Quadratic Energy Model¶
In this model, the energy is approximated as a quadratic function of the number of electrons:
\begin{eqnarray} E\left(N\right) = a + b N + c {N^2} \end{eqnarray}
As it contains three parameters, \(a\), \(b\) and \(c\), this model requires three values of \(E\left(N\right)\) to interpolate the energy. Commonly, the energies of the system with \(N_0 - 1\), \(N_0\) and \(N_0 + 1\) electrons are provided. Fitting the energy expression to the given energy values results in three equations,
\[\begin{split}\begin{cases} E\left(N_0 - 1\right) &= a + b \left(N_0 - 1\right) + c {\left(N_0 - 1\right) ^2} \\ E \left(N_0\right) &= a + b \left(N_0\right) + c {\left(N_0\right) ^2} \\ E\left(N_0 + 1\right) &= a + b \left(N_0 + 1\right) + c {\left(N_0 + 1\right) ^2} \end{cases}\end{split}\]
which can be solved for the three unknowns,
\[\begin{split}a &= E\left(N_0\right) - b N_0 - c {N_0 ^2} \\ b &= \frac{E\left(N_0 + 1\right) - E\left(N_0 - 1\right)}{2} - 2 N_0 c \\ c &= \frac{E\left(N_0 - 1\right) -2 E\left(N_0\right) + E\left(N_0 + 1\right)}{2} \\\end{split}\]
Substituting the obtained parameters \(a\), \(b\) and \(c\) into the energy expression, Eq. (1), gives the fitted energy model as:
\[\begin{split}E\left(N\right) = E\left(N_0\right) &+ \left(\frac{E\left(N_0 + 1\right) - E\left(N_0 - 1\right)}{2}\right) \left(N - N_0\right) \\ &+ \left(\frac{E\left(N_0 - 1\right) - 2 E\left(N_0\right) + E\left(N_0 + 1\right)}{2}\right) \left(N - N_0\right)^2\end{split}\]
or equivalently,
\[E\left(N\right) = E\left(N_0\right) - \left(\frac{IP + EA}{2}\right) \left(N - N_0\right) + \left(\frac{IP - EA}{2}\right) \left(N - N_0\right)^2\]
At this stage, the energy expression can be evaluated for any given number of electrons as
implemented in chemtools.tool.globaltool.QuadraticGlobalTool.energy. By rearranging
the obtained quadratic energy expression, the energy change \(\Delta E = E(N) - E(N_0)\) due to
the electron transfer \(\Delta N = N - N_0\), when the external potential \(v(\mathbf{r})\)
is fixed, is given by:
\[\Delta E = -\left(\frac{IP + EA}{2}\right) \Delta N + \left(\frac{IP - EA}{2}\right) (\Delta N)^2\]
As detailed below, the prefactor of \(\Delta N\) is the first derivative of energy with respect to \(N\) and the prefactor of \((\Delta N)^2\) is one-half the second order derivatives of the energy with respect to \(N\) at fixed external potential \(v(\mathbf{r})\) evaluated at \(N = N_0\). As a result, this energy model is equivalent to the second-order Taylor expansion of the energy as a function of \(N\) around the reference state \(N_0\).
To obtain the fundamental global reactivity indicators for the quadratic energy model, the derivatives of the energy with respect to the number of electrons at fixed external potential \(v(\mathbf{r})\) should be calculated. These are given by:
\[\begin{split}\left( \frac{\partial E}{\partial N} \right)_{v(\mathbf{r})} &= b + 2cN \\ &= \frac{E(N_0 + 1) - E(N_0 - 1)}{2} + \left(E(N_0 - 1) - 2 E(N_0) + E(N_0 + 1)\right) \left(N - N_0\right) \\ &= -\frac{IP + EA}{2} + (IP - EA) \left(N - N_0\right) \\ \left( \frac{\partial^2 E}{\partial N^2} \right)_{v(\mathbf{r})} &= 2c \\ &= E(N_0 - 1) - 2 E(N_0) + E(N_0 + 1) \\ &= IP - EA \\ \left( \frac{\partial^{n+1} E}{\partial N^{n+1}} \right)_{v(\mathbf{r})} &= 0 \text{ for } n \geq 2\end{split}\]
These derivatives can be evaluated for any number of electrons as implemented
in chemtools.tool.globaltool.QuadraticGlobalTool.energy_derivative.
