# chemtools.conceptual.rational.RationalGlobalTool¶

class chemtools.conceptual.rational.RationalGlobalTool(dict_energy)[source]

Class of global conceptual DFT reactivity descriptors based on the rational energy model.

The energy is approximated as a 3-parameter rational function of the number of electrons,

$E(N) = \frac{a_0 + a_1 N}{1 + b_1 N}$

Given $$E(N_0 - 1)$$, $$E(N_0)$$ and $$E(N_0 + 1)$$ values, the unknown parameters of the energy model are obtained by interpolation.

The $$n^{\text{th}}$$-order derivatives of the rational energy model with respect to the number of electrons at fixed external potential is given by:

$\left(\frac{\partial^n E}{\partial N^n} \right)_{v(\mathbf{r})} = \frac{b_1^{n - 1} (a_1 - a_0 b_1) n!}{(1 + b_1 N)^{2n}}$

Initialize rational energy model to compute global reactivity descriptors.

Parameters: dict_energy (dict) – Dictionary of number of electrons (keys) and corresponding energy (values). This model expects three energy values corresponding to three consecutive number of electrons differing by one, i.e. $$\{(N_0 - 1): E(N_0 - 1), N_0: E(N_0), (N_0 + 1): E(N_0 + 1)\}$$. The $$N_0$$ value is considered as the reference number of electrons.
params

Parameter $$a_0$$, $$a_1$$ and $$b_1$$ of energy model.

energy(n_elec)[source]

Return the energy model $$E(N)$$ evaluated for the specified number of electrons.

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

Return the $$n^{\text{th}}$$-order derivative of energy w.r.t. the number of electrons.

This returns the $$n^{\text{th}}$$-order derivative of energy model $$E(N)$$ w.r.t. to the number of electrons, at fixed chemical potential, evaluated for the specified number of electrons.

$\left. \left(\frac{\partial^n E}{\partial N^n} \right)_{v(\mathbf{r})}\right|_{N = N_{\text{elec}}}$
Parameters: n_elec (float) – Number of electrons, $$N_{\text{elec}}$$. order (int, default=1) – The order of derivative denoted by $$n$$ in the formula.

Note

For $$N_{\text{elec}} = N_0$$ the first, second and higher order derivatives are equal to the BaseGlobalTool.chemical_potential, BaseGlobalTool.chemical_hardness and BaseGlobalTool.hyper_hardness, respectively.

chemical_hardness

Chemical hardness of the $$N_0$$-electron system.

This chemical hardness is defined as the second derivative of the energy model w.r.t. the number of electrons, at fixed external potential, evaluated at $$N_0$$.

$\eta = \left. \left(\frac{\partial^2 E}{\partial N^2} \right)_{v(\mathbf{r})} \right|_{N = N_0}$
chemical_potential

Chemical potential of the $$N_0$$-electron system.

The chemical potential is defined as the first derivative of the energy model w.r.t. the number of electrons, at fixed external potential, evaluated at $$N_0$$,

$\mu = \left. \left(\frac{\partial E}{\partial N} \right)_{v(\mathbf{r})} \right|_{N = N_0}$
convert_mu_to_n(mu, guess=None)

Return the number of electrons $$N$$ matching the given chemical potential $$\mu$$.

Chemical potential is a function of the number of electrons, $$\mu(N)$$, as it is the first derivative of energy model $$E(N)$$ with respect to the number of electrons at fixed external potential,

$\mu(N) = \left(\frac{\partial E(N)}{\partial N}\right)_{v(\mathbf{r})}$

Here we solve for $$N$$ which results in the specified $$\mu$$ according to the equation above, i.e. $$N(\mu) = \mu^{-1}(N)$$, using scipy.optimize.newton.

Parameters: mu (float) – Chemical potential, $$\mu$$. guess (float, default=None) – Initial guess used for solving for $$N$$. If None, the reference number of electrons $$N_0$$ is used as an initial guess.
ea

The same as electron_affinity.

electrofugality

Electrofugalityof the $$N_0$$-electron system.

$\nu_{\text{electrofugality}} = \text{sgn}\left(N_{\text{max}} - N_0 + 1\right) \times \left(E(N_0 - 1) - E(N_{\text{max}})\right)$
electron_affinity

Electron affinity (EA) of the $$N_0$$-electron system.

