# Pair Potentials

The most simple interaction that can be used are pair potentials. They are assumed to be isotropic, i.e. they only depend on the pair distance $r_{ij}=|\boldsymbol{r}_j-\boldsymbol{r}_i|$ of the two atoms $i$ and $j$. The potential energy for a system described by pair interactions is given as $$ E_\text{total} = \frac{1}{2} \sum_{i\neq j}V_2(\boldsymbol{r}_i,\boldsymbol{r}_j) = \sum_{i<j}V_2(r_{ij}) $$ and the force $\boldsymbol{F}_i$, acting on atom $i$, as $$ \boldsymbol{F}_i = - \sum_{j}\nabla_iV_2(r_{ij}).$$

## Tabulated Pair Potentials

For use in *potfit*, the potential function $V_2$ needs to be specified at a limited number of
sampling points. Position and value of these sampling points are given in the file
specified with the parameter `potfile`

. The format used in this file is described
here. It is essential that the potential is tabulated equidistant
in the the atom distance, not in the square (as it is done in
IMD). Note that if the potential does not
extend to radii small enough to cover all pair distances appearing in the configurations,
the program will exit with a short distance error.

## Analytic Pair Potentials

In order to use analytic potentials in *potfit*, the potential function $V_2$ has to be
specified by a functional name and the values of its parameters in the potential file.
The format used in this file is described here. With analytic potentials
there is also the option to add an additional energy term to the fit, the so called chemical potential.

### Chemical Potentials

For fitting simple pair potentials the use of the chemical potentials will always be disabled unless the `enable_cp`

switch in the parameter file is set to 1.
This option effectively adds a second energy term into the calculations, the energy will be calculated as

$$E_{tot}=\sum_{\text{species}}\mu_iN_i,+\sum_{\text{pairs}}V(r)$$

With this feature enabled, the potential file has to be altered, an additional block with the chemical potentials $\mu_i$ has to be added right after the header.

#F 0 3 #C Mg Zn #I 0 0 0 #E cp_Mg -1 -10 0 cp_Zn -1 -10 0

There has to be one line per element, starting with `cp_`

, the rest of the name will be replaced by the element name (if the `#C`

line is available) or the element number.
The following three values are the starting value, the minimum and maximum for that parameter.

**This feature is only available for analytic pair potentials.**

## 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:

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