Tersoff potential (1989)

The Tersoff (1989) potential supports systems with one or two atom types [Tersoff1989]. It is less general than the Tersoff (1988) potential but faster.

Potential form

Below we use \(i,j,k,\cdots\) for atom indices and \(I,J,K,\cdots\) for atom types.

The site potential can be written as

\[U_i = \frac{1}{2} \sum_{j \neq i} f_\mathrm{C}(r_{ij}) \left[ f_\mathrm{R}(r_{ij}) - b_{ij} f_\mathrm{A}(r_{ij}) \right].\]

The function \(f_\mathrm{C}\) is a cutoff function, which is 1 when \(r_{ij}<R_{IJ}\) and 0 when \(r_{ij}>S_{IJ}\) and takes the following form in the intermediate region:

\[f_\mathrm{C}(r_{ij}) = \frac{1}{2} \left[ 1 + \cos \left( \pi \frac{r_{ij} - R_{IJ}}{S_{IJ} - R_{IJ}} \right) \right].\]

The repulsive function \(f_\mathrm{R}\) and the attractive function \(f_\mathrm{A}\) take the following forms:

\[\begin{split}f_\mathrm{R}(r) &= A_{IJ} e^{-\lambda_{IJ} r_{ij}} \\ f_\mathrm{A}(r) &= B_{IJ} e^{-\mu_{IJ} r_{ij}}.\end{split}\]

The bond-order function is

\[b_{ij} = \chi_{IJ} \left(1 + \beta_{I}^{n_{I}} \zeta^{n_{I}}_{ij}\right)^{-\frac{1}{2n_{I}}},\]

where

\[\begin{split}\zeta_{ij} &= \sum_{k\neq i, j} f_\mathrm{C}(r_{ik}) g_{ijk} \\ g_{ijk} &= 1 + \frac{c_{I}^2}{d_{I}^2} - \frac{c_{I}^2}{d_{I}^2+(h_{I}-\cos\theta_{ijk})^2}.\end{split}\]

Parameter	Unit
\(A_{IJ}\)	eV
\(B_{IJ}\)	eV
\(\lambda_{IJ}\)	Å\(^{-1}\)
\(\mu_{IJ}\)	Å\(^{-1}\)
\(\beta_I\)	dimensionless
\(n_I\)	dimensionless
\(c_I\)	dimensionless
\(d_I\)	dimensionless
\(h_I\)	dimensionless
\(R_{IJ}\)	Å
\(S_{IJ}\)	Å
\(\chi_{IJ}\)	dimensionless

File format

Single-element systems

In this case, \(\chi_{IJ}\) is irrelevant. The potential file reads:

tersoff_1989 1 <element>
A B lambda mu beta n c d h R S

Here, element is the chemical symbol of the element.

Two-element systems

In this case, there are two sets of parameters, one for each atom type. The following mixing rules are used to determine some parameters between the two atom types \(i\) and \(j\):

\[\begin{split}A_{IJ} &= \sqrt{A_{II} A_{JJ}} \\ B_{IJ} &= \sqrt{B_{II} B_{JJ}} \\ R_{IJ} &= \sqrt{R_{II} R_{JJ}} \\ S_{IJ} &= \sqrt{S_{II} S_{JJ}} \\ \lambda_{IJ} &= (\lambda_{II} + \lambda_{JJ})/2 \\ \mu_{IJ} &= (\mu_{II} + \mu_{JJ})/2.\end{split}\]

Here, the parameter \(\chi_{01}=\chi_{10}\) needs to be provided. \(\chi_{00}=\chi_{11}=1\) by definition.

The potential file reads:

tersoff_1989 2 <list of the 2 elements>
A_0 B_0 lambda_0 mu_0 beta_0 n_0 c_0 d_0 h_0 R_0 S_0
A_1 B_1 lambda_1 mu_1 beta_1 n_1 c_1 d_1 h_1 R_1 S_1
chi_01