class chemtools.denstools.densbased.DensGradLapKedTool(dens, grad, lap, ked)[source]

Local descriptive tools based on density, gradient, Laplacian & kinetic energy density.

Initialize class.

Parameters: dens (np.ndarray) – Electron density evaluated on a set of grid points, $$\rho(\mathbf{r})$$. grad (np.ndarray) – Gradient vector of electron density evaluated on a set of grid points, $$\nabla \rho(\mathbf{r})$$. lap (np.ndarray) – Laplacian of electron density evaluated on a set of grid points, $$\nabla^2 \rho(\mathbf{r})$$. ked (np.ndarray) – Positive-definite or Lagrangian kinetic energy density evaluated on a set of grid points; $$\tau_\text{PD} (\mathbf{r})$$ or $$G(\mathbf{r})$$.
ked_positive_definite

Positive definite or Lagrangian kinetic energy density, $$G(\mathbf{r})$$.

$\tau_\text{PD} \left(\mathbf{r}\right) = \tfrac{1}{2} \sum_i^N n_i \rvert \nabla \phi_i \left(\mathbf{r}\right) \lvert^2$
ked_hamiltonian

Hamiltonian kinetic energy density denoted by $$K(\mathbf{r})$$.

$\tau_\text{ham} \left(\mathbf{r}\right) = \tau_\text{PD} \left(\mathbf{r}\right) - \tfrac{1}{4} \nabla^2 \rho\left(\mathbf{r}\right)$

This is a special case of ked_general() with $$a=0$$.

ked_general(a)[source]

Compute general(ish) kinetic energy density.

$\tau_\text{G} \left(\mathbf{r}, \alpha\right) = \tau_\text{PD} \left(\mathbf{r}\right) + \tfrac{1}{4} (a - 1) \nabla^2 \rho\left(\mathbf{r}\right)$
Parameters: a (float) – Value of parameter $$a$$.
density

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

gradient

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

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

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

Norm of the gradient of electron density.

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

Gradient expansion approximation of kinetic energy density.

$\tau_\text{GEA} \left(\mathbf{r}\right) = \tau_\text{TF} \left(\mathbf{r}\right) + \tfrac{1}{9} \tau_\text{W} \left(\mathbf{r}\right) + \tfrac{1}{6} \nabla^2 \rho\left(\mathbf{r}\right)$

This is a special case of ked_gradient_expansion_general() with $$a=\tfrac{1}{9}$$ and $$b=\tfrac{1}{6}$$.

ked_gradient_expansion_empirical

Empirical gradient expansion approximation of kinetic energy density.

$\tau_\text{empGEA} \left(\mathbf{r}\right) = \tau_\text{TF} \left(\mathbf{r}\right) + \tfrac{1}{5} \tau_\text{W} \left(\mathbf{r}\right) + \tfrac{1}{6} \nabla^2 \rho\left(\mathbf{r}\right)$

This is a special case of ked_gradient_expansion_general() with $$a=\tfrac{1}{5}$$ and $$b=\tfrac{1}{6}$$.

ked_gradient_expansion_general(a, b)

General gradient expansion approximation of kinetic energy density.

$\tau_\text{genGEA} \left(\mathbf{r}\right) = \tau_\text{TF} \left(\mathbf{r}\right) + a \, \tau_\text{W} \left(\mathbf{r}\right) + b \, \nabla^2 \rho\left(\mathbf{r}\right)$
Parameters: a (float) – Value of parameter $$a$$. b (float) – Value of parameter $$b$$.
ked_thomas_fermi

Thomas-Fermi kinetic energy density.

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

Weizsacker kinetic energy density.

$\tau_\text{W} \left(\mathbf{r}\right) = \tfrac{1}{8} \frac{\lvert \nabla\rho\left(\mathbf{r}\right) \rvert^2}{\rho\left(\mathbf{r}\right)}$
laplacian

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:

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

$s\left(\mathbf{r}\right) = \frac{1}{2\left(3\pi ^2 \right)^{1/3}} \frac{\lvert \nabla\rho\left(\mathbf{r}\right) \rvert}{\rho\left(\mathbf{r}\right)^{4/3}}$
shannon_information
Shannon information defined as $$\rho(r) \ln \rho(r)$$.