3.1. Supported Model Force Fields

The interactions between atoms can be approximated by various kinds of model potentials, which are preferred by physicists. For chemists, they can also be used but results must be interpreted with care. Here “atom” is used to indicate any particle like an ion. ABCluster supports many kinds of model potentials, which are listed below:

Tip

I have collected a lot of force field parameters in misc/atomic-force-field.txt. If there is no what you need, you will have to search in the literature or fit the parameters by yourself.

\[U_{\mathrm{CBM}}=\sum_{i=1}^{N}\sum_{i < j}^{N}\left(\frac{e^{2}}{4\pi\epsilon_{0}}\frac{q_{i}q_{j}}{r_{ij}}+B_{ij}\exp\left(-\frac{r_{ij}}{\rho_{ij}}\right)\right)\]
\[U_{\mathrm{LJ}}=\sum_{i=1}^{N}\sum_{i < j}^{N}4\epsilon_{ij}\left(\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{6}\right)\]

  • Coulomb-Lennard-Jones potential (CLJ). It models both Coulomb and dispersion interactions.

\[U_{\mathrm{CLJ}}=\sum_{i=1}^{N}\sum_{i < j}^{N}\left(\frac{q_{i}q_{j}}{r_{ij}}+4\epsilon_{ij}\left(\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{6}\right)\right)\]
\[U_{\mathrm{M}}=\sum_{i=1}^{N}\sum_{i < j}^{N}\epsilon_{ij}\left(\exp\left(-n\beta_{ij}\left(r_{ij}-r^{0}_{ij}\right)\right)-n\exp\left(-\beta_{ij}\left(r_{ij}-r^{0}_{ij}\right)\right)\right)\]
\[\begin{split}\begin{split} U_{\mathrm{CMR}}= & \sum_{i=1}^{N}\sum_{i < j}^{N} \left(\frac{q_{i}q_{j}}{r_{ij}}+D_{ij}\left(\exp\left(-2\alpha_{ij}\left(r_{ij}-\rho_{ij}\right)\right)\right.\right. \\ & \left.\left.+2\exp\left(-\alpha_{ij}\left(r_{ij}-\rho_{ij}\right)\right)\right)+\frac{C_{ij}}{r_{ij}^{12}}\right) \\ \end{split}\end{split}\]

  • Z potential (Z). This is used to study glass formation. Unlike CBM, LJ and M potentials, the potential contains maxima as well as minima, which make the close-packing energetically unfavorable and lead to amorphous structures.

\[U_{\mathrm{Z}}=\sum_{i=1}^{N}\sum_{i < j}^{N}\left(a\frac{e^{\alpha r_{ij}}}{r_{ij}^{3}}\cos\left(2k_{\mathrm{F}}r_{ij}\right)+b\left(\frac{\sigma}{r_{ij}}\right)^{m}+V_{0}\right)\theta\left(r_{\mathrm{c}}-r_{ij}\right)\]
\[\begin{split}\begin{split} U_{\mathrm{Gf}}= & \sum_{i=1}^{N}\sum_{i < j}^{N} \left(-\alpha\left(\frac{1}{s_{ij}\left(s_{ij}-1\right)^3}+\frac{1}{s_{ij}\left(s_{ij}+1\right)^3}-\frac{2}{s_{ij}^{4}}\right)\right. \\ & \left.+\beta\left(\frac{1}{s_{ij}\left(s_{ij}-1\right)^9}+\frac{1}{s_{ij}\left(s_{ij}+1\right)^9}-\frac{2}{s_{ij}^{10}}\right)\right) \\ \end{split}\end{split}\]
\[ \begin{align}\begin{aligned}s_{ij}=\frac{r_{ij}}{2d}\\\alpha=\frac{N_{\mathrm{C}}^{2}A}{12\left(2d\right)^{6}}\\\beta=\frac{N_{\mathrm{C}}^{2}B}{90\left(2d\right)^{12}}\end{aligned}\end{align} \]

  • Gupta potential (G). A very important many-body potential for modelling metallic clusters. They are not suitable for small clusters!

