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.

$$\mathrm{\Phi }(\mathbf{r})=\sum _{i=1}^{N_{atom}}\frac{Z_A}{|\mathbf{r}-{\mathbf{R}}_A|}-\int \frac{\rho ({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}d{\mathbf{r}}^{\prime },$$

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.

$${\tau }_{EGEA}(\mathbf{r})={\tau }_{TF}(\mathbf{r})+\frac{1}{5}{\tau }_W(\mathbf{r})+\frac{1}{6}{\mathrm{\nabla }}^2\rho (\mathbf{r})$$

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.

$${\tau }_{GGE}(\mathbf{r})={\tau }_{TF}(\mathbf{r})+\alpha {\tau }_W(\mathbf{r})+\beta {\mathrm{\nabla }}^2\rho (\mathbf{r})$$

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.

$${\tau }_G(\mathbf{r},\alpha )={\tau }_{PD}(\mathbf{r})+\frac{1}{4}(\alpha -1){\mathrm{\nabla }}^2\rho (\mathbf{r}),$$

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.

$$t_{\alpha }(r)=t_{+}(r)+\alpha {\mathrm{\nabla }}^2\rho (r)$$

where $t_{+}(r)$ is the positive definite kinetic energy density and ${\mathrm{\nabla }}^2$ 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.

$$\mathrm{\nabla }\rho (\mathbf{r})=(\frac{\partial \rho }{\partial x},\frac{\partial \rho }{\partial y},\frac{\partial \rho }{\partial z}),$$

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

$${\tau }_{GEA}(\mathbf{r})={\tau }_{TF}(\mathbf{r})+\frac{1}{9}{\tau }_W(\mathbf{r})+\frac{1}{6}{\mathrm{\nabla }}^2\rho (\mathbf{r})$$

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.

$${\mathrm{\nabla }}^2\rho (\mathbf{r})=\sqrt{(\frac{\partial \rho }{\partial x}{)}^2+(\frac{\partial \rho }{\partial y}{)}^2+(\frac{\partial \rho }{\partial z}{)}^2}$$

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.

$${\tau }_{PD}(\mathbf{r})=\frac{1}{2}\sum _{i=1}^N|\mathrm{\nabla }{\varphi }_i(\mathbf{r}){|}^2,$$

where ${\varphi }_i$ 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.

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

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 $a=0$:

$$K(\mathbf{r})={\tau }_{PD}(\mathbf{r})-\frac{1}{4}{\mathrm{\nabla }}^2\rho (\mathbf{r}),$$

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.

$${\tau }_W(\mathbf{r})=\frac{|\mathrm{\nabla }\rho (\mathbf{r}){|}^2}{8\rho (\mathbf{r})}$$

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 ${\rho }^{\circ }(\mathbf{r})$.

$${\rho }^{\circ }(\mathbf{r})=\sum _A[\sum _i{c}_i{\frac{\alpha }{\pi }}^{3/2}{e}^{-\alpha r_A^2}+\sum _j{d}_j\frac{2{\alpha }^{5/2}}{3{\pi }^{3/2}}{r}_A^2{e}^{-\alpha r_A^2}],$$

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 ${\rho }^{\circ }(\mathbf{r})$.

$$\mathrm{\Phi }(\mathbf{r})=\sum _{i=1}^{N_{atom}}\frac{Z_A}{|\mathbf{r}-{\mathbf{R}}_A|}-\int \frac{{\rho }^{\circ }({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}d{\mathbf{r}}^{\prime },$$

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).