Molecule Class

Molecule class tht reads wavefunction information and computes various quantities.

class cugbasis.Molecule

Bases: pybind11_object

Attributes:

atcoords: Cartesian coordinates of the atomic centers.
atnums: Atomic number of atomic centers.

Methods

`compute_density`	Compute electron density \(\rho(\mathbf{r})\).
`compute_electrostatic_potential`	Compute the molecular electrostatic potential.
`compute_empirical_gradient_expansion_ked`	Compute the empirical gradient expansion approximation of kinetic energy density.
`compute_general_gradient_expansion_ked`	Compute the general gradient expansion approximation of kinetic energy density.
`compute_general_ked`	Compute the general(ish) kinetic energy density.
`compute_general_kinetic_energy_density`	Compute the general kinetic energy density on a grid of points.
`compute_gradient`	Compute the gradient of the electron density.
`compute_gradient_expansion_ked`	Compute the gradient expansion approximation of kinetic energy density
`compute_hessian`	Compute the Hessian of the electron density.
`compute_laplacian`	Compute the Laplacian of the electron density.
`compute_molecular_orbitals`	Compute Molecular Orbitals
`compute_norm_of_vector`	Compute the norm of a three-dimensional vector using GPU.
`compute_positive_definite_kinetic_energy_density`	Compute the positive definite kinetic energy density on a grid of points.
`compute_reduced_density_gradient`	Compute the reduced density gradient.
`compute_schrodinger_ked`	Compute the Schrödinger kinetic energy density.
`compute_shannon_information_density`	Compute the Shannon information density."
`compute_thomas_fermi_ked`	Compute the Thomas-Fermi kinetic energy density.
`compute_weizsacker_ked`	Compute the Weizsacker Kinetic energy density.
`get_file_path`	Get the wavefunction file path.

property atcoords: Cartesian coordinates of the atomic centers.

property atnums: Atomic number of atomic centers.

compute_density()

Compute electron density \(\rho(\mathbf{r})\).

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.
Returns:: The electron density evaluated on each point.
Return type:: ndarray(N,)

compute_electrostatic_potential()

Compute the molecular electrostatic potential.

\[\Phi(\mathbf{r}) = \sum_{i=1}^{N_{atom}} \frac{Z_A}{|\mathbf{r} - \mathbf{R}_A|} - \int \frac{\rho(\mathbf{r}^\prime)}{|\mathbf{r} - \mathbf{r}^\prime| }d\mathbf{r}^\prime,\]

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.

compute_empirical_gradient_expansion_ked()

Compute the empirical gradient expansion approximation of kinetic energy density.

\[\tau_{EGEA}(\mathbf{r}) = \tau_{TF}(\mathbf{r}) + \frac{1}{5} \tau_W(\mathbf{r}) + \frac{1}{6} \nabla^2 \rho(\mathbf{r})\]

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.

compute_general_gradient_expansion_ked()

Compute the general gradient expansion approximation of kinetic energy density.

\[\tau_{GGE}(\mathbf{r}) = \tau_{TF}(\mathbf{r}) + \alpha \tau_{W}(\mathbf{r}) + \beta \nabla^2 \rho(\mathbf{r})\]

Parameters:

points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.
alpha (float) – Constant parameter.
beta (float) – Constant parameter.

compute_general_ked()

Compute the general(ish) kinetic energy density.

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

where \(a\) is a constant parameter. When \(a=0\), then it is the ‘Schrödinger’ kinetic energy, when \(a=1\), then it is the positive-definite kinetic energy.

Parameters:

points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.
alpha (float) – The constant parameter.

compute_general_kinetic_energy_density()

Compute the general kinetic energy density on a grid of points.

\[t_\alpha(r) = t_+(r) + \alpha \nabla^2 \rho(r)\]

where \(t_+(r)\) is the positive definite kinetic energy density and \(\nabla^2\) is the Laplacian.

Parameters:

points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.
alpha (float) – Constant parameter.

compute_gradient()

Compute the gradient of the electron density.

\[\nabla \rho(\mathbf{r}) = \bigg(\frac{\partial \rho}{\partial x}, \frac{\partial \rho}{\partial y}, \frac{\partial \rho}{\partial z} \bigg),\]

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.
Returns:: The gradient of the electron density evaluated on each point.
Return type:: ndarray(N, 3)

compute_gradient_expansion_ked()

Compute the gradient expansion approximation of kinetic energy density

\[\tau_{GEA}(\mathbf{r}) = \tau_{TF}(\mathbf{r}) + \frac{1}{9} \tau_W(\mathbf{r}) + \frac{1}{6} \nabla^2 \rho(\mathbf{r})\]

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.

compute_hessian()

Compute the Hessian of the electron density.

