3.3.12. horton/gbasis/ints.h – Evaluation of integrals of Gaussian basis functions

class

Inherits from GBCalculator

Subclassed by GB2AttractionIntegral, GB2KineticIntegral, GB2MomentIntegral, GB2OverlapIntegral

Public Functions

GB2Integral::GB2Integral(long max_shell_type)

void GB2Integral::reset(long shell_type0, long shell_type1, const double *r0, const double *r1)

virtual void GB2Integral::add(double coeff, double alpha0, double alpha1, const double *scales0, const double *scales1)
= 0

void GB2Integral::cart_to_pure()

const long GB2Integral::get_shell_type0() const

const long GB2Integral::get_shell_type1() const

Protected Attributes

long GB2Integral::shell_type0

long GB2Integral::shell_type1

const double *GB2Integral::r0

const double *GB2Integral::r1

IterPow2 GB2Integral::i2p

class

#include <ints.h>

Compute the overlap integrals in a Gaussian orbital basis.

Inherits from GB2Integral

Public Functions

GB2OverlapIntegral::GB2OverlapIntegral(long max_shell_type)

void GB2OverlapIntegral::add(double coeff, double alpha0, double alpha1, const double *scales0, const double *scales1)

class

#include <ints.h>

Compute the kinetic integrals in a Gaussian orbital basis.

Inherits from GB2Integral

Public Functions

GB2KineticIntegral::GB2KineticIntegral(long max_shell_type)

void GB2KineticIntegral::add(double coeff, double alpha0, double alpha1, const double *scales0, const double *scales1)

class

#include <ints.h>

Compute the nuclear attraction integrals in a Gaussian orbital basis.

Inherits from GB2Integral

Subclassed by GB2ErfAttractionIntegral, GB2GaussAttractionIntegral, GB2NuclearAttractionIntegral

Public Functions

GB2AttractionIntegral::GB2AttractionIntegral(long max_shell_type, double *charges, double *centers, long ncharge)

Initialize a GB2AttractionIntegral object.

Parameters

• max_shell_type -

  Highest angular momentum index to be expected in the reset method.

GB2AttractionIntegral::~GB2AttractionIntegral()

void GB2AttractionIntegral::add(double coeff, double alpha0, double alpha1, const double *scales0, const double *scales1)

Add results for a combination of Cartesian primitive shells to the work array.

Parameters

• coeff -

  Product of the contraction coefficients of the four primitives.

• alpha0 -

  The exponent of primitive shell 0.

• alpha1 -

  The exponent of primitive shell 1.

• scales0 -

  The normalization prefactors for basis functions in primitive shell 0

• scales1 -

  The normalization prefactors for basis functions in primitive shell 1

virtual void GB2AttractionIntegral::laplace_of_potential(double gamma, double arg, long mmax, double *output)
= 0

Evaluate the Laplace transform of the the potential applied to nuclear attraction terms.

For theoretical details and the precise definition of the Laplace transform, we refer to the following paper:

Ahlrichs, R. A simple algebraic derivation of the Obara-Saika scheme for general two-electron interaction potentials. Phys. Chem. Chem. Phys. 8, 3072–3077 (2006). 10.1039/B605188J

For the general definition of this transform, see Eq. (8) in the reference above. Section 5 contains solutions of the Laplace transform for several popular cases.

Parameters

• gamma -

  Sum of the exponents of the two gaussian functions involved in the integral. Similar to the first term in Eq. (3) in Ahlrichs’ paper.

• arg -

  Rescaled distance between the two centers obtained from the application of the Gaussian product theorem. Equivalent to Eq. (5) in Ahlrichs’ paper.

• mmax -

  Maximum derivative of the Laplace transform to be considered.

• output -

  Output array. The size must be at least mmax + 1.

