# chemtools.toolbox.densitybased.DensityLocalTool¶

class chemtools.toolbox.densitybased.DensityLocalTool(density, gradient, hessian=None)[source]

Class of density-based local descriptive tools.

Initialize class with density and gradient.

Parameters: density (np.ndarray) – Electron density of the system evaluated on a grid gradient (np.ndarray) – Gradient vector of electron density evaluated on a grid hessian (np.ndarray) – Hessian matrix of electron density evaluated on a grid
density

Electron density.

Electron density $$\rho\left(\mathbf{r}\right)$$ evaluated on a grid.

gradient

Gradient of electron density.

Gradient vector of electron $$\nabla \rho\left(\mathbf{r}\right)$$ defined as the first-order partial derivatives of electron density w.r.t. coordinate $$\mathbf{r} = \left(x\mathbf{i}, y\mathbf{j}, z\mathbf{k}\right)$$,

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

Hessian of electron density.

Hessian matrix of electron density $$\nabla^2 \rho\left(\mathbf{r}\right)$$ defined as the second-order partial derivatives of electron density w.r.t coordinate $$\mathbf{r} = \left(x\mathbf{i}, y\mathbf{j}, z\mathbf{k}\right)$$.

laplacian

Laplacian of electron density.

Laplacian of electron density $$\nabla ^2 \rho\left(\mathbf{r}\right)$$ defined as the trace of Hessian matrix of electron density which is equal to the sum of $$\left(\lambda_1, \lambda_2, \lambda_3\right)$$ eigen-values of Hessian matrix,

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

Shanon information defined as $$\rho(r) \ln \rho(r)$$.

gradient_norm

Norm of the gradient of electron density.

Gradient norm representing the norm of the gradient vector at every point,

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

Reduced density gradient (RDG) defined as,

$s\left(\mathbf{r}\right) = \frac{1}{2\left(3\pi ^2 \right)^{1/3}} \frac{\lvert \nabla\rho\left(\mathbf{r}\right) \rvert}{\rho\left(\mathbf{r}\right)^{4/3}}$
weizsacker_kinetic_energy_density

Weizsacker kinetic energy density.

Weizsacker kinetic energy/local steric energy/Fisher information density defined as,

$T\left(\mathbf{r}\right) = \frac{\lvert \nabla \rho\left(\mathbf{r}\right) \rvert ^2}{8 \rho\left(\mathbf{r}\right)}$
thomas_fermi_kinetic_energy_density

Thomas-Fermi kinetic energy density.

Thomas-Fermi kinetic energy density defined as,

$T\left(\mathbf{r}\right) = \frac{3}{10} \left(6 \pi^2 \right)^{2/3} \left(\frac{\rho\left(\mathbf{r}\right)}{2}\right)^{5/3}$
electrostatic_potential(numbers, coordinates, int_weights, int_points, points)[source]

Electrostatic potential.

Electrostatic potential defined as,

$\Phi\left(\mathbf{r}\right) = - v \left(\mathbf{r}\right) - \int \frac{\rho\left(\mathbf{r}'\right)}{|\mathbf{r} - \mathbf{r}'|} d \mathbf{r}'$
Parameters: numbers (np.ndarray) – The atomic numbers of the system coordinates (np.ndarray) – The coordinates of the (nuclear) charges of the system int_weights (np.ndarray) – The integration weights. This should have the same dimension as the density used to initialize the class. int_points (np.ndarray) – The coordinates of the integration points. points (np.ndarray) – The coordinates of the point(s) on which to calculate the electrostatic potential.