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.
Returns:: The Hessian of the electron density evaluated on each point.
Return type:: ndarray(N, 3, 3)

compute_laplacian()

Compute the Laplacian of the electron density.

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

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.

compute_molecular_orbitals()

Compute Molecular Orbitals

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.
Returns:: The M molecular orbitals evaluated on each point.
Return type:: ndarray(M, N)

compute_norm_of_vector()

Compute the norm of a three-dimensional vector using GPU.

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.

compute_positive_definite_kinetic_energy_density()

Compute the positive definite kinetic energy density on a grid of points.

\[\tau_{PD}(\mathbf{r}) = \frac{1}{2} \sum_{i=1}^{N} \big|\nabla \phi_i (\mathbf{r})\big|^2,\]

where \(\phi_i\) is the molecular orbitals. In the QTAIM community, this is denoted as \(G(\mathbf{r})\).

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.

compute_reduced_density_gradient()

Compute the reduced density gradient.

\[s(\mathbf{r}) = \frac{1}{2(3 \pi^2)^{1/3}}\frac{|\nabla \rho(\mathbf{r})|}{\rho(\mathbf{r})^{4/3}}.\]

Note it may be more efficient to compute each component individually, then using Numpy to compute this quanity.

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.

compute_schrodinger_ked()

Compute the Schrödinger kinetic energy density.

This is the general(sh) kinetic energy density when \(a=0\):

\[K(\mathbf{r}) = \tau_{PD}(\mathbf{r}) - \frac{1}{4} \nabla^2 \rho(\mathbf{r}),\]

In the QTAIM community, this is denoted as \(K(\mathbf{r})\).

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.

compute_shannon_information_density()

Compute the Shannon information density.”

\[s(\mathbf{r}) = -\rho(\mathbf{r}) \log \rho(\mathbf{r}),\]

where the integral gives the Shannon entropy of the electron density.

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.

compute_thomas_fermi_ked()

Compute the Thomas-Fermi kinetic energy density.

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

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.

compute_weizsacker_ked()

Compute the Weizsacker Kinetic energy density.

\[\tau_W(\mathbf{r}) = \frac{\big|\nabla \rho(\mathbf{r}) \big|^2}{8\rho(\mathbf{r})}\]

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.

get_file_path(): Get the wavefunction file path.

class cugbasis.Promolecule

Bases: pybind11_object

Attributes:

atcoords: Cartesian coordinates of the atomic centers.
atnums: Atomic number of atomic centers.
promol_coeffs: Get the coefficients as a dictionary with keys element_parameter_type (e.g.
promol_exps: Get the exponents as a dictionary with keys element_parameter_type (e.g.

Methods

`compute_density`	Compute promolecular density \(\rho^\circ(\mathbf{r})\).
`compute_esp`	Compute the electrostatic potential on \(\rho^\circ(\mathbf{r})\).

property atcoords: Cartesian coordinates of the atomic centers.

property atnums: Atomic number of atomic centers.

compute_density()

Compute promolecular density \(\rho^\circ(\mathbf{r})\).

\[\rho^\circ(\mathbf{r}) = \sum_A \bigg[ \sum_i c_i \frac{\alpha}{\pi}^{3/2} e^{-\alpha r_{A} ^2} + \sum_j d_j \frac{2 \alpha^{5/2}}{3 \pi^{3/2}} r_A^2 e^{-\alpha r_A^2} \bigg],\]

where \(A\) is an atom, \(c_i, d_j\) are the s-type, p-type coefficients respectively. Note this is normalized. Please see the paper “An information-theoretic approach to basis-set fitting of electron densities and other non-negative functions”.

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.
Returns:: The promolecular density evaluated on each point.
Return type:: ndarray(N,)

compute_esp()

Compute the electrostatic potential on \(\rho^\circ(\mathbf{r})\).

\[\Phi(\mathbf{r}) = \sum_{i=1}^{N_{atom}} \frac{Z_A}{|\mathbf{r} - \mathbf{R}_A|} - \int \frac{\rho^\circ(\mathbf{r}^\prime)}{|\mathbf{r} - \mathbf{r}^\prime| }d\mathbf{r}^\prime,\]

where \(Z_A\) is the atomic number of atom A.

Parameters:: points (ndarray(N, 3)) – Cartesian coordinates of \(N\) points in three-dimensions.
Returns:: The promolecular electrostatic potential evaluated on each point.
Return type:: ndarray(N,)

property promol_coeffs: Get the coefficients as a dictionary with keys element_parameter_type (e.g. f_coeffs_p).

property promol_exps: Get the exponents as a dictionary with keys element_parameter_type (e.g. f_coeffs_p).