Private Members

double *GB2AttractionIntegral::charges

Array with values of the nuclear charges.

double *GB2AttractionIntegral::centers

The centers where the charges are located.

long GB2AttractionIntegral::ncharge

Number of nuclear charges.

double *GB2AttractionIntegral::work_g0

Temporary array to store intermediate results.

double *GB2AttractionIntegral::work_g1

Temporary array to store intermediate results.

double *GB2AttractionIntegral::work_g2

Temporary array to store intermediate results.

double *GB2AttractionIntegral::work_boys

Temporary array to store the laplace of the interaction potential.

class

#include <ints.h>

Nuclear Electron Attraction two-center integrals.

The potential is 1/r.

Inherits from GB2AttractionIntegral

Public Functions

GB2NuclearAttractionIntegral::GB2NuclearAttractionIntegral(long max_shell_type, double *charges, double *centers, long ncharge)

Initialize a GB2NuclearAttractionIntegral object.

Parameters

• max_shell_type -

  Highest angular momentum index to be expected in the reset method.

• charges -

  Array with values of the charges.

• centers -

  The centers [[C1_x, C1_y, C1_z],[…],] around which the moment integrals are computed.

• ncharge -

  Number of nuclear charges.

void GB2NuclearAttractionIntegral::laplace_of_potential(double gamma, double arg, long mmax, double *output)

Evaluate the Laplace transform of the ordinary Coulomb potential.

See Eq. (39) in Ahlrichs’ paper. This is basically a rescaled Boys function.

See base class for more details.

class

#include <ints.h>

Short-range electron repulsion four-center integrals.

The potential is erf(mu*r)/r.

Inherits from GB2AttractionIntegral

Public Functions

GB2ErfAttractionIntegral::GB2ErfAttractionIntegral(long max_shell_type, double *charges, double *centers, long ncharge, double mu)

Initialize a GB2ErfAttractionIntegral object.

Parameters

• max_shell_type -

  Highest angular momentum index to be expected in the reset method.

• charges -

  Array with values of the charges.

• centers -

  The centers [[C1_x, C1_y, C1_z],[…],] around which the moment integrals are computed.

• ncharge -

  Number of nuclear charges.

• mu -

  The range-separation parameter.

void GB2ErfAttractionIntegral::laplace_of_potential(double gamma, double arg, long mmax, double *output)

Evaluate the Laplace transform of the long-range Coulomb potential. (The short-range part is damped away using an error function.) See (52) in Ahlrichs’ paper.

See base class for more details.

const double GB2ErfAttractionIntegral::get_mu() const

The range-separation parameter.

Private Members

double GB2ErfAttractionIntegral::mu

The range-separation parameter.

class

#include <ints.h>

Gaussian nuclear electron attraction two-center integrals.

The potential is c exp(-alpha r^2).

Inherits from GB2AttractionIntegral

Public Functions

GB2GaussAttractionIntegral::GB2GaussAttractionIntegral(long max_shell_type, double *charges, double *centers, long ncharge, double c, double alpha)

Initialize a GB2GaussAttractionIntegral object.

Parameters

• max_shell_type -

  Highest angular momentum index to be expected in the reset method.

• charges -

  Array with values of the charges.

• centers -

  The centers [[C1_x, C1_y, C1_z],[…],] around which the moment integrals are computed.

• ncharge -

  Number of nuclear charges.

• c -

  Coefficient of the gaussian.

• alpha -

  Exponential parameter of the gaussian.

void GB2GaussAttractionIntegral::laplace_of_potential(double gamma, double arg, long mmax, double *output)

Evaluate the Laplace transform of the Gaussian potential.

See Ahlrichs’ paper for details. This type of potential is used in the papers of P.M.W Gill et al. and J. Toulouse et al.:

Gill, P. M. W., & Adamson, R. D. (1996). A family of attenuated Coulomb operators. Chem. Phys. Lett., 261(1-2), 105–110. http://doi.org/10.1016/0009-2614(96)00931-1

Toulouse, J., Colonna, F., & Savin, A. (2004). Long-range-short-range separation of the electron-electron interaction in density-functional theory. Phys. Rev. A, 70, 62505. http://doi.org/10.1103/PhysRevA.70.062505

See base class for more details.

const double GB2GaussAttractionIntegral::get_c() const

Coefficient of the gaussian.

const double GB2GaussAttractionIntegral::get_alpha() const

Exponential parameter of the gaussian.

