chemtools.conceptual.base.BaseLocalTool

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

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)

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)

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.