Generalized Quadratic Model for Charge Transfer

This library is based on the work published in Phys. Chem. Chem. Phys 27, 11318 (2025).

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Table of Contents

1. Reading Outputs
   1. NWChem
2. Quadratic Interpolations
   1. One-Parabola Model
   2. Two-Parabola Model
   3. Generalized Quadratic Model
3. Charge Transfer
   1. One-Parabola Model
   2. Two-Parabola Model
   3. Generalized Quadratic Model

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1. Reading Outputs

Currently we only include a parser for NWChem outputs. But please reach out to us if you are interested in other electronic structure codes, we are happy to help you.

1.1 NWChem

To read NWChem outputs, we can use the read module from the package. We need to provide the path to the output files for the chemical species.

from cdft import read

cation  = read.nwchem( "path/to/cation/output.out" )
neutral = read.nwchem( "path/to/neutral/output.out" )
anion   = read.nwchem( "path/to/anion/output.out" )

Please note that we can use whatever extension we wish for our outputs for as long as we include the complete file name. The read module will go through the information in the file and extract the total energy and orbital energies. These attributes are readily accesible from the returned objects.

# Getting the attributes from the 'neutral' parsed output

total_energy = neutral.energy
homo_energy  = neutral.homo
lumo_energy  = neutral.lumo

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2. Quadratic Interpolations

We include three different models, namely, the One-Parabola Model by Parr and Pearson, the Two-Parabola Model by Gazquez, Cedillo and Vela, and the Generalized Quadratic Model by Albavera-Mata, Gazquez and Vela.

2.1 One-Parabola Model

The One-Parabola Model is based on the assumption that the energy of a system as a function of the number of electrons can be approximated by a single parabola based on the chemical potential $\mu$ and chemical hardness $\eta$. This model is particularly useful for describing systems where the addition or removal of electrons leads to significant changes in energy.

$$\Delta E(\Delta N) = \mu(\Delta N) + \frac{1}{2} \eta (\Delta N)^2$$

where $\mu = -(I + A)/2$ and $\eta = I - A$ are defined in terms of the ionization potential $I$ and electron affinity $A$.

from cdft import model

mu, eta = model.one_parabola( cation=cation, neutral=neutral, anion=anion )

2.2 Two-Parabola Model

The Two-Parabola Model considers two separate parabolas to describe the energy changes associated with electron addition ($\mu^+$ and $\eta^+$) and removal ($\mu^-$ and $\eta^-$). This model provides a more accurate representation for systems where the energy landscape is more complex.

$$\Delta E^\pm(\Delta N) = \mu^\pm(\Delta N) + \frac{1}{2} \eta (\Delta N)^2$$

where $\mu^- = -(3I + A)/4$, $\mu^+ = -(I + 3A)/4$, and $\eta = (I - A)/2$.

from cdft import model

mu_minus, mu_plus, eta = model.two_parabola( cation=cation, neutral=neutral, anion=anion )

2.3 Generalized Quadratic Model

The Generalized Quadratic Model further refines the previous models by incorporating information from the highest-occupied and lowest-unnocupied frontier molecular orbitals, HOMO and LUMO, respectively, to better capture the nuances of charge transfer processes. This model is versatile and can be adapted to a wide range of chemical systems.

$$\Delta E^\pm(\Delta N) = \mu^\pm(\Delta N) + \frac{1}{2} \eta^\pm (\Delta N)^2$$

where $\mu^- = \varepsilon_\mathrm{HOMO}$, $\mu^+ = \varepsilon_\mathrm{LUMO}$, $\eta^- = 2(I + \varepsilon_\mathrm{HOMO})$, and $\eta^+ = -2(A + \varepsilon_\mathrm{LUMO})$

from cdft import model

mu_minus, mu_plus, eta_minus_eta_plus = model.generalized( cation=cation, neutral=neutral, anion=anion )

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3 Charge Transfer

For the sake of consistency and comparison, we include the same three quadratic interpolations. These commonly are used to discern charge trasnfer trends for donor-acceptor reactions, granted that a reactant A is donating charge while reactant B accepts it. For illustrative purposes, here we will assume that species B is our target reagent, where ${\Delta N}_{A}^{-}$ and ${\Delta N}_{A}^{+}$ denote the nucleophilic and electrophilic charge transfer, respectively.

3.1 One-Parabola Model

The charge transfer channels for the One-Parabola Model are given by

$$\begin{aligned} {\Delta N}_{A}^{-} & = +\frac{1}{2} \frac{A_\mathbf{A} - I_\mathbf{B}}{\eta_\mathbf{A} + \eta_\mathbf{B}} \\\ {\Delta N}_{A}^{+} & = -\frac{1}{2} \frac{A_\mathbf{B} - I_\mathbf{A}}{\eta_\mathbf{A} + \eta_\mathbf{B}} \end{aligned}$$

where $I$, $A$ and $\eta$ are the ionization potential, electron affinity and chemical hardness, respectively, for both interacting species.

from cdft import charge

delta_N_minus, delta_N_plus = charge.one_parabola( cation=cation_A, neutral=neutral_A, anion=anion_A,
                                                   ref_cation=cation_B, ref_neutral=neutral_B, ref_anion=anion_B )

3.2 Two-Parabola Model

The charge transfer channels for the Two-Parabola Model are given by

$$\begin{aligned} {\Delta N}_{A}^{-} & = \frac{{\mu}^{+}_\mathbf{B} - {\mu}^{-}_\mathbf{A}}{\eta_\mathbf{A} + \eta_\mathbf{B}} \\\ {\Delta N}_{A}^{+} & = \frac{{\mu}^{-}_\mathbf{B} - {\mu}^{+}_\mathbf{A}}{\eta_\mathbf{A} + \eta_\mathbf{B}} \end{aligned}$$

where $\mu^\pm$ and $\eta$ are the chemical potential and chemical hardness, respectively, for both interacting species.

from cdft import charge

delta_N_minus, delta_N_plus = charge.one_parabola( cation=cation_A, neutral=neutral_A, anion=anion_A,
                                                   ref_cation=cation_B, ref_neutral=neutral_B, ref_anion=anion_B )

3.3 Generalized Quadratic Model

The charge transfer channels for the Generalized Quadratic Model are given by

$$\begin{aligned} {\Delta N}_{A}^{-} & = \frac{{\mu}^{+}_\mathbf{B} - {\mu}^{-}_\mathbf{A}}{{\eta}^{-}_\mathbf{A} + {\eta}^{+}_\mathbf{B}} \\\ {\Delta N}_{A}^{+} & = \frac{{\mu}^{-}_\mathbf{B} - {\mu}^{+}_\mathbf{A}}{{\eta}^{+}_\mathbf{A} + {\eta}^{-}_\mathbf{B}} \end{aligned}$$

where $\mu^\pm$ and $\eta^\pm$ are the chemical potential and chemical hardness, respectively, for both interacting species.

from cdft import charge

delta_N_minus, delta_N_plus = charge.generalized( cation=cation_A, neutral=neutral_A, anion=anion_A,
                                                  ref_cation=cation_B, ref_neutral=neutral_B, ref_anion=anion_B )