Private Members

double GB2GaussAttractionIntegral::c

Coefficient of the gaussian.

double GB2GaussAttractionIntegral::alpha

Exponential parameter of the gaussian.

class

#include <ints.h>

Compute the (multipole) moment integrals in a Gaussian orbital basis. < gto_a | (x - C_x)^l (y - C_y)^m (z - C_z)^n | gto_b >.

Inherits from GB2Integral

Public Functions

GB2MomentIntegral::GB2MomentIntegral(long max_shell_type, long *xyz, double *center)

Initialize Moment integral calculator.

Parameters

• max_shell_type -

  The highest angular momentum index suported

• xyz -

  The powers of x,y,z in the integrals (l, m, n).

• center -

  The center [C_x, C_y, C_z] around which the moment integrals arecomputed

void GB2MomentIntegral::add(double coeff, double alpha0, double alpha1, const double *scales0, const double *scales1)

Add integrals for a pair of primite shells to the current contraction.

Parameters

• coeff -

  The contraction coefficient for the current primitive.

• alpha0 -

  The exponent of the primitive shell 0.

• alpha1 -

  The exponent of the primitive shell 1.

• scales0 -

  The normalization constants for the basis functions in primitive shell 0.

• scales1 -

  The normalization constants for the basis functions in primitive shell 1.

Private Members

long *GB2MomentIntegral::xyz

Powers for x, y and z of the multipole moment.

double *GB2MomentIntegral::center

The origin w.r.t. to which the multipole moment is computed.

class

#include <ints.h>

Base class for four-center integrals.

Inherits from GBCalculator

Subclassed by GB4IntegralLibInt

Public Functions

GB4Integral::GB4Integral(long max_shell_type)

Initialize a GB4Integral object.

Parameters

• max_shell_type -

  Highest angular momentum index to be expected in the reset method.

void GB4Integral::reset(long shell_type0, long shell_type1, long shell_type2, long shell_type3, const double *r0, const double *r1, const double *r2, const double *r3)

Set internal parameters for a new group of four contractions.

Parameters

• shell_type0 -

  Angular momentum index for contraction 0.

• shell_type1 -

  Angular momentum index for contraction 1.

• shell_type2 -

  Angular momentum index for contraction 2.

• shell_type3 -

  Angular momentum index for contraction 3.

• r0 -

  Cartesian coordinates of center 0.

• r1 -

  Cartesian coordinates of center 1.

• r2 -

  Cartesian coordinates of center 2.

• r3 -

  Cartesian coordinates of center 3.

virtual void GB4Integral::add(double coeff, double alpha0, double alpha1, double alpha2, double alpha3, const double *scales0, const double *scales1, const double *scales2, const double *scales3)
= 0

Add results for a combination of Cartesian primitive shells to the work array.

Parameters

• coeff -

  Product of the contraction coefficients of the four primitives.

• alpha0 -

  The exponent of primitive shell 0.

• alpha1 -

  The exponent of primitive shell 1.

• alpha2 -

  The exponent of primitive shell 2.

• alpha3 -

  The exponent of primitive shell 3.

• scales0 -

  The normalization prefactors for basis functions in primitive shell 0

• scales1 -

  The normalization prefactors for basis functions in primitive shell 1

• scales2 -

  The normalization prefactors for basis functions in primitive shell 2

• scales3 -

  The normalization prefactors for basis functions in primitive shell 3

void GB4Integral::cart_to_pure()

Transform the results in the work array from Cartesian to pure functions where needed.

const long GB4Integral::get_shell_type0() const

Shell type of contraction 0.

const long GB4Integral::get_shell_type1() const

Shell type of contraction 1.

const long GB4Integral::get_shell_type2() const

Shell type of contraction 2.

const long GB4Integral::get_shell_type3() const

Shell type of contraction 3.

