The Tersoff-1989 potential

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Potential form

  • Conventions:
    • Use [math]i,j,k,\cdots[/math] for atom indices.
    • Use [math]I,J,K,\cdots[/math] for atom types.


  • The site potential can be written as

\begin{equation} 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]. \end{equation}

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

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

  • The repulsive function [math]f_{\rm R}[/math] and the attractive function [math]f_{\rm A}[/math] take the following forms:

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

  • The bond-order function is

\begin{equation} b_{ij} = \chi_{IJ} \left(1 + \beta_{I}^{n_{I}} \zeta^{n_{I}}_{ij}\right)^{-\frac{1}{2n_{I}}}, \end{equation} where \begin{equation} \zeta_{ij} = \sum_{k\neq i, j} f_C(r_{ik}) g_{ijk}; \end{equation} \begin{equation} 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). \end{equation}

Parameters

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

Potential file format

Tersoff-1989 potential for single-element systems

  • In this case, [math]\chi_{IJ}[/math] 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 [math]i[/math] and [math]j[/math]:

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

  • Here, the parameter [math]\chi_{01}=\chi_{10}[/math] needs to be provided. [math]\chi_{00}=\chi_{11}=1[/math] by definition.
  • The potential file reads
   tersoff_1989 2
   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

Reference

  • [1] J. Tersoff, Modeling solid-state chemistry: Interatomic potentials for multicomponent systems, Phys. Rev. B 39, 5566(R) (1989).