class chemtools.toolbox.conceptuallocal.QuadraticLocalTool(density_zero, density_plus, density_minus, n0)[source]

Class of local conceptual DFT reactivity descriptors based on the quadratic energy model.

Considering the interpolated quadratic energy expression,

$\begin{split}E\left(N\right) = E\left(N_0\right) &+ \left(\frac{E\left(N_0 + 1\right) - E\left(N_0 - 1\right)}{2}\right) \left(N - N_0\right) \\ &+ \left(\frac{E\left(N_0 - 1\right) - 2 E\left(N_0\right) + E\left(N_0 + 1\right)}{2}\right) \left(N - N_0\right)^2 \\\end{split}$

and its first and second derivatives with respect to the number of electrons at constant external potential,

$\begin{split}\mu\left(N\right) &= \left(\frac{\partial E\left(N\right)}{\partial N} \right)_{v(\mathbf{r})} \\ &= \left(\frac{E\left(N_0 + 1\right) - E\left(N_0 - 1\right)}{2}\right) + \left[E\left(N_0 - 1\right) - 2 E\left(N_0\right) + E\left(N_0 + 1\right) \right] \left(N - N_0\right) \\ \eta\left(N\right) &= \left(\frac{\partial^2 E\left(N\right)}{\partial^2 N} \right)_{v(\mathbf{r})} = E\left(N_0 - 1\right) - 2 E\left(N_0\right) + E\left(N_0 + 1\right)\end{split}$

the quadratic local tools are obtained by taking the functional derivative of these expressions with respect to external potential $$v(\mathbf{r})$$ at fixed number of electrons.

Initialize class.

Parameters: density_zero (np.ndarray) – Electron density of $$N_0$$-electron system, i.e. $$\rho_{N_0}\left(\mathbf{r}\right)$$. density_plus (np.ndarray) – Electron density of $$(N_0 + 1)$$-electron system, i.e. $$\rho_{N_0 + 1}\left(\mathbf{r}\right)$$. density_minus (np.ndarray) – Electron density of $$(N_0 - 1)$$-electron system, i.e. $$\rho_{N_0 - 1}\left(\mathbf{r}\right)$$. n0 (float) – Reference number of electrons, i.e. $$N_0$$, which corresponds to the integral of density_zero over all space.
density(number_electrons=None)[source]

Return quadratic electron density of $$N$$-electron system $$\rho_{N}(\mathbf{r})$$.

This is defined as the functional derivative of quadratic energy model w.r.t. external potential at fixed number of electrons,

$\begin{split}\rho_{N}(\mathbf{r}) = \rho_{N_0}\left(\mathbf{r}\right) &+ \left(\frac{\rho_{N_0 + 1}\left(\mathbf{r}\right) - \rho_{N_0 - 1}\left(\mathbf{r}\right)}{2}\right) \left(N - N_0\right) \\ &+ \left(\frac{\rho_{N_0 - 1}\left(\mathbf{r}\right) - 2 \rho_{N_0}\left(\mathbf{r}\right) + \rho_{N_0 + 1}\left(\mathbf{r}\right)}{2}\right) \left(N - N_0\right)^2\end{split}$
Parameters: number_electrons (float, default=None) – Number of electrons. If None, the $$\rho_{N_0}\left(\mathbf{r}\right)$$ is returned.
fukui_function(number_electrons=None)[source]

Return quadratic Fukui function of $$N$$-electron system, $$f_{N}(\mathbf{r})$$.

This is defined as the functional derivative of quadratic chemical potential w.r.t. external potential at fixed number of electrons,

$f_{N}(\mathbf{r}) = \left(\frac{\rho_{N_0 + 1}\left(\mathbf{r}\right) - \rho_{N_0 - 1}\left(\mathbf{r}\right)}{2} \right) + \left[\rho_{N_0 - 1}\left(\mathbf{r}\right) - 2 \rho_{N_0}\left(\mathbf{r}\right) + \rho_{N_0 + 1}\left(\mathbf{r}\right) \right] \left(N - N_0\right)$
Parameters: number_electrons (float, default=None) – Number of electrons. If None, the $$f_{N_0}\left(\mathbf{r}\right)$$ is returned.
dual_descriptor()[source]

Quadratic dual descriptor of $$N$$-electron system, $$\Delta f_{N}(\mathbf{r})$$.

This is defined as the functional derivative of quadratic chemical hardness w.r.t. external potential at fixed number of electrons,

$\Delta f_{N}(\mathbf{r}) = \rho_{N_0 - 1}\left(\mathbf{r}\right) - 2 \rho_{N_0}\left(\mathbf{r}\right) + \rho_{N_0 + 1}\left(\mathbf{r}\right)$

The quadratic dual descriptor is independent of the number electrons.

softness(global_softness, number_electrons=None)[source]

Return quadratic softness of $$N$$-electron system, $$s_{N}(\mathbf{r})$$.

$s_N\left(\mathbf{r}\right) = S \cdot f_N\left(\mathbf{r}\right)$
Parameters: global_softness (float) – The value of global softness. number_electrons (float, default=None) – Number of electrons. If None, the $$s_{N_0}\left(\mathbf{r}\right)$$ is returned.
hyper_softness(global_softness)[source]

Quadratic hyper-softness of $$N$$-electron system, $$s_N^{(2)}(\mathbf{r})$$.

$s_N^{(2)}\left(\mathbf{r}\right) = S^2 \cdot \Delta f_N\left(\mathbf{r}\right)$

The quadratic hyper-softness is independent of the number electrons.

Parameters: global_softness (float) – The value of global softness.