Protected Attributes

long GB4Integral::shell_type0

Shell type of contraction 0.

long GB4Integral::shell_type1

Shell type of contraction 1.

long GB4Integral::shell_type2

Shell type of contraction 2.

long GB4Integral::shell_type3

Shell type of contraction 3.

const double *GB4Integral::r0

Center of contraction 0.

const double *GB4Integral::r1

Center of contraction 1.

const double *GB4Integral::r2

Center of contraction 2.

const double *GB4Integral::r3

Center of contraction 3.

struct

#include <ints.h>

Arguments associated with one primitive shell in LibInt conventions.

Public Members

unsigned int libint_arg_t::am

Shell type in LibInt conventions.

const double *libint_arg_t::r

Center of a primitive shell.

double libint_arg_t::alpha

Exponent of a primitive shell.

class

#include <ints.h>

Base class for four-center integrals that use LibInt.

Inherits from GB4Integral

Subclassed by GB4ElectronRepulsionIntegralLibInt, GB4ErfIntegralLibInt, GB4GaussIntegralLibInt, GB4RAlphaIntegralLibInt

Public Functions

GB4IntegralLibInt::GB4IntegralLibInt(long max_shell_type)

Initialize a GB4IntegralLibInt object.

Parameters

• max_shell_type -

  Highest angular momentum index to be expected in the reset method.

GB4IntegralLibInt::~GB4IntegralLibInt()

void GB4IntegralLibInt::reset(long shell_type0, long shell_type1, long shell_type2, long shell_type3, const double *r0, const double *r1, const double *r2, const double *r3)

Set internal parameters for a new group of four contractions.

See base class for details.

void GB4IntegralLibInt::add(double coeff, double alpha0, double alpha1, double alpha2, double alpha3, const double *scales0, const double *scales1, const double *scales2, const double *scales3)

Add results for a combination of Cartesian primitive shells to the work array.

See base class for details.

virtual void GB4IntegralLibInt::laplace_of_potential(double prefac, double rho, double t, long mmax, double *output)
= 0

Evaluate the Laplace transform of the the potential.

For theoretical details and the precise definition of the Laplace transform, we refer to the following paper:

Ahlrichs, R. A simple algebraic derivation of the Obara-Saika scheme for general two-electron interaction potentials. Phys. Chem. Chem. Phys. 8, 3072–3077 (2006). 10.1039/B605188J

For the general definition of this transform, see Eq. (8) in the reference above. Section 5 contains solutions of the Laplace transform for several popular cases.

Parameters

• prefac -

  Prefactor with which all results in the output array are multiplied.

• rho -

  See Eq. (3) in Ahlrichs’ paper.

• t -

  Rescaled distance between the two centers obtained from the application of the Gaussian product theorem. See Eq. (5) in Ahlrichs’ paper.

• mmax -

  Maximum derivative of the Laplace transform to be considered.

• output -

  Output array. The size must be at least mmax + 1.

Private Members

Libint_eri_t GB4IntegralLibInt::erieval

LibInt runtime object.

libint_arg_t GB4IntegralLibInt::libint_args[4]

Arguments (shell info) for libint.

long GB4IntegralLibInt::order[4]

Re-ordering of shells for compatibility with LibInt.

double GB4IntegralLibInt::ab[3]

Relative vector from shell 2 to 0 (LibInt order).

double GB4IntegralLibInt::cd[3]

Relative vector from shell 3 to 1 (LibInt order).

double GB4IntegralLibInt::ab2

Norm squared of ab.

double GB4IntegralLibInt::cd2

Norm squared of cd.

class

#include <ints.h>

Electron repulsion four-center integrals.

The potential is 1/r.

Inherits from GB4IntegralLibInt

Public Functions

GB4ElectronRepulsionIntegralLibInt::GB4ElectronRepulsionIntegralLibInt(long max_shell_type)

Initialize a GB4ElectronRepulsionIntegralLibInt object.