$EA = E\left(N_0\right) - E\left(N_0 + 1\right)$
electronegativity

Mulliken electronegativity defined as negative chemical_potential.

$\chi_{\text{Mulliken}} = - \mu$
electrophilicity

Electrophilicity of the $$N_0$$-electron system.

$\omega_{\text{electrophilicity}} = \text{sgn}\left(N_{\text{max}} - N_0\right) \times \left(E(N_0) - E(N_{\text{max}})\right)$
eta

The same as chemical_hardness.

grand_potential(n_elec)

Return the grand potential model evaluated for the specified number of electrons.

$\begin{split}\Omega [\mu(N_{\text{elec}}); v(\mathbf{r})] &= E(N_{\text{elec}}) - \mu(N_{\text{elec}}) \times N_{\text{elec}} \\ &= E(N_{\text{elec}}) - \left.\left(\frac{\partial E(N)}{\partial N} \right)_{v(\mathbf{r})}\right|_{N = N_{\text{elec}}} \times N_{\text{elec}}\end{split}$
Parameters: n_elec (float) – Number of electrons, $$N_{\text{elec}}$$.
grand_potential_derivative(n_elec, order=1)

Evaluate the $$n^{\text{th}}$$-order derivative of grand potential at the given n_elec.

This returns the $$n^{\text{th}}$$-order derivative of grand potential model w.r.t. to the chemical potential, at fixed external potential, evaluated for the specified number of electrons $$N_{\text{elec}}$$.

That is,

$\left. \left(\frac{\partial^n \Omega}{\partial \mu^n} \right)_{v(\mathbf{r})} \right|_{N = N_{\text{elec}}} = \left. \left(\frac{\partial^{n-1}}{\partial \mu^{n-1}} \frac{\partial \Omega}{\partial \mu} \right)_{v(\mathbf{r})} \right|_{N = N_{\text{elec}}} = - \left. \left(\frac{\partial^{n-1} N}{\partial \mu^{n-1}} \right)_{v(\mathbf{r})} \right|_{N = N_{\text{elec}}} \quad n = 1, 2, \dots$

These derivatives can be computed using the derivative of energy model w.r.t. number of electrons, at fixed external potential, evaluated at $$N_{\text{elec}}$$. More specifically,

$\begin{split}\left. \left(\frac{\partial \Omega}{\partial \mu} \right)_{v(\mathbf{r})} \right|_{N = N_{\text{elec}}} &= - N_{\text{elec}} \\ \left. \left(\frac{\partial^2 \Omega}{\partial \mu^2} \right)_{v(\mathbf{r})} \right|_{N = N_{\text{elec}}} &= -\frac{1}{\eta^{(1)}}\end{split}$

where $$\eta^{(n)}$$ denotes the $$(n+1)^{\text{th}}$$-order derivative of energy w.r.t. number of electrons evaluated at $$N_{\text{elec}}$$, i.e.

$\eta^{(n)} = \left. \left(\frac{\partial^{n+1} E}{\partial N^{n+1}} \right)_{v(\mathbf{r})} \right|_{N = N_{\text{elec}}} \qquad n = 1, 2, \dots$

To compute higher-order derivatives, Faa di Bruno formula which generalizes the chain rule to higher derivatives can be used. i.e. for $$n \geq 2$$,

$\left. \left(\frac{\partial^n \Omega}{\partial \mu^n} \right)_{v(\mathbf{r})} \right|_{N = N_{\text{elec}}} = \frac{-\displaystyle\sum_{k=1}^{n-2} \left.\left(\frac{\partial^k \Omega}{\partial \mu^k} \right)_{v(\mathbf{r})} \right|_{N = N_{\text{elec}}} \cdot B_{n-1,k} \left(\eta^{(1)}, \eta^{(2)}, \dots, \eta^{(n-k)} \right)} {B_{n-1,n-1} \left(\eta^{(1)}\right)}$

where $$B_{n-1,k} \left(x_1, x_2, \dots, x_{n-k}\right)$$ denotes the Bell polynomials.

Parameters: n_elec (float) – Number of electrons, $$N_{\text{elec}}$$. order (int, default=1) – The order of derivative denoted by $$n$$ in the formula.
grand_potential_mu(mu)

Evaluate the grand potential model for the specified chemical potential $$\mu$$.

