Density-Based Local DescriptorsΒΆ

All the tools for calculations based on the electron density \(\rho\left(\mathbf{r}\right)\), gradient and hessian of the \(N\) electron reference state.

Electron density \(\rho\left(\mathbf{r}\right)\) represents the probability of finding an electron within a certain volume element \(d\boldsymbol{r}\). It is found by integrating the square of the wave function over all variables except the spatial coordinates of one electron:

\[\rho(\boldsymbol{r}) = N\int \ldots \int \vert \Psi(\boldsymbol{x}_1 , \boldsymbol{x}_2 , \ldots , \boldsymbol{x}_N) \vert^2 d\sigma_1 , d\boldsymbol{x}_2 , \ldots , d\boldsymbol{x}_N ,\]

where the vectors \(\boldsymbol{x}_i\) include the space coordinates, \(\boldsymbol{r}_i\), and the spin coordinate, \(\sigma_i\), of the i-th electron.

Gradient of electron density \(\nabla \rho\left(\mathbf{r}\right)\) represents the first-order partial derivatives of the electron density with respect to the coordinates:

\[\nabla \rho\left(\mathbf{r}\right) = \left( \frac{\partial}{\partial x} \rho\left(\mathbf{r}\right), \frac{\partial}{\partial y} \rho\left(\mathbf{r}\right), \frac{\partial}{\partial z} \rho\left(\mathbf{r}\right)\right)\]

Hessian of electron density \(\nabla^2 \rho\left(\mathbf{r}\right)\) represents the second-order partial derivative of the electron density with respect to coordinates:

\[\begin{split}\nabla^2 \rho\left(\mathbf{r}\right) = \begin{bmatrix} \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial x^2} & \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial x \partial y} & \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial x \partial z} \\ \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial y \partial x} & \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial y^2} & \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial y \partial z} \\ \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial z \partial x} & \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial z \partial y} & \frac{\partial^2 \rho\left(\mathbf{r}\right)}{\partial z^2} \\ \end{bmatrix}\end{split}\]

Todo

Elaborate on the different properties and their meaning

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

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 (RDG) defined as:

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

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 defined as:

\[T\left(\mathbf{r}\right) = \frac{3}{10} \left( 6 \pi ^2 \right)^{2/3} \left( \frac{\rho\left(\mathbf{r}\right)}{2} \right)^{5/3}\]