# chemtools.conceptual.linear.LinearCondensedTool¶

class chemtools.conceptual.linear.LinearCondensedTool(dict_population, n_max=None, global_softness=None)[source]

Condensed conceptual DFT reactivity descriptors class based on the linear energy model.

Considering the interpolated linear energy expression,

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

and its derivative with respect to the number of electrons at constant external potential,

$\begin{split}\mu\left(N\right) = \begin{cases} \mu^- &= E\left(N_0\right) - E\left(N_0 - 1\right) = - IP && N < N_0 \\ \mu^0 &= 0.5 \left(E\left(N_0 + 1\right) - E\left(N_0 - 1\right)\right) = -0.5 (IP + EA) && N = N_0 \\ \mu^+ &= E\left(N_0 + 1\right) - E\left(N_0\right) = - EA && N > N_0 \\ \end{cases}\end{split}$

the linear local tools are obtained by taking the functional derivative of these expressions with respect to external potential $$v(\mathbf{r})$$ at fixed number of electrons.

Initialize linear density model to compute local reactivity descriptors.

Parameters: dict_population (dict) – Dictionary of number of electrons (keys) and corresponding atomic populations array (values). This model expects three energy values corresponding to three consecutive number of electrons differing by one, i.e. $$\{(N_0 - 1): {N_A \left(N_0 - 1\right)}, N_0: {N_A \left(N_0\right)}, (N_0 + 1): {N_A \left(N_0 + 1\right)}$$. The $$N_0$$ value is considered as the reference number of electrons. n_max (float, optional) – Maximum number of electrons that system can accept, i.e. $$N_{\text{max}}$$. See BaseGlobalTool.n_max. global_softness (float, optional) – Global softness. See BaseGlobalTool.softness.
ff_plus

Atomic Fukui Function from above, $$f_A^+(\mathbf{r})$$.

$f_A^+\left(\mathbf{r}\right) = \rho_{N_0 + 1}\left(\mathbf{r}\right) - \rho_{N_0}\left(\mathbf{r}\right)$
ff_minus

Atomic Fukui Function from below, $$f_A^-(\mathbf{r})$$.

$f_A^-\left(\mathbf{r}\right) = \rho_{N_0}\left(\mathbf{r}\right) - \rho_{N_0 - 1}\left(\mathbf{r}\right)$
ff_zero

Atomic Fukui Function from center, $$f_A^0(\mathbf{r})$$.

This is defined as the average of ff_plus and ff_minus,

$f_A^0\left(\mathbf{r}\right) = \frac{f^+\left(\mathbf{r}\right) + f^-\left(\mathbf{r}\right)}{2} = \frac{\rho_{N_0 + 1}\left(\mathbf{r}\right) - \rho_{N_0 - 1}\left(\mathbf{r}\right)}{2}$
population(n_elec)[source]

Evaluate atomic populations at the given number of electrons $$N_{\text{elec}}$$.

..math::
N_A = Z_A - int w_A(mathbf{r}) rho_N(mathbf{r}) dmathbf{r}

where $$w_A(\mathbf{r})$$ is the atomic weight of atom $$A$$ at point mathbf{r}.

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

Evaluate n-th derivative of atomic populations w.r.t. number of electrons.

The n-th order derivative of atomic populations $$\rho_N(\mathbf{r})$$ w.r.t. the number of electrons, at fixed chemical potential, evaluated at the given number of electrons $$N_{\text{elec}}$$ is:

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

Note

For $$N_{\text{elec}} = N_0$$ the first, second and higher order density derivatives correspond to the condensed fukui function, dual descriptor and hyper fukui function, respectively.

dual_descriptor

Atomic dual descriptor of $$N_0$$-electron system.

This is defined as the 2nd derivative of density model $$\rho_N(\mathbf{r})$$ w.r.t. the number of electrons, at fixed external potential, evaluated at $$N_0$$, or the functional derivative of chemical hardness w.r.t. external potential, at fixed number of electrons, i.e.

$\Delta f_{N_0}(\mathbf{r}) = {\left( \frac{\delta \eta}{\delta v(\mathbf{r})} \right)_N} = \left. \left(\frac{\partial^2 \rho_N(\mathbf{r})}{\partial N^2} \right)_{v(\mathbf{r})}\right|_{N = N_0}$

where $$\eta$$ is the chemical hardness.

fukui_function

Atomic Fukui function of $$N_0$$-electron system.

This is defined as the 1st derivative of density model $$\rho_N(\mathbf{r})$$ w.r.t. the number of electrons, at fixed external potential, evaluated at $$N_0$$, or the functional derivative of chemical potential w.r.t. external potential, at fixed number of electrons, i.e.

$f_{N_0}(\mathbf{r}) = {\left( \frac{\delta \mu}{\delta v(\mathbf{r})} \right)_N} = \left. \left(\frac{\partial \rho_N(\mathbf{r})}{\partial N} \right)_{v(\mathbf{r})}\right|_{N = N_0}$

where $$\mu$$ is the chemical potential.

global_softness

Global softness.

hyper_softness

Chemical hyper-softness of $$N_0$$-electron system, $$s_N^{(2)}(\mathbf{r})$$.

$s_N^{(2)}\left(\mathbf{r}\right) = S^2 \cdot \Delta f_N\left(\mathbf{r}\right)$

where $$S$$ is the global softness and $$\Delta f_N$$ is the dual descriptor.

n_max

Maximum number of electrons that the system accepts, i.e. $$N_{\text{max}}$$.

n_ref

Reference number of electrons, i.e. $$N_0$$.

softness

Atomic chemical softness of $$N_0$$-electron system.

$s_A\left(N\right) = S \cdot f_A\left(\mathbf{r}\right)$

where $$S$$ is the global softness.