3. API

3.1. pyscf.ita.promolecule module

class pyscf.ita.promolecule.ProMolecule(method, element_methods=None, pro_charge=None, pro_multiplicity=None)

Bases: object

Promolecule class.

A promolecule is an approximation of a molecule constructed from a linear combination of atomic and/or ionic species. Properties of this promolecule can be computed from those of the atomic and/or ionic species, depending on whether the property is an extensive one or an intensive one.

Examples:

>>> mol = gto.M(atom='H 0 0 0; H 0 0 1.1')
>>> mf = scf.HF(mol)
>>> mf.kernel()
>>> pro_charge, pro_multiplicity = {'H':0, 'O':0}, {'H':2, 'O':3}
>>> promol = ProMolecule.build(mf, pro_charge, pro_multiplicity)

build(method=None, pro_charge=None, pro_multiplicity=None)

Initialize a instance.

Parameters:

• method (PyscfMethod) – Pyscf method.

• pro_charge (Dict{str:int}, optional) – Dictionary of charge for each element, by default None.

• pro_multiplicity (Dict{str:int}, optional) – Dictionary of multiplicity for each element, by default None.

Returns:

Instance of ProMolecule class.

Return type:

ProMolecule

electron_density(grids_coords, spin='ab', deriv=2)

Compute the electron density of the promolecule at the desired points.

Parameters:

• grid_coords (np.ndarray((N, 3), dtype=float)) – Points at which to compute the density.

• spin (('ab' | 'a' | 'b' | 'm')) – Type of density to compute; either total, alpha-spin, beta-spin, or magnetization density, by default ‘ab’.

• deriv (int) – List of molecule and promolecule derivative+1 level, by default 2.

Returns:

Instance of PartitionDensity class.

Return type:

PartitionDensity

get_atom(element)

Get the Pyscf Mole object for given element.

Parameters:

element (str) – Symbol of element.

Returns:

Pyscf Mole object for free atom.

Return type:

Mole

get_method(element)

Get the Pyscf method for given element.

Parameters:

element (str) – Symbol of element.

Returns:

Pyscf method.

Return type:

PyscfMethod

get_scf(element)

Get the mean field Pyscf method for given element.

Parameters:

element (str) – Symbol of element.

Returns:

Mean field pyscf method.

Return type:

PyscfMethod

get_xc(element)

Get the exchange correlation functional for given element.

Parameters:

element (str) – Symbol of element.

Returns:

Name of exchange correlation functional.

Return type:

str

identical_method(method)

_summary_

Parameters:

method (PyscfMethod) – Pyscf scf method or post-scf method instance.

Returns:

_description_

Return type:

_type_

spheric_average_method(method)

_summary_

Parameters:

method (PyscfMethod) – Pyscf scf method or post-scf method instance.

Returns:

_description_

Return type:

_type_

3.2. pyscf.ita.aim module

class pyscf.ita.aim.Becke(mol=None, grids=None)

Bases: AIM

Becke’s fuzzy atom approach.

sharing_function(mol=None, grids=None)

Becke sharing funcition \(\omega_A(\mathbf{r})\) defined as:

\[\omega_A(\mathbf{r}) = \frac{V_A (\mathbf{r}) }{\sum_A V_A (\mathbf{r})}\]

Parameters:

• mol (Mole, optional) – Pyscf Mole instance, by default None.

• grids (Grid, optional) – Pyscf Grids instance, by default None.

Returns:

omega – Sharing functions for a list of atoms.

Return type:

List[np.ndarray((N,), dtype=float)]

class pyscf.ita.aim.Hirshfeld(promoldens=None)

Bases: AIM

Hirshfeld’s stockholder approach.

Examples:

>>> mol = gto.M(atom='H 0 0 0; H 0 0 1.1')
>>> mf = scf.HF(mol)
>>> mf.kernel()
>>> grids = dft.Grids(mol)
>>> grids.build()
>>> pro_charge, pro_multiplicity = {'H':0, 'O':0}, {'H':2, 'O':3}
>>> aim = Hirshfeld.build(mf, grids.coords, pro_charge, pro_multiplicity)
>>> omega = aim.sharing_function()

classmethod build(method, grids_coords, pro_charge=None, pro_multiplicity=None, spin='ab', deriv=0)

Class method to build the class.

Parameters:

• method (PyscfMethod) – Pyscf scf method or post-scf method instance.

• grids_coords (np.ndarray((N,), dtype=float)) – Grids coords of N points.

