# chemtools.toolbox.kinetic.KED¶

class chemtools.toolbox.kinetic.KED(dens, grad, lap=None, ked=None)[source]

Kinetic Energy Density Class.

Initialize class.

Parameters: dens (np.ndarray) – Electron density evaluated on a set of points, $$\rho(\mathbf{r})$$. grad (np.ndarray) – Gradient vector of electron density evaluated on a set of points, $$\nabla \rho(\mathbf{r})$$. lap (np.ndarray, optional) – Laplacian of electron density evaluated on a set of points, $$\nabla^2 \rho(\mathbf{r})$$. ked (np.ndarray, optional) – Positive-definite or Lagrangian kinetic energy density evaluated on a set of points; $$\tau_\text{PD} (\mathbf{r})$$ or $$G(\mathbf{r})$$.
classmethod from_molecule(molecule, points, spin='ab', index=None)[source]

Initialize class using instance of Molecule and points.

Parameters: molecule (Molecule) – An instance of Molecule class. points (np.ndarray) – The (npoints, 3) array of cartesian coordinates of points. spin (str, optional) – Type of occupied spin orbitals; options are ‘a’, ‘b’ & ‘ab’. index (sequence, optional) – Sequence of integers representing the index of spin orbitals.
classmethod from_file(fname, points, spin='ab', index=None)[source]

Initialize class from file.

Parameters: fname (str) – Path to molecule’s files. points (np.ndarray) – The (npoints, 3) array of cartesian coordinates of points. spin (str, optional) – Type of occupied spin orbitals; options are ‘a’, ‘b’ & ‘ab’. index (sequence, optional) – Sequence of integers representing the index of spin orbitals.
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 $$\mathbf{r} = \left(x\mathbf{i}, y\mathbf{j}, z\mathbf{k}\right)$$,

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

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

$\tau_\text{W} \left(\mathbf{r}\right) = \tfrac{1}{8} \frac{\lvert \nabla\rho\left(\mathbf{r}\right) \rvert^2}{\rho\left(\mathbf{r}\right)}$
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(alpha, beta)[source]

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_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(alpha)[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 $$a$$.