# 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.

tersoff_1989 2 <list of the 2 elements>