• pro_charge (Dict{str:int}, optional) – Dictionary of charge for each element, by default None.

• pro_multiplicity (Dict{str:int}, optional) – Dictionary of multiplicity for each element, by default None.

• spin (('ab' | 'a' | 'b' | 'm')) – Type of density to compute; either total, alpha-spin, beta-spin, or magnetization density, by default ‘ab’.

• deriv (int) – List of molecule and promolecule derivative+1 level, by default 0.

Returns:

Instance of Hirshfeld class.

Return type:

obj

sharing_function(promoldens=None)

Hirshfeld sharing funcition \(\omega_A(\mathbf{r})\) defined as:

\[\omega_A(\mathbf{r}) = \frac{\rho^0_A (\mathbf{r}) }{\sum_A \rho^0_A (\mathbf{r})}\]

Parameters:

promoldens (PartitionDensity) – PartitionDensity instance for promolecule, by default None.

Returns:

omega – Sharing functions for a list of atoms.

Return type:

List[np.ndarray((N,), dtype=float)]

3.3. pyscf.ita.dens module

class pyscf.ita.dens.ElectronDensity(rhos=None)

Bases: ndarray

Class to generate the electron density of the molecule at the desired points.

Examples:

>>> mol = gto.M(atom='H 0 0 0; H 0 0 1.1')
>>> grids = dft.gen_grid.Grids(mol)
>>> grids.build()
>>> mf = scf.HF(mol)
>>> mf.kernel()
>>> ed = ita.dens.ElectronDenstiy.build(mf, grids)

classmethod build(method, grids_coords, spin='ab', deriv=2)

Compute the electron density of the molecule at the desired points.

Parameters:

• method (PyscfMethod) – Pyscf scf method or post-scf method instance.

• grid_coords (np.ndarray((N, 3), dtype=float)) – Points at which to compute the density.

• spin (('ab' | 'a' | 'b' | 'm')) – Type of density to compute; either total, alpha-spin, beta-spin, or magnetization density, by default ‘ab’.

• deriv (int) – List of molecule and promolecule derivative+1 level, by default 2.

Returns:

obj – Instance of ElectronDensity class.

Return type:

ElectronDensity

density(mask=False, threshold=1e-30)

Electron density \(\rho\left(\mathbf{r}\right)\).

Parameters:

• mask (Bool) – If mask the corresponding element of the associated array which is less than given the threshold.

• threshold (float) – Threshold for array element to mask or fill.

Returns:

rho – Electron density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

gradient(mask=False, threshold=1e-30)

Gradient of electron density \(\nabla \rho\left(\mathbf{r}\right)\).

This is the first-order partial derivatives of electron density w.r.t. coordinate \(\mathbf{r} = \left(x\mathbf{i}, y\mathbf{j}, z\mathbf{k}\right)\),

\[\nabla\rho\left(\mathbf{r}\right) = \left(\frac{\partial}{\partial x}\mathbf{i}, \frac{\partial}{\partial y}\mathbf{j}, \frac{\partial}{\partial z}\mathbf{k}\right) \rho\left(\mathbf{r}\right)\]

Parameters:

• mask (Bool) – If mask the corresponding element of the associated array which is less than given the threshold.

• threshold (float) – Threshold for array element to mask or fill.

Returns:

rho_grad

Return type:

np.ndarray((3,N), dtype=float)

gradient_norm(mask=False, threshold=1e-30)

Norm of the gradient of electron density.

\[\lvert \nabla \rho\left(\mathbf{r}\right) \rvert = \sqrt{ \left(\frac{\partial\rho\left(\mathbf{r}\right)}{\partial x}\right)^2 + \left(\frac{\partial\rho\left(\mathbf{r}\right)}{\partial y}\right)^2 + \left(\frac{\partial\rho\left(\mathbf{r}\right)}{\partial z}\right)^2 }\]

Parameters:

• mask (Bool) – If mask the corresponding element of the associated array which is less than given the threshold.

• threshold (float) – Threshold for array element to mask or fill.

Returns:

grad_rho_norm

Return type:

np.ndarray((N,), dtype=float)

laplacian(mask=False, threshold=1e-30)

Laplacian of electron density \(\nabla ^2 \rho\left(\mathbf{r}\right)\).

This is defined as the trace of Hessian matrix of electron density which is equal to the sum of its \(\left(\lambda_1, \lambda_2, \lambda_3\right)\) eigen-values:

\[\nabla^2 \rho\left(\mathbf{r}\right) = \nabla\cdot\nabla\rho\left(\mathbf{r}\right) = \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial x^2} + \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial y^2} + \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial z^2} = \lambda_1 + \lambda_2 + \lambda_3\]

Parameters:

• mask (Bool) – If mask the corresponding element of the associated array which is less than given the threshold.