Parameters

• max_shell_type -

  Highest angular momentum index to be expected in the reset method.

void GB4ElectronRepulsionIntegralLibInt::laplace_of_potential(double prefac, double rho, double t, long mmax, double *output)

Evaluate the Laplace transform of the ordinary Coulomb potential.

See Eq. (39) in Ahlrichs’ paper. This is basically a rescaled Boys function.

See base class for more details.

class

#include <ints.h>

Short-range electron repulsion four-center integrals.

The potential is erf(mu*r)/r.

Inherits from GB4IntegralLibInt

Public Functions

GB4ErfIntegralLibInt::GB4ErfIntegralLibInt(long max_shell_type, double mu)

Initialize a GB4ErfIntegralLibInt object.

Parameters

• max_shell_type -

  Highest angular momentum index to be expected in the reset method.

• mu -

  The range-separation parameter

void GB4ErfIntegralLibInt::laplace_of_potential(double prefac, double rho, double t, long mmax, double *output)

Evaluate the Laplace transform of the long-range Coulomb potential. (The short-range part is damped away using an error function.) See (52) in Ahlrichs’ paper.

See base class for more details.

const double GB4ErfIntegralLibInt::get_mu() const

The range-separation parameter.

Private Members

double GB4ErfIntegralLibInt::mu

The range-separation parameter.

class

#include <ints.h>

Gaussian electron repulsion four-center integrals.

The potential is c exp(-alpha r^2).

Inherits from GB4IntegralLibInt

Public Functions

GB4GaussIntegralLibInt::GB4GaussIntegralLibInt(long max_shell_type, double c, double alpha)

Initialize a GB4GaussIntegralLibInt object.

Parameters

• max_shell_type -

  Highest angular momentum index to be expected in the reset method.

• c -

  Coefficient of the gaussian.

• alpha -

  Exponential parameter of the gaussian.

void GB4GaussIntegralLibInt::laplace_of_potential(double prefac, double rho, double t, long mmax, double *output)

Evaluate the Laplace transform of the Gaussian potential.

See Ahlrichs’ paper for details. This type of potential is used in the papers of P.M.W Gill et al. and J. Toulouse et al.:

Gill, P. M. W., & Adamson, R. D. (1996). A family of attenuated Coulomb operators. Chem. Phys. Lett., 261(1-2), 105–110. http://doi.org/10.1016/0009-2614(96)00931-1

Toulouse, J., Colonna, F., & Savin, A. (2004). Long-range-short-range separation of the electron-electron interaction in density-functional theory. Phys. Rev. A, 70, 62505. http://doi.org/10.1103/PhysRevA.70.062505

See base class for more details.

const double GB4GaussIntegralLibInt::get_c() const

Coefficient of the gaussian.

const double GB4GaussIntegralLibInt::get_alpha() const

Exponential parameter of the gaussian.

Private Members

double GB4GaussIntegralLibInt::c

Coefficient of the gaussian.

double GB4GaussIntegralLibInt::alpha

Exponential parameter of the gaussian.

class

#include <ints.h>

Gaussian electron repulsion four-center integrals.

The potential is r^alpha.

Inherits from GB4IntegralLibInt

Public Functions

GB4RAlphaIntegralLibInt::GB4RAlphaIntegralLibInt(long max_shell_type, double alpha)

Initialize a GB4RAlphaIntegralLibInt object.

Parameters

• max_shell_type -

  Highest angular momentum index to be expected in the reset method.

• alpha -

  The power of r in the potential.

void GB4RAlphaIntegralLibInt::laplace_of_potential(double prefac, double rho, double t, long mmax, double *output)

Evaluate the Laplace transform of the r^alpha potential. See Eq. (49) in Ahlrichs’ paper.

See base class for more details.

const double GB4RAlphaIntegralLibInt::get_alpha() const

The power of r.

Private Members

double GB4RAlphaIntegralLibInt::alpha

The power of r.