> For the complete documentation index, see [llms.txt](https://ehjc.gitbook.io/kallisto/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://ehjc.gitbook.io/kallisto/features/eeq.md).

# Partial Charges

## Introduction

Classical electronegativity equilibration (EEQ) partial charges are determined by minimising the following energy expression of the isotropic electrostatic interaction, which is dependent on atomic charges *q*

$$
E\_{IES} = \sum\limits\_{i=1}^N\left( \chi\_iq\_i + \frac{1}{2}\left(J\_{ii} + \frac{2\gamma\_{ii}}{\sqrt{\pi}}\right)q\_i^2\right) + \frac{1}{2}\sum\limits\_{i=1}^N\sum\limits\_{j\ne i}^Nq\_iq\_j\frac{\text{erf}(\gamma\_{ij}R\_{ij})}{R\_{ij}} \\\[1em]\quad \text{with}\quad \chi\_i = EN\_i - \kappa\_i\sqrt{CN\_i}.
$$

The first part of the equation above describes the on-side interaction of atom `i` in terms of a Taylor expansion expressed in atomic partial charges. Within `chii` , we apply the Pauling electronegativity (`EN`) and the atomic coordination number (`CN`) scaled by a parameter `kappai`to introduce an environment dependency into the partial-charge approach.The second part of the equation describes the (pairwise) interactions between the atom `i` and all `j`particles as obtained for interacting charge densities (for a deeper understanding check the references, [Goedecker et al.](https://doi.org/10.1103/PhysRevB.92.045131) and [Caldeweyher et al.](https://doi.org/10.26434/chemrxiv.7430216.v2); **Note**: Eq. 12 in [Caldeweyher et al.](https://doi.org/10.26434/chemrxiv.7430216.v2) is *erroneous* while the definition above is correct as it is also given in the [JCP publication](https://doi.org/10.1063/1.5090222)).

To obtain EEQ partial charges under the constraint that the partial charges conserve the total charge of the system, the method of constrained [Lagrangian optimisation](https://en.wikipedia.org/wiki/Lagrange_multiplier) is used

$$
\mathcal{L} = E\_{IES} + \lambda\left(\sum\limits\_{k=1}^Nq\_k - q\_{total}\right) \ \quad \text{with}\quad \frac{\partial\mathcal{L}}{\partial q} = \mathbf{0} \quad\text{and} \quad \frac{\partial\mathcal{L}}{\partial \lambda}= \sum\limits\_{k=1}^N q\_k-q\_{total} =0,
$$

which leads to a set of (N + 1) linear equations that can be rewritten in matrix form as

$$
\begin{pmatrix}
\mathcal{A} & \mathbf{1}\\
\mathbf{1} & \mathbf{0}
\end{pmatrix}
\begin{pmatrix}
\mathbf{q} \\
\mathbf{\lambda}
\end{pmatrix} =
\begin{pmatrix}
\mathcal{X} \\
q\_{total}
\end{pmatrix}  \quad \text{with}\quad \mathcal{X}\_i = -\chi\_i
$$

Let's define the interaction matrix (the definition of the right-hand side is given [above](/kallisto/features/eeq.md#introduction))

$$
\mathcal{A}*{ij} =     \begin{cases}
J*{ii} + \frac{2\gamma\_{ii}}{\sqrt{\pi}}, &         \text{if } i=j\\
\frac{\text{erf}(\gamma\_{ij}R\_{ij})}{R\_{ij}}, &         \text{if } i\neq j
\end{cases},
$$

Overall five parameter exist per element (fitted against PBE0/def2-TZVP Hirshfeld charges):

| Parameter | Meaning for atom i                                                                                     |
| --------- | ------------------------------------------------------------------------------------------------------ |
| `Jii`     | [Chemical Hardness](https://doi.org/10.1021/j100023a006)                                               |
| `gammaii` | Atomic radii dependent term                                                                            |
| `ENi`     | [Pauling electronegativity](https://en.wikipedia.org/wiki/Electronegativity#Pauling_electronegativity) |
| `Rcovi`   | [Covalent atomic radius](https://doi.org/10.1002/chem.200800987)                                       |
| `Kappai`  | Scaling factor for `chii`                                                                              |

## Define the Subcommand

{% tabs %}
{% tab title="eeq" %}

```bash
> kallisto eeq options arguments
```

{% endtab %}

{% tab title="options" %}

```markup
--chrg <int>
(optional, default: 0)
description:
 absolute charge (qtotal) of the input structure (Lagrangian constraint)
 
--out <string> 
(optional)
description: 
 write output to file
```

{% endtab %}

{% tab title="arguments" %}

```
input file is given as (positional) argument
```

{% endtab %}
{% endtabs %}

## Application

To calculate atomic EEQ charges for a neutral charged Alanine-Glycine molecule, we call the subcommand `eeq`

```bash
> kallisto eeq alanine-glycine.xyz
0.059704461728256275
0.2626494653657499
-0.4965512448739412
0.047991263003576215
0.2744196871227069
-0.38881537388038867
-0.6402141334784498
-0.3928319273751289
-0.47582669447302023
0.2790624116759344
0.2891096702066603
-0.1845935570451514
0.12417598515268524
0.27028773655342087
0.14722120980685544
0.14502596136754614
0.2992214711225029
0.13031907222773667
0.1342346682463232
0.11540986754612566
# Save output to file 'eeq'
> kallisto eeq --out eeq alanine-glycine.xyz
> cat eeq
0.059704461728256275
0.2626494653657499
-0.4965512448739412
0.047991263003576215
0.2744196871227069
-0.38881537388038867
-0.6402141334784498
-0.3928319273751289
-0.47582669447302023
0.2790624116759344
0.2891096702066603
-0.1845935570451514
0.12417598515268524
0.27028773655342087
0.14722120980685544
0.14502596136754614
0.2992214711225029
0.13031907222773667
0.1342346682463232
0.11540986754612566
```

Now we obtain a list of atomic EEQ charges, which sum up to a total charge of zero. However, we can furthermore calculate atomic EEQ charges for the cationic (or anionic) Alanine-Glycine molecule by incorporating the `chrg` option as described in the subcommand definition.