• threshold (float) – Threshold for array element to mask or fill.

Returns:

rho_lapl

Return type:

np.ndarray((N,), dtype=float)

static spin_reduction(tensor, spin)

Unified the shape of mo_coeff to square matirx, mo_occ to vector according to spin type.

Parameters:

• tensor (np.ndarray) – Tensor of mo_coeff or mo_occ.

• spin (('ab' | 'a' | 'b' | 'm'), default='ab') – Type of density to compute; either total, alpha-spin, beta-spin, or magnetization density.

Returns:

tensor – Square matrix mo_coeff or vector mo_occ.

Return type:

np.ndarray

tau(mask=False, threshold=1e-30)

Laplacian of electron density \(\nabla ^2 \rho\left(\mathbf{r}\right)\).

This is defined as the trace of Hessian matrix of electron density which is equal to the sum of its \(\left(\lambda_1, \lambda_2, \lambda_3\right)\) eigen-values:

\[\nabla^2 \rho\left(\mathbf{r}\right) = \nabla\cdot\nabla\rho\left(\mathbf{r}\right) = \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial x^2} + \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial y^2} + \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial z^2} = \lambda_1 + \lambda_2 + \lambda_3\]

Parameters:

• mask (Bool) – If mask the corresponding element of the associated array which is less than given the threshold.

• threshold (float) – Threshold for array element to mask or fill.

Returns:

rho_tau

Return type:

np.ndarray((N,), dtype=float)

class pyscf.ita.dens.PartitionDensity(rhos=None)

Bases: list

Class to generate the partition electron density of the molecule at the desired points.

Examples:

>>> mol = gto.M(atom='H 0 0 0; H 0 0 1.1')
>>> grids = dft.gen_grid.Grids(mol)
>>> grids.build()
>>> mf = scf.HF(mol)
>>> mf.kernel()
>>> pd = ita.dens.PartitionDenstiy.orbital(mf, grids)

density(mask=False, threshold=1e-30)

Electron density \(\rho\left(\mathbf{r}\right)\).

Parameters:

• mask (Bool) – If mask the corresponding element of the associated array which is less than given the threshold.

• threshold (float) – Threshold for array element to mask or fill.

Returns:

rho

Return type:

np.ndarray((N,), dtype=float)

gradient(mask=False, threshold=1e-30)

Gradient of electron density \(\nabla \rho\left(\mathbf{r}\right)\).

This is the first-order partial derivatives of electron density w.r.t. coordinate \(\mathbf{r} = \left(x\mathbf{i}, y\mathbf{j}, z\mathbf{k}\right)\),

\[\nabla\rho\left(\mathbf{r}\right) = \left(\frac{\partial}{\partial x}\mathbf{i}, \frac{\partial}{\partial y}\mathbf{j}, \frac{\partial}{\partial z}\mathbf{k}\right) \rho\left(\mathbf{r}\right)\]

Parameters:

• mask (Bool) – If mask the corresponding element of the associated array which is less than given the threshold.

• threshold (float) – Threshold for array element to mask or fill.

Returns:

rho_grad

Return type:

np.ndarray((3,N), dtype=float)

gradient_norm(mask=False, threshold=1e-30)

Norm of the gradient of electron density.

\[\lvert \nabla \rho\left(\mathbf{r}\right) \rvert = \sqrt{ \left(\frac{\partial\rho\left(\mathbf{r}\right)}{\partial x}\right)^2 + \left(\frac{\partial\rho\left(\mathbf{r}\right)}{\partial y}\right)^2 + \left(\frac{\partial\rho\left(\mathbf{r}\right)}{\partial z}\right)^2 }\]

Parameters:

• mask (Bool) – If mask the corresponding element of the associated array which is less than given the threshold.

• threshold (float) – Threshold for array element to mask or fill.

Returns:

grad_norm

Return type:

np.ndarray((N,), dtype=float)

laplacian(mask=False, threshold=1e-30)

Laplacian of electron density \(\nabla ^2 \rho\left(\mathbf{r}\right)\).

