chemtools.toolbox.kinetic.KED¶
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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 \(G(\mathbf{r})\).
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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.
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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.
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density¶ Electron density \(\rho\left(\mathbf{r}\right)\).
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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 \(\mathbf{r} = \left(x\mathbf{i}, y\mathbf{j}, z\mathbf{k}\right)\),
\[\nabla\rho\left(\mathbf{r}\right) = \left(\frac{\partial}{\partial x}\mathbf{i}, \frac{\partial}{\partial y}\mathbf{j}, \frac{\partial}{\partial z}\mathbf{k}\right) \rho\left(\mathbf{r}\right)\]
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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:
\[\nabla^2 \rho\left(\mathbf{r}\right) = \nabla\cdot\nabla\rho\left(\mathbf{r}\right) = \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial x^2} + \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial y^2} + \frac{\partial^2\rho\left(\mathbf{r}\right)}{\partial z^2} = \lambda_1 + \lambda_2 + \lambda_3\]
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ked_positive_definite¶ Positive definite or Lagrangian kinetic energy density, \(G(\mathbf{r})\).
\[\tau_\text{PD} \left(\mathbf{r}\right) = \tfrac{1}{2} \sum_i^N n_i \rvert \nabla \phi_i \left(\mathbf{r}\right) \lvert^2\]
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ked_thomas_fermi¶ Thomas-Fermi kinetic energy density.
\[\tau_\text{TF} \left(\mathbf{r}\right) = \tfrac{3}{10} \left(6 \pi^2 \right)^{2/3} \left(\frac{\rho\left(\mathbf{r}\right)}{2}\right)^{5/3}\]
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ked_weizsacker¶ Weizsacker kinetic energy density.
\[\tau_\text{W} \left(\mathbf{r}\right) = \tfrac{1}{8} \frac{\lvert \nabla\rho\left(\mathbf{r}\right) \rvert^2}{\rho\left(\mathbf{r}\right)}\]
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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 \(b=\tfrac{1}{6}\).
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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 \(b=\tfrac{1}{6}\).
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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 \(b\).
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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 \(a=0\).