Tersoff potential (1988)

The implementation of the Tersoff (1988) potential in GPUMD mimics the one in lammps [Tersoff1988]. The Tersoff-1988 potential has a more general form than the Tersoff (1989) potential. When possible, it is, however, recommended to use the Tersoff (1989) potential as it is faster.

Potential form

The site potential can be written as

\[U_i = \frac{1}{2} \sum_{j \neq i} f_C(r_{ij}) \left[ f_R(r_{ij}) - b_{ij} f_A(r_{ij}) \right].\]

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

\[f_{C}(r_{ij}) = \frac{1}{2} \left[ 1 + \cos \left( \pi \frac{r_{ij} - R}{S - R} \right) \right].\]

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

\[\begin{split}f_{R}(r) &= A e^{-\lambda r_{ij}} \\ f_{A}(r) &= B e^{-\mu r_{ij}}.\end{split}\]

The bond-order is

\[b_{ij} = \left(1 + \beta^{n} \zeta^{n}_{ij}\right)^{-\frac{1}{2n}},\]

where

\[\begin{split}\zeta_{ij} &= \sum_{k\neq i, j}f_C(r_{ik}) g_{ijk} e^{\alpha(r_{ij} - r_{ik})^{m}} \\ g_{ijk} &= \gamma\left( 1 + \frac{c^2}{d^2} - \frac{c^2}{d^2+(h-\cos\theta_{ijk})^2} \right).\end{split}\]

Parameter	Unit
\(A\)	eV
\(B\)	eV
\(\lambda\)	Å\(^{-1}\)
\(\mu\)	Å\(^{-1}\)
\(\beta\)	dimensionless
\(n\)	dimensionless
\(c\)	dimensionless
\(d\)	dimensionless
\(h\)	dimensionless
\(R\)	Å
\(S\)	Å
\(m\)	dimensionless
\(\alpha\)	Å\(^{-m}\)
\(\gamma\)	dimensionless

File format

We have adopted a file format that similar but not identical to that used by lammps.

The potential file for a single-element system reads:

tersoff_1988 1 element
A_000 B_000 lambda_000 mu_000 beta_000 n_000 c_000 d_000 h_000 R_000 S_000 m_000 alpha_000 gamma_000

Here, element is the chemical symbol of the element.

The potential file for a double-element system reads:

tersoff_1988 2 <list of the 2 elements>
A_000 B_000 lambda_000 mu_000 beta_000 n_000 c_000 d_000 h_000 R_000 S_000 m_000 alpha_000 gamma_000
A_001 B_001 lambda_001 mu_001 beta_001 n_001 c_001 d_001 h_001 R_001 S_001 m_001 alpha_001 gamma_001
A_010 B_010 lambda_010 mu_010 beta_010 n_010 c_010 d_010 h_010 R_010 S_010 m_010 alpha_010 gamma_010
A_011 B_011 lambda_011 mu_011 beta_011 n_011 c_011 d_011 h_011 R_011 S_011 m_011 alpha_011 gamma_011
A_100 B_100 lambda_100 mu_100 beta_100 n_100 c_100 d_100 h_100 R_100 S_100 m_100 alpha_100 gamma_100
A_101 B_101 lambda_101 mu_101 beta_101 n_101 c_101 d_101 h_101 R_101 S_101 m_101 alpha_101 gamma_101
A_110 B_110 lambda_110 mu_110 beta_110 n_110 c_110 d_110 h_110 R_110 S_110 m_110 alpha_110 gamma_110
A_111 B_111 lambda_111 mu_111 beta_111 n_111 c_111 d_111 h_111 R_111 S_111 m_111 alpha_111 gamma_111

The extension to more than two components is accordingly.