This is defined as the trace of Hessian matrix of electron density which is equal to the sum of its \(\left(\lambda_1, \lambda_2, \lambda_3\right)\) eigen-values:

\[\nabla^2 \rho\left(\mathbf{r}\right) = \nabla\cdot\nabla\rho\left(\mathbf{r}\right) = \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial x^2} + \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial y^2} + \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial z^2} = \lambda_1 + \lambda_2 + \lambda_3\]

Parameters:

• mask (Bool) – If mask the corresponding element of the associated array which is less than given the threshold.

• threshold (float) – Threshold for array element to mask or fill.

Returns:

rho_lapl

Return type:

np.ndarray((N,), dtype=float)

classmethod orbital(method, grids_coords, spin='ab', deriv=1)

Compute the orbital electron density of the molecule at the desired points.

Parameters:

• method (PyscfMethod) – Pyscf scf method or post-scf method instance.

• grid_coords (np.ndarray((N, 3), dtype=float)) – Points at which to compute the density.

• spin (('ab' | 'a' | 'b' | 'm')) – Type of density to compute; either total, alpha-spin, beta-spin, or magnetization density, by default ‘ab’.

• deriv (int) – List of molecule and promolecule derivative+1 level, by default 1.

Returns:

obj – Instance of PartitionDensity class.

Return type:

PartitionDensity

tau(mask=False, threshold=1e-30)

Laplacian of electron density \(\nabla ^2 \rho\left(\mathbf{r}\right)\).

This is defined as the trace of Hessian matrix of electron density which is equal to the sum of its \(\left(\lambda_1, \lambda_2, \lambda_3\right)\) eigen-values:

\[\nabla^2 \rho\left(\mathbf{r}\right) = \nabla\cdot\nabla\rho\left(\mathbf{r}\right) = \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial x^2} + \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial y^2} + \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial z^2} = \lambda_1 + \lambda_2 + \lambda_3\]

Parameters:

• mask (Bool) – If mask the corresponding element of the associated array which is less than given the threshold.

• threshold (float) – Threshold for array element to mask or fill.

Returns:

rho_tau

Return type:

np.ndarray((N,), dtype=float)

3.4. pyscf.ita.ked module

class pyscf.ita.ked.KineticEnergyDensity(dens=None, orbdens=None)

Bases: object

The kinetic-energy density class.

dirac(rho=None, omega=None)

Dirac kinetic energy density defined as:

\[\tau_\text{D} = \tfrac{3}{10} \left(6 \pi^2 \right)^{2/3} \left(\frac{\rho\left(\mathbf{r}\right)}{2}\right)^{5/3}\]

Parameters:

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ked – Kinetic energy density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

general(a=0.0, rho=None, rho_lapl=None, omega=None)

Compute general(ish) kinetic energy density defined as:

\[\tau_\text{G} (\mathbf{r}, \alpha) = \tau_\text{PD} (\mathbf{r}) + \frac{1}{4} (a - 1) \nabla^2 \rho (\mathbf{r})\]

Parameters:

• a (float) – Value of parameter \(a\).

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• rho_lapl (np.ndarray((N,), dtype=float), optional) – Electron density laplacian on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ked – Kinetic energy density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

gradient_expansion(a=None, b=None, rho=None, rho_grad_norm=None, rho_lapl=None, omega=None)

Gradient expansion approximation of kinetic energy density defined as:

\[\tau_\text{GEA} \left(\mathbf{r}\right) = \tau_\text{TF} \left(\mathbf{r}\right) + a* \tau_\text{W} \left(\mathbf{r}\right) + b* \nabla^2 \rho\left(\mathbf{r}\right)\]

There is a special case of gradient_expansion() with \(a=\tfrac{1}{9}\) and \(b=\tfrac{1}{6}\).

Parameters:

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• rho_grad_norm (np.ndarray((N,), dtype=float), optional) – Electron density graident norm on grid of N points, by default None.

• rho_lapl (np.ndarray((N,), dtype=float), optional) – Electron density laplacian on grid of N points, by default None.

• a (float, optional) – Value of parameter \(a\), by default None.

• b (float, optional) – Value of parameter \(b\), by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ked – Kinetic energy density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

positive_definite(tau=None, omega=None)

Positive definite or Lagrangian kinetic energy density defined as:

\[\tau_\text{PD} (\mathbf{r}) = \frac{1}{2} \sum_i^N n_i \rvert \nabla \phi_i (\mathbf{r}) \lvert^2\]

Parameters:

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ked – Kinetic energy density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

single_particle(rho_lapl=None, orbrho=None, orbrho_grad_norm=None, omega=None)

Single-particle kinetic energy density defined as:

\[\tau_\text{s} = \sum_i \frac{1}{8} \frac{\nabla \rho_i \nabla \rho_i}{\rho_i} -\frac{1}{8} \nabla^2 \rho\]

where \(\rho_i\) are the Kohn-Sham orbital densities.

