

Returns: double Inner radial boundary

◆ lmg_outer()

double occ::numint::lmg_outer	(	const double	max_error,
		const double	alpha_outer,
		const int	l,
		const double	guess )

Helper function to calculate LMG outer radial boundary.

Parameters

max_error	Maximum error parameter
alpha_outer	Outer alpha parameter
l	Angular momentum
guess	Initial guess for the boundary

Returns: double Outer radial boundary

◆ nearest_grid_level_at_or_above()

uint_fast16_t occ::numint::nearest_grid_level_at_or_above ( uint_fast16_t n )

Find the nearest Lebedev grid level at or above the requested number of points.

Parameters

n	Requested number of angular points

Returns: uint_fast16_t The actual number of angular points in the Lebedev grid

◆ nearest_grid_level_below()

uint_fast16_t occ::numint::nearest_grid_level_below ( uint_fast16_t n )

Find the nearest Lebedev grid level below the requested number of points.

Parameters

n	Requested number of angular points

Returns: uint_fast16_t The actual number of angular points in the Lebedev grid

◆ orca_cosx_atomic_grid()

OrcaCOSXAtomicGrid occ::numint::orca_cosx_atomic_grid	(	int	atomic_number,
		int	level )

ORCA 6.1 COSX atomic grid for an element and GRIDX level (1-3).

Measured from the grids ORCA 6.1.1 writes for single atoms (exact to 1e-9 in radii and weights). Returns num_radial == 0 for Z > 86.

◆ print_cosx_grid_summary()

void occ::numint::print_cosx_grid_summary	(	const GridSettings &	settings,
		COSXGridLevel	level,
		size_t	num_points,
		size_t	num_batches,
		size_t	num_atoms )

Print ORCA-style grid generation summary.

Prints a summary similar to ORCA's "COSX GRID GENERATION" output:

IntAcc, angular grid level, pruning method
Total grid points, batches, average points per batch/atom

Parameters

settings	Grid settings used
level	COSX grid level (Grid1, Grid2, Grid3)
num_points	Total number of grid points
num_batches	Number of batches
num_atoms	Number of atoms

◆ prune_numgrid_scheme()

IVec occ::numint::prune_numgrid_scheme	(	size_t	atomic_number,
		size_t	max_angular,
		size_t	min_angular,
		const Vec &	radii )

Apply the NumGrid pruning scheme to a radial grid.

Parameters

atomic_number	Atomic number
max_angular	Maximum number of angular points
min_angular	Minimum number of angular points
radii	Radial grid points

Returns: IVec Number of angular points for each radial point

◆ prune_nwchem_scheme()

IVec occ::numint::prune_nwchem_scheme	(	size_t	nuclear_charge,
		size_t	max_angular,
		size_t	num_radial,
		const Vec &	radii )

Apply the NWChem pruning scheme to a radial grid.

Parameters

nuclear_charge	Atomic number
max_angular	Maximum number of angular points
num_radial	Number of radial points
radii	Radial grid points

Returns: IVec Number of angular points for each radial point

◆ prune_orca_scheme()

IVec occ::numint::prune_orca_scheme	(	size_t	atomic_number,
		const std::array< size_t, 5 > &	region_grids,
		const Vec &	radii )

Apply ORCA-style 5-region angular pruning.

ORCA divides the radial grid into 5 regions, each with a different number of angular points. The region boundaries are scaled by the atomic radius and are optimized for each element.

Parameters

atomic_number	Atomic number
region_grids	Array of 5 Lebedev grid sizes for each region
radii	Radial grid points in Bohr

Returns: IVec Number of angular points for each radial point

◆ stratmann_scuseria_partition()

Vec occ::numint::stratmann_scuseria_partition ( const Vec & w )

Apply the Stratmann-Scuseria partition function.

Parameters

w	Input values to partition

Returns: Vec Partitioned values

Variable Documentation

◆ lebedev_grid_levels

const std::array<uint_fast16_t, 33> occ::numint::lebedev_grid_levels

extern

Array of available Lebedev grid levels.

Each element in the array represents a valid number of Lebedev grid points.

Namespaces

Classes

Enumerations

Functions

Variables

Enumeration Type Documentation

◆ COSXGridLevel

◆ GridQuality

◆ PartitionFunction

◆ PruningScheme

◆ RadialGridMethod

Function Documentation

◆ angular_point_count()

◆ becke_partition()

◆ becke_weight_gradient()

◆ becke_weight_hessian()

◆ calculate_atomic_grid_weights()

◆ calculate_interatomic_distances()

◆ calculate_radial_points_orca()

◆ cosx_grid_int_acc()

◆ cosx_grid_level_to_string()

◆ generate_atom_grid()

◆ generate_becke_radial_grid()

◆ generate_euler_maclaurin_radial_grid()

◆ generate_gauss_chebyshev_m3_radial_grid()

◆ generate_gauss_chebyshev_radial_grid()

◆ generate_lmg_radial_grid()

◆ generate_mura_knowles_radial_grid()

◆ generate_perez_jorda_m3_radial_grid()

◆ generate_treutler_alrichs_radial_grid()

◆ get_atomic_radius()

◆ get_grid_settings()

◆ grid_quality_from_string()

◆ grid_quality_to_string()

◆ lmg_h()

◆ lmg_inner()

◆ lmg_outer()

◆ nearest_grid_level_at_or_above()

◆ nearest_grid_level_below()

◆ orca_cosx_atomic_grid()

◆ print_cosx_grid_summary()

◆ prune_numgrid_scheme()

◆ prune_nwchem_scheme()

◆ prune_orca_scheme()

◆ stratmann_scuseria_partition()

Variable Documentation

◆ lebedev_grid_levels