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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
$$E_{\text{total}}=\frac{1}{2}\sum_{i,j}^{N}\Phi_{ij}\left(r_{ij}\right)= \frac{1}{2}\sum_{i,j}f_c(r_{ij})\left(A_{ij}e^{-\lambda_{ij}r_{ij}}+b_{ij}B_{ij} e^{-\mu_{ij}r_{ij}}\right)$$
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
LAMMPS output
1. J. Tersoff, Phys. Rev. B 39, 5566 (1989)