Angular dependent potential

The angular dependent potential (ADP) was proposed by Mishin et al. [MisMehPap05] and extends the EAM formalism with angular dependent dipole and quadrupole terms.

\[E = \frac{1}{2} \sum_{ij} V(r_{ij}) + \sum_i F(\rho_i) + \frac{1}{2} \sum_{i,\alpha} (\mu_i^{\alpha})^2 + \frac{1}{2} \sum_{i,\alpha,\beta} (\lambda_i^{\alpha,\beta})^2 - \frac{1}{6} \sum_i \eta_i^2\]

where

\[\rho_i = \sum_j \rho_j(r_{ij}),\]

\[\mu_i^{\alpha} = \sum_{j\neq i} u_{ij}(r_{ij})r_{ij}^{\alpha},\]

\[\lambda_i^{\alpha,\beta} = \sum_{j\neq i} w_{ij}(r_{ij})r_{ij}^{\alpha}r_{ij}^{\beta}\]

and

\[\eta_i = \sum_{\alpha} \lambda_i^{\alpha\alpha}\]

The functions \(V\), \(F\), \(\rho\), \(u\) and \(w\) can be chosen freely using a math parser compose functional forms, similar to EAM potentials. The current implementation allows the electron density of EAM potentials to be defined for each pair of elements separately (i.e. for 2 Elements 3 functions can be defined). For ADPs the electron density at the position of some atom is only defined by the distance and species of the atoms surrounding it. Consequently, \(\rho\) requires to set only one species. Furthermore, the additional function \(u\) and \(w\) have to be defined. The following code block illustrates the differences compared to setting up an EAM potential of a rather simple functional form. The main block is called <adp>.

<adp id="ADP_SAMPLE" species-a="*" species-b="*">
  <export-adp-file resolution='10000' rho-range-factor='2.0'>output.adp</export-adp-file>

  <mapping>
    <pair-interaction species-a="*" species-b="*" function="V" />
    <electron-density species="*" function="rho" />
    <embedding-energy species="*" function="F" />
    <u-function species-a="*" species-b="*" function="u" />
    <w-function species-a="*" species-b="*" function="w" />
  </mapping>

  <functions>
    <user-function id="V">
      Same as EAM
    </user-function>

    <user-function id="rho">
      Same as EAM, but only 1 per species
    </user-function>

    <user-function id="F">
      Same as EAM
    </user-function>

    <user-function id="i">
      <input-var>r</input-var>
      <expression> exp(-twomu*r) </expression>
      <derivative> -twomu*exp(-twomu*r) </derivative>
      <param name="twomu">1.14</param>
      <fit-dof> <twomu/> </fit-dof>
      <screening>
        <user-function id="rho_screening">
          <cutoff>6.5</cutoff>
          <input-var>r</input-var>
          <expression> 1 - 1/(1 + ((r - cutoff) / h)^4) </expression>
          <derivative> 4 * h^4 * (r - cutoff)^3 / ((h^4 + (r - cutoff)^4)^2) </derivative>
          <param name="h">3</param>
        </user-function>
      </screening>
    </user-function>

    <user-function id="w">
      <input-var>r</input-var>
      <expression> exp(-twomu*r) </expression>
      <derivative> -twomu*exp(-twomu*r) </derivative>
      <param name="twomu">1.14</param>
      <fit-dof> <twomu/> </fit-dof>
      <screening>
        <user-function id="rho_screening">
          <cutoff>6.5</cutoff>
          <input-var>r</input-var>
          <expression> 1 - 1/(1 + ((r - cutoff) / h)^4) </expression>
          <derivative> 4 * h^4 * (r - cutoff)^3 / ((h^4 + (r - cutoff)^4)^2) </derivative>
          <param name="h">3</param>
        </user-function>
      </screening>
    </user-function>

  </functions>
</adp>

Further information

For further information on ADPs refer to the original publication [MisMehPap05].