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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) 