Parameters:

• rho_lapl (np.ndarray((N,), dtype=float), optional) – Electron density laplacian on grid of N points, by default None.

• orbrho (np.ndarray((N,), dtype=float), optional) – Electron density of orbitals on grid of N points, by default None.

• orbrho_grad_norm (np.ndarray((N,), dtype=float), optional) – Electron density graident norm of orbitals on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ked – Kinetic energy density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

thomas_fermi(rho=None, omega=None)

Thomas-Fermi kinetic energy density defined as:

\[\tau_\text{TF} \left(\mathbf{r}\right) = \tfrac{3}{10} \left(6 \pi^2 \right)^{2/3} \left(\frac{\rho\left(\mathbf{r}\right)}{2}\right)^{5/3}\]

Parameters:

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ked – Kinetic energy density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

weizsacker(rho=None, rho_grad_norm=None, omega=None)

Weizsacker kinetic energy density defined as:

\[\tau_\text{W} (\mathbf{r}) = \frac{1}{8} \frac{\lvert \nabla\rho (\mathbf{r}) \rvert^2}{\rho (\mathbf{r})}\]

Parameters:

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• rho_grad_norm (np.ndarray((N,), dtype=float), optional) – Electron density graident norm on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ked – Kinetic energy density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

3.5. pyscf.ita.itad module

class pyscf.ita.itad.ItaDensity(dens=None, prodens=None, keds=None)

Bases: object

Information-Theoretic Approch (ITA) Density class.

G1(rho=None, prorho=None, rho_lapl=None, omega=None)

G1 density defined as:

\[g_1(\mathbf{r}) \equiv {}^r_F i^{\prime}(\mathbf{r}) = \nabla^2 \rho(\mathbf{r}) \ln \frac{\rho(\mathbf{r})}{\rho_0(\mathbf{r})}\]

Parameters:

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• prorho (np.ndarray((N,), dtype=float), optional) – Electron density of promolecule on grid of N points, by default None.

• rho_lapl (np.ndarray((N,), dtype=float), optional) – Electron density laplacian on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ita_density – Information theory density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

G2(rho=None, prorho=None, rho_lapl=None, prorho_lapl=None, omega=None)

G2 density defined as:

\[g_2(\mathbf{r}) = \rho(\mathbf{r}) \left[ \frac{\nabla^2 \rho(\mathbf{r})}{\rho(\mathbf{r})} -\frac{\nabla^2 \rho_0(\mathbf{r})}{\rho_0(\mathbf{r})} \right]\]

Parameters:

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• prorho (np.ndarray((N,), dtype=float), optional) – Electron density of promolecule on grid of N points, by default None.

• rho_lapl (np.ndarray((N,), dtype=float), optional) – Electron density laplacian on grid of N points, by default None.

• prorho_lapl (np.ndarray((N,), dtype=float), optional) – Electron density laplacian of promolecule on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ita_density – Information theory density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

G3(rho=None, prorho=None, rho_grad=None, prorho_grad=None, omega=None)

G3 density defined as:

\[g_3(\mathbf{r}) \equiv {}^r_F i(\mathbf{r}) = \rho(\mathbf{r}) \left\vert \frac{\nabla \rho(\mathbf{r})}{\rho(\mathbf{r})} -\frac{\nabla \rho_0(\mathbf{r})}{\rho_0(\mathbf{r})} \right\vert^2\]

Parameters:

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• prorho (np.ndarray((N,), dtype=float), optional) – Electron density of promolecule on grid of N points, by default None.

• rho_grad (np.ndarray((N,), dtype=float), optional) – Electron density graident on grid of N points, by default None.

• prorho_grad (np.ndarray((N,), dtype=float), optional) – Electron density graident of promolecule on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ita_density – Information theory density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

GBP_entropy(rho=None, ts=None, tTF=None, k=1.0, omega=None)

Ghosh-Berkowitz-Parr (GBP) entropy density \(s_{GBP}\) defined as:

\[s_{GBP} = \frac{3}{2}k\rho(\mathbf{r}) \left[ c+\ln \frac{t(\mathbf{r};\rho)}{t_{TF}(\mathbf{r};\rho)} \right]\]

Parameters:

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• ts (np.ndarray((N,), dtype=float), optional) – Single particle kenetic Electron density on grid of N points, by default None.

