Analytic bond-order potentials

The ABOP potential [Bre90] is a reformulation of the Tersoff potential [Ter88b] as implemented in atomicrex and Lammps. The difference lies in the two-body parameters \(A\), \(B\), \(\lambda_1\), and \(\lambda_2\), which have been rewritten in terms of the more physical choices, \(r_0\), \(D_0\), \(\beta\), and \(S\), which relate to dimer properties and the Pauling relation.

The implementation supports multi-component systems with all parameters being dependent on the types of the atoms involved, i.e. the two-body terms dependent on the types of atoms \(i\) and \(j\), whereas the three-body terms depend on the types of atoms \(i\), \(j\), and \(k\). Note that the three-body terms are usually dependent on the types of atoms \(i\) and \(k\), only (see e.g., [AlbNorAve02], [AlbNorNor02], [ErhAlb05], [JusErhTra05]). This behavior can be obtained by setting the classic-mode attribute to True (default: False).

In the ABOP format, the potential can be written

\[E = \frac{1}{2} \sum_i \sum_{j\neq i} V_{ij}\]

with

\[V_{ij} = f_c(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right].\]

Here, \(V_R(r) = \frac{D_0}{S-1} \exp[-\beta\sqrt{2S} (r-r_0)]\) and \(V_A(r) = -\frac{SD_0}{S-1}\exp[-\beta \sqrt{2/S}(r-r_0)]\) are repulsive and attractive pair potential branches, respectively, and \(f_c(r_{ij})\) is a cut-off function that is unity and decays sinusodially in the interval \((R-D,R+D)\), beyond which it vanishes. The three-body contributions arise due to the bond-order parameter

\[b_{ij} =\left( 1+\beta_2^n\zeta_{ij}^n \right)^{-\frac{1}{2n}}\]

where

\[\zeta_{ij} = \sum_{k\neq i,j} f_c(r_{ij})g(\theta_{ijk})\exp \left[ \lambda_3^m(r_{ij}-r_{ik})^m \right].\]

The angular dependence is due to the factor

\[g(\theta) = \gamma_{ijk} \left( 1+\frac{c^2}{d^2} - \frac{c^2}{d^2 + (h + \cos \theta)^2}\right).\]

Parameter files are written in the Lammps Tersoff format. In the output files, the \(r_0\), \(D_0\), \(\beta\), and \(S\) parameters are included in comment blocks.

Note that the \(h\) parameter in the ABOP form has the opposite sign than in the Tersoff formulation.

The following code snippet, to be inserted in the potentials block, illustrates the definition of this potential type in the input file.

<abop id="Si" species-a="*" species-b="*">
   <param-file>ABOP_potential.tersoff</param-file>
   <export-potential>ABOP_potential_fitted.tersoff</export-potential>
   <fit-dof>
      <r0 tag="SiSiSi" enabled="true" />
      <D0 tag="SiSiSi" enabled="true" />
      <beta tag="SiSiSi" enabled="true" />
      <S tag="SiSiSi" enabled="true" />
      <gamma tag="SiSiSi" enabled="true" />
      <c tag="SiSiSi" enabled="true" />
      <d tag="SiSiSi" enabled="true" />
      <h tag="SiSiSi" enabled="true" />
      <twomu tag="SiSiSi" enabled="false" />
      <beta2 tag="SiSiSi" enabled="false" />
      <powern tag="SiSiSi" enabled="false" />
 </fit-dof>
</abop>

Warning

Note that the species-a and species-b attributes should always be set to *. The assignment of atom types is handled using the information in the tag attribute.

Elements and attributes

  • <param-file>: Name of file with input parameters in Lammps/Tersoff format.

  • <export-potential> (optional): Name of file, to which potential parameters are being written in Lammps/Tersoff format [Default: no file is written].

  • <classic-mode> (optional): Enforce “classic” coupling of three-body parameters, which then depend only on the types of atoms \(i\) and \(k\) (rather than \(i\), \(j\), and \(k\)) [Default: False]

Further information

An extensive list of references is provided here.