# chemtools.conceptual.base.BaseLocalTool¶

class chemtools.conceptual.base.BaseLocalTool(n0, n_max=None, global_softness=None)[source]

Base class of local conceptual DFT reactivity descriptors.

Initialize class.

Parameters: n0 (float) – Reference number of electrons, i.e. $$N_0$$. 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.
n0

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

n_max

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

global_softness

Global softness.

density(n_elec)[source]

Evaluate density model $$\rho_N(\mathbf{r})$$ at the $$N_{\text{elec}}$$.

The functional derivative of energy model $$E(N)$$ w.r.t. external potential at fixed number of electrons, evaluated at the given number of electrons $$N_{\text{elec}}$$, i.e.

$\left.\rho_N(\mathbf{r}) = {\left(\frac{\delta E(N)}{\delta v(\mathbf{r})}\right)_N} \right|_{N = N_{\text{elec}}}$
Parameters: n_elec (float) – Number of electrons, $$N_{\text{elec}}$$.
density_derivative(n_elec, order=1)[source]

Evaluate n-th derivative of density w.r.t. number of electrons at $$N_{\text{elec}}$$.

The n-th order derivative of density model $$\rho_N(\mathbf{r})$$ w.r.t. the number of electrons, at fixed external 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 fukui function, dual descriptor and hyper fukui function, respectively.

fukui_function

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.

dual_descriptor

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.

softness

Chemical softness of $$N_0$$-electron system.

$s_{N_0}\left(\mathbf{r}\right) = S \cdot f_{N_0}\left(\mathbf{r}\right)$

where $$S$$ is the global softness, and $$f_{N_0}\left(\mathbf{r}\right)$$ is fukui function.

hyper_softness

Chemical hyper-softness of $$N_0$$-electron system.

$s_{N_0}^{(2)}\left(\mathbf{r}\right) = S^2 \cdot \Delta f_{N_0}\left(\mathbf{r}\right)$

where $$S$$ is the global softness, and $$\Delta f_{N_0}\left(\mathbf{r}\right)$$ is the dual descriptor.