# chemtools.toolbox.conceptualglobal.LinearGlobalTool¶

class chemtools.toolbox.conceptualglobal.LinearGlobalTool(energy_zero, energy_plus, energy_minus, n0)[source]

Class of global conceptual DFT reactivity descriptors based on the linear energy model.

The energy is approximated as a piece-wise linear function of the number of electrons,

$E(N) = a + b N$

Given $$E(N_0 - 1)$$, $$E(N_0)$$ and $$E(N_0 + 1)$$ values, the unknown parameters of the energy model are obtained by interpolation.

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

Because the energy model is not differentiable at integer number of electrons, the first derivative of the energy w.r.t. number if electrons is calculated from above, from below and averaged:

$\begin{split}\mu^{-} &= -IP \\\ \mu^{0} &= \frac{\mu^{+} + \mu^{-}}{2} \\ \mu^{+} &= -EA \\\end{split}$
mu_minus

Chemical potential from below.

$\mu^{-} = E\left(N_0\right) - E\left(N_0 - 1\right) = -IP$
mu_plus

Chemical potential from above.

$\mu^{+} = E\left(N_0 + 1\right) - E\left(N_0\right) = -EA$
mu_zero

Chemical potential averaged of $$N_0^{+}$$ and $$N_0^{-}$$.

$\mu^{0} = \frac{\mu^{+} + \mu^{-}}{2} = \frac{E\left(N_0 + 1\right) - E\left(N_0 - 1\right)}{2} = - \frac{IP + EA}{2}$
energy(n_elec)[source]

Return the energy model $$E(N)$$ evaluated for the specified number of electrons.

Parameters: n_elec (float) – Number of electrons, $$N_{\text{elec}}$$.
energy_derivative(n_elec, order=1)[source]

Return the $$n^{\text{th}}$$-order derivative of energy w.r.t. the number of electrons.

This returns the $$n^{\text{th}}$$-order derivative of energy model $$E(N)$$ w.r.t. to the number of electrons, at fixed chemical potential, evaluated for the specified number of electrons.

$\left. \left(\frac{\partial^n E}{\partial N^n} \right)_{v(\mathbf{r})}\right|_{N = N_{\text{elec}}}$
Parameters: n_elec (float) – Number of electrons, $$N_{\text{elec}}$$. order (int, default=1) – The order of derivative denoted by $$n$$ in the formula.

Note

For $$N_{\text{elec}} = N_0$$ the first, second and higher order derivatives are equal to the BaseGlobalTool.chemical_potential, BaseGlobalTool.chemical_hardness and BaseGlobalTool.hyper_hardness, respectively.