In the quadratic model, evaluating the first-, second-, and higher-order derivatives of
energy evaluated at \(N_0\) gives the following expressions for the chemical potential,
chemical hardness, and hyper-hardnesses,
\[\begin{split}\mu = \left. \left(\frac{\partial E}{\partial N} \right)_{v(\mathbf{r})} \right|_{N = N_0} &= \frac{E(N_0 + 1) - E(N_0 - 1)}{2} = - \frac{{IP + EA}}{2} \\ \eta = \left. \left( \frac{\partial^2 E}{\partial N^2} \right)_{v(\mathbf{r})} \right|_{N = N_0} &= E(N_0 - 1) - 2 E(N_0) + E(N_0 + 1) = IP - EA \\ \eta^{(n)} = \left. \left( \frac{\partial^{n+1} E}{\partial N^{n+1}} \right)_{v(\mathbf{r})} \right|_{N = N_0} &= 0 \text{ for } n \geq 2\end{split}\]
These are implemented in chemtools.tool.globaltool.QuadraticGlobalTool.chemical_potential
and chemtools.tool.globaltool.QuadraticGlobalTool.chemical_hardness.
Accordingly, within the quadratic energy model, the chemical softness and hyper-softnesses are given by the expressions,
\[\begin{split}S = - \left. \left( \frac{\partial^2\Omega}{\partial\mu^2} \right)_{v(\mathbf{r})} \right|_{N = N_0} &= \frac{1}{\eta} = \frac{1}{IP - EA} \\ S^{(n)} = - \left. \left( \frac{\partial^{n+1}\Omega}{\partial\mu^{n+1}} \right)_{v(\mathbf{r})} \right|_{N = N_0} &= 0 \text { for } n \geq 2\end{split}\]
To obtain the derived global reactivity indicators for the quadratic energy model, the maximum number of electrons to saturate the system should be calculated. This is obtained by setting the first derivative of the energy with respect to the number of electrons equal to zero,
\[\begin{split}\left( \frac{\partial E}{\partial N} \right)_{v(\mathbf{r})} = 0 &= b + 2cN = -\frac{IP + EA}{2} + (IP - EA)(N - N_0) \\ & \to N_{\text{max}} = \frac{-b}{2c} = N_{0} + \frac{IP + EA}{2 \left(IP - EA \right)} = N_{0} - \frac{\mu}{\eta} \\ & \to \Delta N_{\text{max}} = N_0 - N_{\text{max}} = \frac{IP + EA}{2 \left(IP - EA \right)} = - \frac{\mu}{\eta}\end{split}\]
The related derived global reactivity indicators for the quadratic energy model are:
Todo
include the generalized signed definitions.
\[\begin{split}\omega_{\text{electrophilicity}} &= E\left(N_0\right) - E\left(N_{\text{max}}\right) &&= \frac{\left(IP + EA\right)^2}{8\left(IP - EA\right)} &&&= \frac{\mu^2}{2 \eta} \\ \nu_{\text{nucleofugality}} &= E\left(N_0 + 1\right) - E\left(N_{\text{max}}\right) &&= \frac{\left(IP - 3 \cdot EA \right)^2}{8 \left(IP - EA \right)} &&&= \frac{\left(\mu + \eta\right)^2}{2\eta} = -EA + \omega_{\text{electrophilicity}} \\ \nu_{\text{electrofugality}} &= E\left(N_0 - 1\right) - E\left(N_{\text{max}}\right) &&= \frac{\left(3 \cdot IP - EA \right)^2}{8 \left(IP - EA \right)} &&&= \frac{\left(\mu - \eta\right)^2}{2\eta} = IP + \omega_{\text{electrophilicity}}\end{split}\]
References:
- R.G. Parr and R.G. Pearson. Absolute hardness: companion parameter to absolute electronegativity. J. Am. Chem. Soc., 105(26):7512–7516, 1983. doi:10.1021/ja00364a005.