The Tersoff-1989 potential

Potential form

• Conventions:
• Use $i,j,k,\cdots$ for atom indices.
• Use $I,J,K,\cdots$ for atom types.

• The site potential can be written as

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

• The function $f_{\rm C}$ is a cutoff function, which is 1 when $r_{ij}\lt R_{IJ}$ and 0 when $r_{ij}\gt S_{IJ}$ and takes the following form in the intermediate region:

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

• The repulsive function $f_{\rm R}$ and the attractive function $f_{\rm A}$ take the following forms:

$$f_{\rm R}(r) = A_{IJ} e^{-\lambda_{IJ} r_{ij}};$$ $$f_{\rm A}(r) = B_{IJ} e^{-\mu_{IJ} r_{ij}}.$$

• 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 $$\zeta_{ij} = \sum_{k\neq i, j} f_C(r_{ik}) g_{ijk};$$ $$g_{ijk} = \left( 1 + \frac{c_{I}^2}{d_{I}^2} - \frac{c_{I}^2}{d_{I}^2+(h_{I}-\cos\theta_{ijk})^2} \right).$$

Parameters

 Parameter Units $A_{IJ}$ eV $B_{IJ}$ eV $\lambda_{IJ}$ A$^{-1}$ $\mu_{IJ}$ A$^{-1}$ $\beta_{I}$ dimensionless $n_{I}$ dimensionless $c_{I}$ dimensionless $d_{I}$ dimensionless $h_{I}$ dimensionless $R_{IJ}$ A $S_{IJ}$ A $\chi_{IJ}$ dimensionless

Potential file format

Tersoff-1989 potential for single-element systems

• In this case, $\chi_{IJ}$ is irrelevant. The potential file reads
   tersoff_1989 1
A B lambda mu beta n c d h R S


Tersoff-1989 potential for double-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 $j$:

$$A_{IJ} = \sqrt{A_{II} A_{JJ}};$$ $$B_{IJ} = \chi_{IJ}\sqrt{B_{II} B_{JJ}};$$ $$R_{IJ} = \sqrt{R_{II} R_{JJ}};$$ $$S_{IJ} = \sqrt{S_{II} S_{JJ}};$$ $$\lambda_{IJ} = (\lambda_{II} + \lambda_{JJ})/2;$$ $$\mu_{IJ} = (\mu_{II} + \mu_{JJ})/2.$$

• Here, the parameter $\chi_{01}=\chi_{10}$ needs to be provided. $\chi_{00}=\chi_{11}=1$ by definition.
   tersoff_1989 2