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interactions:tersoff

# Tersoff Potentials

The Tersoff potential1) is available when potfit is compiled with the tersoff flag, which also implies the apot flag for analytic potentials.

## Basic Theory

The total potential energy is defined as

$$E_{\text{total}}= \frac{1}{2}\sum_{i,j}f_c(r_{ij})\left(A_{ij}e^{-\lambda_{ij}r_{ij}}+b_{ij}B_{ij} e^{-\mu_{ij}r_{ij}}\right)$$

where

$$b_{ij} = \chi_{ij}\left(1+\gamma_{ij}^{n_i}\zeta_{ij}^{n_i}\right)^{-\frac{1}{2n_i}},$$

$$\zeta_{ij} = \sum_{k\neq i,j}f_c(r_{ik})\omega_{ik}g(\theta_{ijk})$$

and

$$g(\theta_{ijk}) = 1 + \frac{c_{ij}^2}{d_{ij}^2}-\frac{c_{ij}^2}{d_{ij}^2+(h_{ij}-\cos\theta_{ijk})^2}.$$

The cutoff function $f_c$ is given as

$$f_c(r_{ij}) = \begin{cases} 1 & r_{ij} \le R_{ij} \\ \frac{1}{2} + \frac{1}{2}\cos\left(\frac{\pi(r_{ij}-R_{ij})}{S_{ij}-R_{ij}}\right) & R_{ij} < r_{ij} < S_{ij} \\ 0 & S_{ij} \le r_{ij} \end{cases}.$$

### Analytic functions

To defined an analytic Tersoff potential in potfit, there are 2 functional forms required. They are available as the tersoff_pot and tersoff_mix function types. Out of the 13 free parameters for each interaction, the first eleven are specified in the tersoff_pot potential. The other two, which are only relevant for mixing potentials, are defined in the tersoff_mix potential.

## Number of potential functions

To describe a system with $N$ atom types you need $N^2$ potentials. They are not directly related to functions in the potential energy, they are divided into pure and mixing potentials. For more details take a look at the examples.

# atom types tersoff_pot tersoff_mix Total # potentials
$N$ $N(N+1)/2$ $N(N-1)/2$ $N^2$
1 1 0 1
2 3 1 4
3 6 3 9
4 10 6 16

## 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 potentials in the potential file for $N$ atom types is:

$P_{00}, \ldots, P_{0N}, P_{11}, \ldots, P_{1N}, \ldots, P_{NN}$
$M_{01}, \ldots, M_{0N}, M_{12}, \ldots, M_{1N}, \ldots, M_{N-1,N}$

where $P$ stands for tersoff_pot and $M$ for tersoff_mix potentials.

### IMD output

An analytic IMD potential which can be used with the tersoff2 option of IMD is written to *.imd.tersoff.pot. This file, however, may not be used as a potential file for IMD. Instead, its contents need to be copied INTO the IMD parameter file.

### LAMMPS output

The potential can also be written in LAMMPS format. The name of the output file is *.lammps.tersoff.

1)
J. Tersoff, Phys. Rev. B 39, 5566 (1989)