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

and its first and second derivatives with respect to $$N$$ at fixed external potential,

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

alongside the electron density of systems with $$N_0 - 1$$, $$N_0$$ and $$N_0 + 1$$ electrons, namely $${\{\rho_{N_0 - 1}\left(\mathbf{r}\right), \rho_{N_0}\left(\mathbf{r}\right), \rho_{N_0 + 1}\left(\mathbf{r}\right)\}}$$, are used to calculate quadratic local descriptors. These local tools include:

Quadratic Electron Density: According to Eq. ???, the change in quadratic energy expression with respect to external potential at fixed number of electrons yields the density of $$N$$-electron system, that is,

$\begin{split}\rho_{N}(\mathbf{r}) =& \left( \frac{\delta E\left(N\right)}{\delta v(\mathbf{r})} \right)_N \\ =& \left( \frac{\delta E\left(N_0\right)}{\delta v(\mathbf{r})} \right)_N &&+ \left[\left( \frac{\delta E\left(N_0 +1\right)}{\delta v(\mathbf{r})} \right)_N - \left( \frac{\delta E\left(N_0 - 1\right)}{\delta v(\mathbf{r})} \right)_N \right] \frac{N - N_0}{2} \\ & &&+ \left[\left(\frac{\delta E\left(N_0 + 1\right)}{\delta v(\mathbf{r})} \right)_N - 2 \left(\frac{\delta E\left(N_0\right)}{\delta v(\mathbf{r})} \right)_N + \left(\frac{\delta E\left(N_0 - 1\right)}{\delta v(\mathbf{r})} \right)_N \right] \frac{\left(N - N_0\right)^2}{2}\end{split}$

So,

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

As expected, the obtained expression for density equals $$\rho_{N_0 - 1}\left(\mathbf{r}\right)$$, $$\rho_{N_0}\left(\mathbf{r}\right)$$ and $$\rho_{N_0 + 1}\left(\mathbf{r}\right)$$ when setting $$N$$ equal to $$N_0-1$$, $$N_0$$ and $$N_0+1$$, respectively. Also, integrating the quadratic electron density over all space gives $$N$$ which confirms that the density expression properly integrates to the number of electrons.

It is important to note that the obtained expression for quadratic electron density in Eq. ??? can be perceived as the second-order Taylor expansion of density around the $$\rho_{N_0}(\mathbf{r})$$ as the reference within the quadratic energy model, that is,

$\rho_{N}(\mathbf{r}) = \rho_{N_0}\left(\mathbf{r}\right) + \left. \left(\frac{\partial \rho_{N}(\mathbf{r})}{\partial N}\right)_{v(\mathbf{r})} \right|_{N = N_0} \left(N - N_0\right) + \frac{1}{2} \left. \left(\frac{\partial^2 \rho_{N}(\mathbf{r})}{\partial N^2}\right)_{v(\mathbf{r})} \right|_{N = N_0} \left(N - N_0\right)^2$

where,

$\begin{split}\left. \left(\frac{\partial \rho_{N}(\mathbf{r})}{\partial N}\right)_{v(\mathbf{r})} \right|_{N = N_0} &= \left(\frac{\rho_{N_0 + 1}\left(\mathbf{r}\right) - \rho_{N_0 - 1}\left(\mathbf{r}\right)}{2}\right) = f_{N_0}(\mathbf{r}) \\ \left. \left(\frac{\partial^2 \rho_{N}(\mathbf{r})}{\partial N^2}\right)_{v(\mathbf{r})} \right|_{N = N_0} &= \rho_{N_0 + 1}\left(\mathbf{r}\right) - 2 \rho_{N_0}\left(\mathbf{r}\right) + \rho_{N_0 - 1}\left(\mathbf{r}\right) = \Delta f_{N_0}(\mathbf{r})\end{split}$

The first and second derivatives of electron density with respect to number of electrons evaluated at $$N_0$$ are the quadratic Fukui function and dual descriptor of $$N_0$$-electron system. The quadratic density of $$N$$-electron system is implemented in chemtools.tool.localtool.QuadraticLocallTool.density.

Quadratic Fukui Function: According to Eq. ???, the change in quadratic chemical potential with respect to external potential at fixed number of electrons yields the Fukui function of $$N$$-electron system. Equivalently, the Fukui function of $$N$$-electron system can be viewed as the change in quadratic electron density $$\rho_N\left(\mathbf{r}\right)$$ with respect to number of electrons $$N$$ at fixed external potential. In other words,

$\begin{split}f_{N}(\mathbf{r}) = \left( \frac{\delta \mu\left(N\right)}{\delta v(\mathbf{r})} \right)_N = \left( \frac{\delta}{\delta v(\mathbf{r})} \left(\frac{\partial E\left(N\right)}{\partial N}\right)_{v(\mathbf{r})} \right)_N = \left( \frac{\partial}{\partial N} \left(\frac{\delta E\left(N\right)}{\delta v(\mathbf{r})}\right)_{N} \right)_{v(\mathbf{r})} = \left(\frac{\partial \rho_{N}(\mathbf{r})}{\partial N}\right)_{v(\mathbf{r})} \\\end{split}$

where,

\begin{align}\begin{aligned}\begin{split}\left( \frac{\delta \mu\left(N\right)}{\delta v(\mathbf{r})} \right)_N = \frac{1}{2} && \left[\left( \frac{\delta E\left(N_0 +1\right)}{\delta v(\mathbf{r})} \right)_N - \left( \frac{\delta E\left(N_0 - 1\right)}{\delta v(\mathbf{r})} \right)_N \right] + \\ && \left[\left(\frac{\delta E\left(N_0 + 1\right)}{\delta v(\mathbf{r})} \right)_N - 2 \left(\frac{\delta E\left(N_0\right)}{\delta v(\mathbf{r})} \right)_N + \left(\frac{\delta E\left(N_0 - 1\right)}{\delta v(\mathbf{r})} \right)_N \right] \left(N - N_0\right) \\\end{split}\\\begin{split}\left( \frac{\partial \rho_{N}(\mathbf{r})}{\partial N} \right)_{v(\mathbf{r})} = \frac{\partial}{\partial N} \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}\end{aligned}\end{align}

