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interactions:coulomb

# Coulomb Interactions

Electrostatics can be enabled with the coulomb option. A charge $q$, which has to be given in the potential file, is assigned to each atom type. The coulomb interaction of a set of $N$ ions is calculated as $$E_\text{total}^\text{Coul} = \frac{1}{2} \sum_{i\neq j} E(r_{ij}) \qquad \text{with} \qquad E(r_{ij}) \sim \frac{q_{i}q_{j}}{r_{ij}}.$$

There are three modifications required for a proper numerical calculation.

#### 1. Spherical truncation in real space

$$E_\text{total}^\text{Coul} = \frac{1}{2} \sum_{\substack{i\neq j\\r_{ij} \leq r_\text{cut}}} E(r_{ij})$$

Because electrostatics are long ranged, a cutoff radius $r_{\text{cut}}$ is introduced to achieve convergency.

#### 2. Shifting

$$E (r_{ij}) \longrightarrow E (r_{ij}) - E (r_{\text{cut}}) - (r_{ij} - \left.r_{\text{cut}})\frac{dE}{dr} \right|_{r_{\text{cut}}}$$

The shifted potential enery and it's gradient go to zero at the cutoff sphere. This is equivalent to the idea of putting mirror charges on the sphere which screen the total net charge within the sphere.

#### 3. Smoothing with damping function

$$E (r_{ij}) \longrightarrow E (r_{ij}) \text{erfc}(\kappa r_{ij})$$

This damping is an approximation, which yields faster convergency in real space.

This approach is the Wolf summation method1). The Coulomb potential can be combined with any analytic (short-range) pair potential. Morse-Stretch (ms) and Buckingham (buck) potentials are adapted to use with Wolf-Summation for dipoles by assuring zero gradient at the cutoff. Coulomb potentials can be combined with EAM or angular-dependent potentials by using the force calculation routines built for this purpose.

## Parameters

Only one additional parameter can be given in the parameter file:

dp_cut - float - 10

## Compatibilities

• coulomb has to be compiled with option apot.
• coulomb can be used with stress, fweight, evo and can also be executed in parallel using option mpi.
• coulomb can optionally be combined with angular (ang) or EAM (eam) potentials. It cannot not be used together with any other force-field approaches (pair, adp, …). Note that pair potentials are available by default and must not be specifically added.

## Potential file

When using coulomb interactions, the following parameters have to given in the potential file, labeled by the keyword elstat:

• ratio stoichiometric ratio is needed, because charges are optimized under the constraint of charge neutrality.
• charge (ntypes-1) charges have to be given, the last one is calculated via ratio.
• kappa damping parameter $\kappa$ of the Wolf summation, it is recommended to keep it constant

Example for the diatomic oxide SiO2

#F 0 3
#C Si O
#I 0 0 0
#E

elstat
ratio     1 2
charge_Si value min max
kappa     value min max

## Number of potential functions

To describe a system with $N$ atom types you need $N(N+1)/2$ potentials.

$N$ $N(N+1)/2$
1 1
2 3
3 6

## Order of potential functions

The potential table is assumed to be symmetric, i.e. the potential for the atom types 1-0 is the same as the potential 0-1.

The order of the pair potentials in the potential file for N atom types is:

$\Phi_{00}, \ldots, \Phi_{0N}, \Phi_{11}, \ldots, \Phi_{1N}, \ldots, \Phi_{NN}$

1)
D. Wolf, P. Keblinski, S. R. Phillpot, and J. Eggenbrecht, J. Chem. Phys. 110, 8254 (1999)