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.