# The Tersoff-1988 potential

## Relationship with LAMMPS

• This potential is made to mimic the Tersoff potential in LAMMPS [1].

## 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 $r_{ij}\lt R$ and 0 when $r_{ij}\gt S$ 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:

$$f_{R}(r) = A e^{-\lambda r_{ij}};$$ $$f_{A}(r) = B e^{-\mu r_{ij}}.$$

• The bond-order is

$$b_{ij} = \left(1 + \beta^{n} \zeta^{n}_{ij}\right)^{-\frac{1}{2n}},$$ where $$\zeta_{ij} = \sum_{k\neq i, j}f_C(r_{ik}) g_{ijk} e^{\alpha(r_{ij} - r_{ik})^{m}};$$ $$g_{ijk} = \gamma\left( 1 + \frac{c^2}{d^2} - \frac{c^2}{d^2+(h-\cos\theta_{ijk})^2} \right).$$

## Parameters

 Parameter Units $A$ eV $B$ eV $\lambda$ A$^{-1}$ $\mu$ A$^{-1}$ $\beta$ dimensionless $n$ dimensionless $c$ dimensionless $d$ dimensionless $h$ dimensionless $R$ A $S$ A $m$ dimensionless $\alpha$ A$^{-m}$ $\gamma$ dimensionless

## Potential file format

• We have adopted a file format similar (but not identical) to that used by LAMMPS [1].
• The potential file for a single-element system reads:
tersoff_1988 1
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

• The potential file for a double-element system reads:
tersoff_1988 2
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

• Can you guess the file format for a triple-element system?