# chemtools.conceptual.base.BaseCondensedTool¶

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

Base class of condensed 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.
n_ref

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.

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.

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.

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.

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.

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.