• tTF (np.ndarray((N,), dtype=float), optional) – Thomas-Fermi kinetic energy density on grid of N points, by default None.

• k (float, optional) – Boltzmann constant, by default 1.0 for convenience.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ita_density – Information theory density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

alternative_fisher_information(rho=None, rho_lapl=None, omega=None)

Alternative Fisher information density defined as:

\[I^{\prime}_F = \nabla^2 \rho(\mathbf{r}) \ln \rho(\mathbf{r})\]

Parameters:

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• rho_lapl (np.ndarray((N,), dtype=float), optional) – Electron density laplacian on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ita_density – Information theory density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

fisher_information(rho=None, rho_grad_norm=None, omega=None)

Fisher information density defined as:

\[i_F = \frac{|\nabla \rho(\mathbf{r})|^2}{\rho(\mathbf{r})}\]

Parameters:

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• rho_grad_norm (np.ndarray((N,), dtype=float), optional) – Electron density graident norm on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ita_density – Information theory density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

relative_alternative_fisher_information(rho=None, prorho=None, rho_lapl=None, omega=None)

Relative alternative Fisher information density defined as:

\[{}^r_F i^{\prime}(\mathbf{r}) = \nabla^2 \rho(\mathbf{r}) \ln \frac{\rho(\mathbf{r})}{\rho_0(\mathbf{r})}\]

Parameters:

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• prorho (np.ndarray((N,), dtype=float), optional) – Electron density of promolecule on grid of N points, by default None.

• rho_lapl (np.ndarray((N,), dtype=float), optional) – Electron density laplacian on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ita_density – Information theory density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

relative_fisher_information(rho=None, prorho=None, rho_grad=None, prorho_grad=None, omega=None)

Relative Fisher information density defined as:

\[{}^r_F i(\mathbf{r}) = \rho(\mathbf{r}) \left\vert \frac{\nabla \rho(\mathbf{r})}{\rho(\mathbf{r})} -\frac{\nabla \rho_0(\mathbf{r})}{\rho_0(\mathbf{r})} \right\vert^2\]

Parameters:

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• prorho (np.ndarray((N,), dtype=float), optional) – Electron density of promolecule on grid of N points, by default None.

• rho_grad (np.ndarray((N,), dtype=float), optional) – Electron density graident on grid of N points, by default None.

• prorho_grad (np.ndarray((N,), dtype=float), optional) – Electron density graident of promolecule on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ita_density – Information theory density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

relative_rho_power(n=2, rho=None, prorho=None, omega=None)

Relative Renyi entropy density defined as:

\[r^r_n = \frac{\rho^n(\mathbf{r})}{\rho^{n-1}_0(\mathbf{r})}\]

Parameters:

• n (int, optional) – Order of relative Renyi entropy, by default 2.

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• prorho (np.ndarray((N,), dtype=float), optional) – Electron density of promolecule on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ita_density – Information theory density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

relative_shannon_entropy(rho=None, prorho=None, omega=None)

Relative Shannon entropy density defined as:

\[s^r_S = -\rho(\mathbf{r})\ln \frac{\rho(\mathbf{r})}{\rho_0(\mathbf{r})}\]

Parameters:

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• prorho (np.ndarray((N,), dtype=float), optional) – Electron density of promolecule on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ita_density – Information theory density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

rho_power(n=2, rho=None, omega=None)

Electron density of power n defined as \(\rho(\mathbf{r})^n\).

Parameters:

• n (int, optional) – Order of rho power, by default 2.

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ita_density – Information theory density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

shannon_entropy(rho=None, omega=None)

Shannon entropy density defined as:

\[s_S = -\rho(\mathbf{r}) \ln \rho(\mathbf{r})\]

Parameters:

• rho (np.ndarray((N,), dtype=float), optional) – Electron density on grid of N points, by default None.

• omega (np.ndarray((N,), dtype=float), optional) – Sharing function of single atom, by default None.

Returns:

ita_density – Information theory density on grid of N points.

Return type:

np.ndarray((N,), dtype=float)

3.6. pyscf.ita.ita module

Information-theoretic approach (ITA) module.

References

The following references are for the atoms-in-molecules module.

class pyscf.ita.ita.ITA(method, grids, rung=[3, 3], spin='ab', promolecule=None)

Bases: object

Information-Theoretic Approch (ITA) class.

