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.