(Note: Fix the missing bracket in the last expression) Simplifying either of the above expressions results in the quadratic Fukui function of $$N$$-electron system:

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

Integrating the Fukui function expression confirms that it is normalized to one for any number of electrons $$N$$. For $$N=N_0$$, the familiar expression of Fukui function in obtained:

$\begin{split}f_{N_0}\left(\mathbf{r}\right) = \frac{\rho_{N_0+1}\left(\mathbf{r}\right) - \rho_{N_0-1}\left(\mathbf{r}\right)}{2} \\\end{split}$

It is important to note that the obtained expression for quadratic Fukui function in Eq. ??? can be perceived as the first-order Taylor expansion of Fukui function around the $$f_{N_0}(\mathbf{r})$$ as the reference within the quadratic energy model, that is,

$f_{N}(\mathbf{r}) = f_{N_0}\left(\mathbf{r}\right) + \left. \left(\frac{\partial f_N(\mathbf{r})}{\partial N}\right) \right|_{N = N_0} \left(N - N_0\right)$

where,

$\left. \left(\frac{\partial f_N(\mathbf{r})}{\partial N}\right) \right|_{N = N_0} = \rho_{N_0 - 1}\left(\mathbf{r}\right) - 2 \rho_{N_0}\left(\mathbf{r}\right) + \rho_{N_0 + 1}\left(\mathbf{r}\right) = \Delta f_{N_0}(\mathbf{r})$

the derivative of Fukui function with respect to the number of electrons is the dual descriptor. The quadratic Fukui function is implemented in chemtools.tool.localtool.QuadraticLocallTool.fukui_function.

Quadratic Dual Descriptor: According to Eq. ???, the change in quadratic chemical hardness with respect to external potential at fixed number of electrons yields the dual descriptor of $$N$$-electron system. Equivalently, the dual descriptor of $$N$$-electron system can be viewed as the change in quadratic Fukui function with respect to the number of electrons $$N$$ at fixed external potential. That is,

$\Delta f_{N}(\mathbf{r}) = \left( \frac{\delta \eta\left(N\right)}{\delta v(\mathbf{r})} \right)_N = \left( \frac{\delta}{\delta v(\mathbf{r})} \left(\frac{\partial \mu\left(N\right)}{\partial N}\right)_{v(\mathbf{r})} \right)_N = \left( \frac{\partial}{\partial N} \left(\frac{\delta \mu\left(N\right)}{\delta v(\mathbf{r})}\right)_{N} \right)_{v(\mathbf{r})} = \left(\frac{\partial f_{N}(\mathbf{r})}{\partial N}\right)_{v(\mathbf{r})}$

where,

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

Simplifying either of the above expressions results in the quadratic dual descriptor of $$N$$ -electron system:

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

The dual descriptor does not depend on $$N$$ as one expects for the quadratic energy model. Also, the obtained expression properly integrates to zero. The dual descriptor is implemented in chemtools.tool.localtool.QuadraticLocallTool.dual_descriptor.

Mention that higher order local descriptors do not exist.

Quadratic Softness: The quadratic local softness is easily found by substituting the quadratic Fukui functions in Eq. (????):

$\begin{split}s_N\left(\mathbf{r}\right) &= S \cdot f_N\left(\mathbf{r}\right) \\ &= \frac{1}{\eta} \cdot \left[ \left(\frac{\rho_{N_0 + 1}\left(\mathbf{r}\right) - \rho_{N_0 - 1}\left(\mathbf{r}\right)}{2} \right) + \left(\rho_{N_0 + 1}\left(\mathbf{r}\right) - 2 \rho_{N_0}\left(\mathbf{r}\right) + \rho_{N_0 - 1}\left(\mathbf{r}\right) \right) \left(N - N_0\right) \right]\end{split}$

For $$N=N_0$$,

$s_{N_0}\left(\mathbf{r}\right) = \frac{\rho_{N_0+1}\left(\mathbf{r}\right) - \rho_{N_0-1}\left(\mathbf{r}\right)}{2 \eta} = \frac{\rho_{N_0+1}\left(\mathbf{r}\right) - \rho_{N_0-1}\left(\mathbf{r}\right)}{E\left(N_0 + 1\right) - 2 E\left(N_0\right) + E\left(N_0 - 1\right)}$

$\begin{split}s^{(2)}\left(\mathbf{r}\right) &= S^{2} \cdot f^{(2)}(\mathbf{r}) + S^{(2)} \cdot f(\mathbf{r}) \\ &= \frac{\rho_{N_0 + 1}\left(\mathbf{r}\right) - 2 \rho_{N_0}\left(\mathbf{r}\right) + \rho_{N_0 - 1}\left(\mathbf{r}\right)}{\eta^2} = \frac{\rho_{N_0 + 1}\left(\mathbf{r}\right) - 2 \rho_{N_0}\left(\mathbf{r}\right) + \rho_{N_0 - 1}\left(\mathbf{r}\right)}{\left[E\left(N_0 + 1\right) - 2 E\left(N_0\right) + E\left(N_0 - 1\right)\right]^2} \\\end{split}$