Examples:

>>> mol = gto.M(atom='H 0 0 0; H 0 0 1.1')
>>> mf = scf.HF(mol)
>>> mf.kernel()
>>> grids = dft.Grids(mol)
>>> grids.build()
>>> ita = ITA()
>>> ita.method = mf
>>> ita.grids = grids
>>> ita.build()

G1(grids_weights=None, ita_density=None)

G1 density defined as:

\[G_1(\mathbf{r}) = \sum_A \int \nabla^2 \rho_A(\mathbf{r}) \ln \frac{\rho_A(\mathbf{r})}{\rho^0_A(\mathbf{r})} d\mathbf{r}\]

Parameters:

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

G2(grids_weights=None, ita_density=None)

G2 density defined as:

\[G_2(\mathbf{r}) = \sum_A \int \rho_A(\mathbf{r}) \left[ \frac{\nabla^2 \rho_A(\mathbf{r})}{\rho_A(\mathbf{r})} -\frac{\nabla^2 \rho^0_A(\mathbf{r})}{\rho^0_A(\mathbf{r})} \right] d\mathbf{r}\]

Parameters:

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

G3(grids_weights=None, ita_density=None)

G3 density defined as:

\[G_3(\mathbf{r}) = \sum_A \int \rho_A(\mathbf{r}) \left\vert \frac{\nabla \rho_A(\mathbf{r})}{\rho_A(\mathbf{r})} -\frac{\nabla \rho^0_A(\mathbf{r})}{\rho^0_A(\mathbf{r})} \right\vert^2 d\mathbf{r}\]

Parameters:

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

GBP_entropy(grids_weights=None, ita_density=None)

Ghosh-Berkowitz-Parr (GBP) entropy \(S_{GBP}\) defined as:

\[S_{GBP} = \int \frac{3}{2}k\rho(\mathbf{r}) \left[ c+\ln \frac{t(\mathbf{r};\rho)}{t_{TF}(\mathbf{r};\rho)} \right] d\mathbf{r}\]

Parameters:

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

alternative_fisher_information(grids_weights=None, ita_density=None)

Alternative Fisher information \(I^{\prime}_F\) defined as:

\[I^{\prime}_F = -\int \nabla^2 \rho(\mathbf{r}) \ln \rho(\mathbf{r}) d\mathbf{r}\]

Parameters:

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

batch_compute(ita_code=[], representation='electron density', partition='hirshfeld', filename='pyita.log')

ITA batch calcuation.

Parameters:

• ita (ITA) – Instance of ITA class.

• ita_code (List[int]) – List of ITA code to calculate.

• representation (('electron density' | 'shape function' | 'atoms in molecules')) – Type of representation, by default ‘electron density’.

• partition (('hirshfeld' | 'bader' | 'becke'), optional) – Atoms in molecule partition method, by default ‘hirshfeld’.

• filename (str, optional) – File path and name of output, by default ‘pyita.log’

build(method=None, grids=None, rung=None, spin=None, promolecule=None)

Method to build the class.

Parameters:

• method (PyscfMethod, optional) – Pyscf scf method or post-scf method instance, by deafult None.

• grids (Grids, optional) – Pyscf Grids instance, by deafult None.

• rung (List[int, int], optional) – List of molecule and promolecule derivate+1 level, by default [3,0].

• spin (('ab' | 'a' | 'b' | 'm'), optional) – Type of density to compute; either total, alpha-spin, beta-spin, or magnetization density, by default None.

• promolecule (ProMolecule, optional) – ProMolecule instance, by default None.

fisher_information(grids_weights=None, ita_density=None)

Fisher information \(I_F\) defined as:

\[I_F = \int \frac{|\nabla \rho(\mathbf{r})|^2}{\rho(\mathbf{r})} d\mathbf{r}\]

Parameters:

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

onicescu_information(n=2, grids_weights=None, ita_density=None)

Onicescu information \(E_n\) defined as:

\[E_n = \frac{1}{n-1} \int \rho(\mathbf{r})^n d\mathbf{r}\]

Parameters:

• n (int, optional) – Order of the Onicescu information, by default 2.

