# Exponential Energy Model¶

In this model, energy is approximated by an exponential function of the number of electrons:

$E(N) = A \exp(-\gamma(N-N_0)) + B$

Containing three parameters, $$A$$, $$B$$ and $$\gamma$$, this model requires three values of $$E(N)$$ to interpolate energy. Commonly, the energy 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:

$\begin{split}\begin{cases} E(N_0-1) &= A \exp(\gamma) + B \\ E(N_0) &= A + B \\ E(N_0+1) &= A \exp(-\gamma) + B \\ \end{cases}\end{split}$

This allows us to solve for the three unknowns:

$\begin{split}A &= \frac{(E\left(N_0 - 1\right) - E\left(N_0\right))(E\left(N_0\right) - E\left(N_0 + 1\right))} {E\left(N_0 - 1\right) - 2 E\left(N_0\right) + E\left(N_0 + 1\right)} = \frac{IP \cdot EA}{IP - EA} \\ B &= E\left(N_0\right) - A \\ \gamma &= \ln \left( 1 - \frac{E\left(N_0 - 1\right) - 2E\left(N_0\right) + E\left(N_0 + 1\right)} {E\left(N_0 + 1\right) - E\left(N_0\right)} \right) \\\end{split}$

Due to the complexity of the obtained parameters, we skip substituting them into the energy expression. However, knowing the parameters of the model, at this stage the energy expression can be evaluated for any given number of electrons as implemented in chemtools.tool.globaltool.ExponentialGlobalTool.energy.

The derivatives of the energy model with respect to the number of electrons at fixed external potential $$v(\mathbf{r})$$ are:

$\begin{split}\left( \frac{\partial E}{\partial N} \right)_{v(\mathbf{r})} &= A \left(-\gamma\right) \exp\left(-\gamma \left(N - N_0\right)\right) \\ \left( \frac{\partial^2 E}{\partial N^2} \right)_{v(\mathbf{r})} &= A {\left(-\gamma\right)^2} \exp\left(-\gamma \left(N - N_0\right)\right) \\ \left( \frac{\partial^n E}{\partial N^n} \right)_{v(\mathbf{r})} &= A {\left(-\gamma\right)^n} \exp\left(-\gamma \left(N - N_0\right)\right) \text{ for } n \geq 1\end{split}$

These derivatives can be evaluated for any number of electrons as implemented in chemtools.tool.globaltool.ExponentialGlobalTool.energy_derivative.

In the exponential 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:

$\begin{split}\mu = \left. \left( \frac{\partial E}{\partial N} \right)_{v(\mathbf{r})} \right|_{N = N_0} &= -A \gamma \\ \eta = \left. \left( \frac{\partial^2 E}{\partial N^2} \right)_{v(\mathbf{r})} \right|_{N = N_0} &= A {\gamma ^2} \\ \eta^{(2)} = \left. \left( \frac{\partial^{3} E}{\partial N^{3}} \right)_{v(\mathbf{r})} \right|_{N = N_0} &= -A \gamma^3 \\ \eta^{(n)} = \left. \left( \frac{\partial^{n+1} E}{\partial N^{n+1}} \right)_{v(\mathbf{r})} \right|_{N = N_0} &= A {(-\gamma)^{(n+1)}} \text{ for } n \geq 2\end{split}$

These are implemented in chemtools.tool.globaltool.ExponentialGlobalTool.chemical_potential and chemtools.tool.globaltool.ExponentialGlobalTool.chemical_hardness.

Using these expressions, one can derive the following expressions for the chemical softness and the low-order hypersoftnesses,

$\begin{split}S = - \left. \left( \frac{\partial^2\Omega}{\partial\mu^2} \right)_{v(\mathbf{r})} \right|_{N = N_0} &= \frac{1}{\eta} = \frac{1}{A \gamma^2} \\ S^{(2)} = - \left. \left( \frac{\partial^{3}\Omega}{\partial\mu^{3}} \right)_{v(\mathbf{r})} \right|_{N = N_0} &= -\eta^{(2)} \cdot S^3 = - \left(-A\gamma^3\right) \left(\frac{1}{A \gamma^2} \right)^3 = \frac{1}{A^2\gamma^3} \\ S^{(3)} = - \left. \left( \frac{\partial^{4}\Omega}{\partial\mu^{4}} \right)_{v(\mathbf{r})} \right|_{N = N_0} &= -\eta^{(3)} \cdot S^4 + 3 \left(\eta^{(2)}\right)^2 \cdot S^5 \\ &= - \left(A\gamma^4\right) \left(\frac{1}{A\gamma^2}\right)^4 + 3 \left(\frac{1}{-A\gamma^3}\right)^2 \left(\frac{1}{A\gamma^2}\right)^5 = \frac{2}{A^3\gamma^4}\\\end{split}$

ChemTools can also compute higher-order hyper-softnesses, using the (extended) inverse function theorem for derivatives.

To obtain the derived global reactivity indicators for the exponential energy model, the maximum number of electrons accepted by the system should be calculated.

Todo

1. Include $$N_{\text{max}}=\infty$$ and derived global reactivity tools

References:

Todo