# Electrostatic interactions#

## Second order#

### Isotropic electrostatics#

The isotropic electrostatic in a shell-resolved formulation is given by the parametrized Coulomb interaction between shellwise partial charges

$E_\text{IES} = \frac12 \sum_{\text{A},\text{B}} \sum_{l,l'}^{s,p,d} q^{l}_\text{A} \gamma^{ll'}_\text{AB} q^{l'}_\text{B}$

The interaction potential is parametrized by a Klopman–Ohno type potential in the xTB Hamiltonian or the γ-functional as used in the DFTB Hamiltonian.

#### Klopman–Ohno kernel#

The interaction kernel for the Klopman–Ohno electrostatic is given by

$\gamma^{ll'}_\text{AB} = \left( R_\text{AB}^g + f_\text{av}(\eta_A^l, \eta_B^{l'})^{-g} \right)^{-\frac1g}$

where η:sub:A/B are the chemical hardness parameters of the respective shells and g is the exponent to manipulate the potential shape.

#### γ-functional kernel#

The interaction kernel for the DFTB γ-functional is derived from the integral of two exponential densities

$\begin{split}\begin{split} \gamma^{ll'}_\text{AB} = \frac1{R_\text{AB}} - \exp[-\tau_\text{A}R] \left( \frac{\tau_\text{B}^4\tau_\text{A}}{2(\tau_\text{A}^2-\tau_\text{B}^2)^2} - \frac{\tau_\text{B}^6\tau_\text{A} - 3\tau_\text{B}^4\tau_\text{A}^2} {(\tau_\text{A}^2-\tau_\text{B}^2)^3 R_\text{AB}} \right) \\ - \exp[-\tau_\text{B}R] \left( \frac{\tau_\text{A}^4\tau_\text{B}}{2(\tau_\text{B}^2-\tau_\text{A}^2)^2} - \frac{\tau_\text{A}^6\tau_\text{B} - 3\tau_\text{A}^4\tau_\text{B}^2} {(\tau_\text{B}^2-\tau_\text{A}^2)^3 R_\text{AB}} \right) \end{split}\end{split}$

where τ:sub:A/B are scaled Hubbard parameters of the respective shells and R is the distance between the atomic sides.

### Anisotropic electrostatics#

The anisotropic electrostatic in an atom-resolved formulation is given by the multipole interactions between the different moments:

$E_\text{AES} = \sum_{\text{A},\text{B}} \sum_{k}^{x,y,z} q_\text{A} \gamma^{k}_\text{AB} \mu^{k}_\text{B} + \frac12 \sum_{\text{A},\text{B}} \sum_{k,k'}^{x,y,z} \mu^{k}_\text{A} \gamma^{kk'}_\text{AB} \mu^{k'}_\text{B} + \sum_{\text{A},\text{B}} \sum_{k,k'}^{x,y,z} q_\text{A} \gamma^{kk'}_\text{AB} \theta^{kk'}_\text{B}$

## Third order#

The isotropic third-order contributions are included as the trace of the on-site shell-resolved Hubbard derivatives.

$E_\text{IXC} = \frac13 \sum_\text{A} \sum_{l} \Gamma^l_\text{A} (q^l_\text{A})^3$