To evaluate grand potential model, first the number of electrons corresponding to the specified $$\mu$$ is found, i.e. $$N(\mu)=\mu^{-1}(N)$$, then the grand potential in computed by,

$\Omega [\mu(N); v(\mathbf{r})] = E(N(\mu)) - \mu \times N(\mu)$
Parameters: mu (float) – Chemical potential $$\mu$$.
grand_potential_mu_derivative(mu, order=1)

Evaluate the $$n^{\text{th}}$$-order derivative of grand potential at the given mu.

This returns the $$n^{\text{th}}$$-order derivative of grand potential model w.r.t. chemical potential, at fixed external potential, evaluated for the specified chemical potential $$\mu$$.

That is,

$\left. \left(\frac{\partial^n \Omega}{\partial \mu^n} \right)_{v(\mathbf{r})} \right|_{N = N\left(\mu\right)} = \left. \left(\frac{\partial^{n-1}}{\partial \mu^{n-1}} \frac{\partial \Omega}{\partial \mu} \right)_{v(\mathbf{r})} \right|_{N = N\left(\mu\right)} = - \left. \left(\frac{\partial^{n-1} N}{\partial \mu^{n-1}} \right)_{v(\mathbf{r})} \right|_{N = N\left(\mu\right)} \quad n = 1, 2, \dots$

To evaluate this expression, the number of electrons corresponding to the specified $$\mu$$ should is found, i.e. $$N(\mu)=\mu^{-1}(N)$$.

Parameters: mu (float) – Chemical potential, $$\mu$$. order (int, default=1) – The order of derivative denoted by $$n$$ in the formula.
hyper_hardness(order=2)

Return the $$n^{\text{th}}$$-order hyper-hardness of the $$N_0$$-electron system.

The $$n^{\text{th}}$$-order hyper-hardness is defined as the $$(n+1)^{\text{th}}$$ -order derivative, where $$n \geq 2$$, of the energy model w.r.t the number of electrons, at fixed external potential, evaluated at $$N_0$$.

$\eta^{(n)} = \left. \left(\frac{\partial^{n+1} E}{\partial N^{n+1}} \right)_{v(\mathbf{r})} \right|_{N = N_0} \quad \text{for} \quad n \geq 2$
Parameters: order (int, default=2) – The order of hyper-hardness denoted by $$n \geq 2$$ in the formula.
hyper_softness(order)

Return the $$n^{\text{th}}$$-order hyper-softness of the $$N_0$$-electron system.

The $$n^{\text{th}}$$-order hyper softness is defined as the $$(n+1)^{\text{th}}$$ -order derivative, where $$n \geq 2$$, of the grand potential model w.r.t the number of electrons at fixed external potential evaluated at $$N_0$$.

$S^{(n)} = - \left. \left(\frac{\partial^{n+1} \Omega}{\partial \mu^{n+1}} \right)_{v(\mathbf{r})} \right|_{N = N_0} \quad \text{for} \quad n \geq 2$
Parameters: order (int, default=2) – The order of hyper-hardness denoted by $$n \geq 2$$ in the formula.
ionization_potential

Ionization potential (IP) of the $$N_0$$-electron system.

$IP = E\left(N_0 - 1\right) - E\left(N_0\right)$
ip

The same as ionization_potential.

mu

The same as chemical_potential.

n0

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

n_max

Maximum number of electrons that the system can accept.

$N_{\text{max}} = \underbrace {\min }_N E(N)$
nucleofugality

Nucleofugality of the $$N_0$$-electron system.

$\nu_{\text{nucleofugality}} = \text{sgn}\left(N_0 + 1 - N_{\text{max}}\right) \times \left(E(N_0 + 1) - E(N_{\text{max}})\right)$
softness

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

The chemical softness is defined as the second derivative of the grand potential model w.r.t the number of electrons, at fixed external potential, evaluated at $$N_0$$. This is equal to the inverse chemical hardness.

$S = - \left. \left(\frac{\partial^2 \Omega}{\partial \mu^2} \right)_{v(\mathbf{r})}\right|_{N = N_0} = \frac{1}{\eta}$