Linear Energy Model chemtools.tool.localtool.LinearLocalToolΒΆ

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:

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