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

relative_alternative_fisher_information(grids_weights=None, ita_density=None)

Relative alternative Fisher information defined as:

\[{}^r_F I^{\prime}(\mathbf{r}) = \int \nabla^2 \rho(\mathbf{r}) \ln \frac{\rho(\mathbf{r})}{\rho_0(\mathbf{r})} dr\]

Parameters:

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

relative_fisher_information(grids_weights=None, ita_density=None)

Relative Fisher information defined as:

\[{}^r_F I(\mathbf{r}) = \int \rho(\mathbf{r}) \left\vert \frac{\nabla \rho(\mathbf{r})}{\rho(\mathbf{r})} -\frac{\nabla \rho_0(\mathbf{r})}{\rho_0(\mathbf{r})} \right\vert^2 dr\]

Parameters:

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

relative_onicescu_information(n=2, grids_weights=None, ita_density=None)

Relative Tsallis entropy \(E^r_n\) defined as:

\[E^r_n = \frac{1}{n-1} \int \frac{\rho^n(\mathbf{r})}{\rho^{n-1}_0(\mathbf{r})} d\mathbf{r}\]

Parameters:

• n (int, optional) – Order of the relative Renyi entropy, by default 2.

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

relative_renyi_entropy(n=2, grids_weights=None, ita_density=None)

Relative Renyi entropy \(R^r_n\) defined as:

\[R^r_n = \frac{1}{1-n} \ln \left[ \int \frac{\rho^n(\mathbf{r})}{\rho^{n-1}_0(\mathbf{r})} d\mathbf{r} \right]\]

Parameters:

• n (int, optional) – Order of the relative Renyi entropy, by default 2.

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

relative_shannon_entropy(grids_weights=None, ita_density=None)

Relative Shannon entropy \(S^r_S\) defined as:

\[S^r_S = \int \rho(\mathbf{r})\ln \frac{\rho(\mathbf{r})}{\rho_0(\mathbf{r})} d\mathbf{r}\]

Parameters:

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

relative_tsallis_entropy(n=2, grids_weights=None, ita_density=None)

Relative Tsallis entropy \(T^r_n\) defined as:

\[T^r_n = \frac{1}{n-1} \left[ 1- \int \frac{\rho^n(\mathbf{r})}{\rho^{n-1}_0(\mathbf{r})} d\mathbf{r} \right]\]

Parameters:

• n (int, optional) – Order of the relative Renyi entropy, by default 2.

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

renyi_entropy(n=2, grids_weights=None, ita_density=None)

Renyi entropy \(R_n\) defined as:

\[R_n = \frac{1}{1-n} \ln \left[ \int \rho(\mathbf{r})^n d\mathbf{r} \right]\]

Parameters:

• n (int, optional) – Order of the Renyi entropy, by default 2.

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

rho_power(n=2, grids_weights=None, ita_density=None)

Electron density of power n defined as \(\int \rho(\mathbf{r})^n d\mathbf{r}\).

Parameters:

• n (int, optional) – Order of rho power, by default 2.

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

shannon_entropy(grids_weights=None, ita_density=None)

Shannon entropy \(S_S\) defined as:

\[S_S = -\int \rho(\mathbf{r}) \ln \rho(\mathbf{r}) dr\]

Parameters:

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

tsallis_entropy(n=2, grids_weights=None, ita_density=None)

Tsallis entropy \(T_n\) defined as:

\[T_n = \frac{1}{n-1} \left[ 1- \int \rho(\mathbf{r})^n d\mathbf{r} \right]\]

Parameters:

• n (int, optional) – Order of the Tsallis entropy, by default 2.

• grids_weights (np.ndarray((N,), dtype=float), optional) – Grids weights on N points, by default None.

• ita_density (np.ndarray((N,), dtype=float), optional) – ITA density, by default None.

Returns:

result – Scalar ITA result.

Return type:

float

3.7. pyscf.ita.script module

pyscf.ita.script.batch_compute(ita, ita_code=[], representation='electron density', partition='hirshfeld', filename='ita.log')

ITA batch calcuation.

Parameters:

• ita (ITA) – Instance of ITA class.

• ita_code (List[int]) – List of ITA code to calculate.

• representation (('electron density' | 'shape function' | 'atoms in molecules')) – Type of representation, by default ‘electron density’.

• partition (('hirshfeld' | 'bader' | 'becke'), optional) – Atoms in molecule partition method, by default ‘hirshfeld’.

• filename (str, optional) – File path and name of output, by default ‘ita.log’

pyscf.ita.script.section_compute(ita, code, log, representation='electron density', partition='hirshfeld', **kwargs)

Function to calculation a single ITA section.

Parameters:

• ita (ITA) – Instance of ITA class.

• code (int) – Single ITA code.

• log (Log) – Instance of Log class.

• representation (('electron density' | 'shape function' | 'atoms in molecules')) – Type of representation, by default ‘electron density’.

• partition (('hirshfeld' | 'bader' | 'becke'), optional) – Atoms in molecule partition method, by default ‘hirshfeld’.