# Linear Energy ModelΒΆ

Complementing the linear global tools, the fitted piece-wise linear energy expression,

$\begin{split}E\left(N\right) = \begin{cases} \left(N - N_0 + 1\right) E\left(N_0\right) - \left(N - N_0\right) E\left(N_0 - 1\right) & \text{ for } N \leqslant N_0 \\ \left(N - N_0\right) E\left(N_0 + 1\right) - \left(N - N_0 - 1\right) E\left(N_0\right) & \text{ for } N \geqslant N_0 \\ \end{cases}\end{split}$

and its derivative with respect to number of electrons $$N$$ at fixed external potential,

$\begin{split}\mu\left(N\right) = \begin{cases} E\left(N_0\right) - E\left(N_0 - 1\right) = - IP &= \mu^- & \text{ for } N < N_0 \\ E\left(N_0 + 1\right) - E\left(N_0\right) = - EA &= \mu^+ & \text{ for } N > N_0 \\ \end{cases}\end{split}$

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 linear local descriptors. These local tools include:

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

$\begin{split}\rho_{N}(\mathbf{r}) = \left( \frac{\delta E\left(N\right)}{\delta v(\mathbf{r})} \right)_N = \begin{cases} \left(N - N_0 + 1\right) \left(\frac{\delta E\left(N_0\right)}{\delta v(\mathbf{r})} \right)_N - \left(N - N_0\right) \left(\frac{\delta E\left(N_0 - 1\right)}{\delta v(\mathbf{r})} \right)_N & \text{ for } N \leqslant N_0 \\ \left(N - N_0\right) \left(\frac{\delta E\left(N_0 + 1\right)}{\delta v(\mathbf{r})} \right)_N - \left(N - N_0 - 1\right) \left(\frac{\delta E\left(N_0\right)}{\delta v(\mathbf{r})} \right)_N & \text{ for } N \geqslant N_0 \\ \end{cases}\end{split}$

So,

$\begin{split}\rho_{N}(\mathbf{r}) = \begin{cases} \left(N - N_0 + 1\right) \rho_{N_0}(\mathbf{r}) - \left(N - N_0\right) \rho_{N_0 - 1}(\mathbf{r}) & \text{ for } N \leqslant N_0 \\ \left(N - N_0\right) \rho_{N_0 + 1}(\mathbf{r}) - \left(N - N_0 - 1\right) \rho_{N_0}(\mathbf{r}) & \text{ for } N \geqslant N_0 \\ \end{cases}\end{split}$

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 $$\left(N_0-1\right)$$, $$N_0$$ and $$\left(N_0+1\right)$$, respectively. Also, integrating the linear electron density over all space results in $$N$$ which confirms that the density expression is properly normalized to the number of electrons.

By rearranging the expression for linear electron density, it can be easily perceived as first-order Taylor expansion of density around the $$\rho_{N_0}(\mathbf{r})$$ as the reference within the linear energy model, that is,

$\begin{split}\rho_{N}(\mathbf{r}) = \begin{cases} \rho_{N_0}(\mathbf{r}) + \left[\rho_{N_0}(\mathbf{r}) - \rho_{N_0 - 1}(\mathbf{r})\right] \left(N - N_0\right) & \text{ for } N \leqslant N_0 \\ \rho_{N_0}(\mathbf{r}) + \left[\rho_{N_0 + 1}(\mathbf{r}) - \rho_{N_0}(\mathbf{r})\right] \left(N - N_0\right) & \text{ for } N \geqslant N_0 \\ \end{cases}\end{split}$

where,

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

Linear Fukui Function: According to Eq. ???, the change in linear 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 linear 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,

$\begin{split}\left( \frac{\delta \mu\left(N\right)}{\delta v(\mathbf{r})} \right)_N &= \begin{cases} \left(\frac{\delta E\left(N_0\right)}{\delta v(\mathbf{r})} \right)_N - \left(\frac{\delta E\left(N_0 - 1\right)}{\delta v(\mathbf{r})} \right)_N & \text{ for } N < N_0 \\ \left(\frac{\delta E\left(N_0 + 1\right)}{\delta v(\mathbf{r})} \right)_N - \left(\frac{\delta E\left(N_0\right)}{\delta v(\mathbf{r})} \right)_N & \text{ for } N > N_0 \\ \end{cases} \\ \left( \frac{\partial \rho_{N}(\mathbf{r})}{\partial N} \right)_{v(\mathbf{r})} &= \begin{cases} \frac{\partial}{\partial N}\left[\left(N - N_0 + 1\right) \rho_{N_0}(\mathbf{r}) - \left(N - N_0\right) \rho_{N_0 - 1}(\mathbf{r})\right] & \text{ for } N < N_0 \\ \frac{\partial}{\partial N}\left[\left(N - N_0\right) \rho_{N_0 + 1}(\mathbf{r}) - \left(N - N_0 - 1\right) \rho_{N_0}(\mathbf{r})\right] & \text{ for } N > N_0 \\ \end{cases}\end{split}$

Simplifying either of the above expressions results in the linear Fukui function of $$N$$-electron system:

$\begin{split}f_{N}(\mathbf{r}) = \begin{cases} \rho_{N_0}(\mathbf{r}) - \rho_{N_0 - 1}(\mathbf{r}) = f^-(\mathbf{r}) & \text{ for } N < N_0 \\ \rho_{N_0 + 1}(\mathbf{r}) - \rho_{N_0}(\mathbf{r}) = f^+(\mathbf{r}) & \text{ for } N > N_0 \\ \end{cases}\end{split}$

Considering the fact that the linear energy model is not differentiable at $$N_0$$, Commonly, the average Fukui function $$f^0\left(\mathbf{r}\right)$$ is used:

$f^0\left(\mathbf{r}\right) = \frac{f^+\left(\mathbf{r}\right) + f^-\left(\mathbf{r}\right)}{2} = \frac{\rho_{N_0 + 1}\left(\mathbf{r}\right) - \rho_{N_0 - 1}\left(\mathbf{r}\right)}{2}$

(TO BE REMOVED) Dual descriptor is defined as:

$d\left(\mathbf{r}\right) = f^+\left(\mathbf{r}\right) - f^-\left(\mathbf{r}\right) = \rho_{N_0 + 1}\left(\mathbf{r}\right) - 2 \rho_{N_0 - 1}\left(\mathbf{r}\right) + \rho_{N_0 - 1}\left(\mathbf{r}\right)$

Todo

• This is not really dual descriptor for linear model. Technically the dual descriptor is zero for linear model, but the dual descriptor for quadratic model happens to be f+(r) - f-(r). Does this need to be clarified?

Linear Softness:

$\begin{split}s_{N}(\mathbf{r}) = S \cdot f_{N}(\mathbf{r}) = \begin{cases} S \cdot \left[\rho_{N_0}(\mathbf{r}) - \rho_{N_0 - 1}(\mathbf{r})\right] &= S \cdot f^-(\mathbf{r}) & \text{ for } N < N_0 \\ S \cdot \left[\rho_{N_0 + 1}(\mathbf{r}) - \rho_{N_0 - 1}(\mathbf{r})\right] \cdot 0.5 &= S \cdot f^0(\mathbf{r}) & \text{ for } N = N_0 \\ S \cdot \left[\rho_{N_0 + 1}(\mathbf{r}) - \rho_{N_0}(\mathbf{r})\right] &= S \cdot f^+(\mathbf{r}) & \text{ for } N > N_0 \\ \end{cases}\end{split}$