\[\begin{split}U_{\mathrm{G}}=\sum_{i=1}^{N}\left(\sum_{\substack{j=1\\j \ne i}}^{N}A_{ij}\exp\left(-p_{ij}\left(\frac{r_{ij}}{d_{ij}}-1\right)\right)-\sqrt{\rho\left(\mathbf{r}_i\right)}\right)\end{split}\]
\[\begin{split}\rho\left(\mathbf{r}_i\right)=\sum_{\substack{j=1\\j \ne i}}^{N}\xi_{ij}^{2}\exp\left(-2q_{ij}\left(\frac{r_{ij}}{d_{ij}}-1\right)\right)\end{split}\]
\[\begin{split}U_{\mathrm{SC}}=\sum_{i=1}^{N}\left(\frac{1}{2}\sum_{\substack{j=1\\j \ne i}}^{N}\epsilon_{ij}\left(\frac{a_{ij}}{r_{ij}}\right)^{p}-\sqrt{\rho\left(\mathbf{r}_i\right)}\right)\end{split}\]
\[\begin{split}\rho\left(\mathbf{r}_i\right)=\sum_{\substack{j=1\\j \ne i}}^{N}c_{ij}^{2}\left(\frac{a_{ij}}{r_{ij}}\right)^{q}\end{split}\]

  • Modified Sutton-Chen potential (SCMJB). In 2015, Januszko and Bose proposed a modified form of Sutton-Chen potential, which shows some improvement in reproducing experimental data. They are not suitable for small clusters!

\[\begin{split}U_{\mathrm{SCMJB}}=\sum_{i=1}^{N}\left(\sum_{\substack{j=1\\j \ne i}}^{N}\epsilon_{ij}\left(\frac{a_{ij}}{r_{ij}}\right)^{p}-\sqrt{\rho\left(\mathbf{r}_i\right)}\right)\end{split}\]
\[\begin{split}\rho\left(\mathbf{r}_i\right)=\sum_{\substack{j=1\\j \ne i}}^{N}c_{ij}^{2}\exp\left(-\alpha_{ij}\left(\frac{r_{ij}}{a_{ij}}-1\right)\right)\end{split}\]

  • Tersoff potential (T). This very complex many-body potential is useful for describing C, Si, Ge, N, B clusters. Its formula contains bond order information, thus it can generate structures with reasonable coordination numbers.

\[\begin{split}U_{\mathrm{T}}=\frac{1}{2}\sum_{i=1}^{N}\sum_{\substack{j=1\\j \ne i}}^{N}f_\mathrm{c}\left(r_{ij}\right)\left(U_{\mathrm{R}}\left(r_{ij}\right)+b_{ij}U_{\mathrm{A}}\left(r_{ij}\right)\right)\end{split}\]

where:

\[ \begin{align}\begin{aligned}f_\mathrm{c}\left(r_{ij}\right)= \begin{cases} 1 &r_{ij} < R_{ij} \cr \displaystyle{\frac{1}{2}+\frac{1}{2}\cos\frac{r_{ij}-R_{ij}}{S_{ij}-R_{ij}}\pi} & R_{ij} < r_{ij} < S_{ij}\cr 0 & S_{ij} < r_{ij} \end{cases}\\U_{\mathrm{R}}\left(r_{ij}\right)=A_{ij}\exp\left(-\lambda_{ij}r_{ij}\right)\\U_{\mathrm{A}}\left(r_{ij}\right)=-B_{ij}\exp\left(-\mu_{ij}r_{ij}\right)\\b_{ij} = \chi_{ij}\left(1+\beta_{i}^{m_i}\xi_{ij}^{m_i}\right)^{-\frac{1}{2m_{i}}}\\\begin{split}\xi_{ij} = \sum_{\substack{k=1\\k\ne i, j}}^{N}f_{\mathrm{c}}\left(r_{ik}\right)\omega_{ik}g_{ijk}\end{split}\\g_{ijk} = 1+\left(\frac{c_{i}}{d_{i}}\right)^{2}-\frac{c_{i}^{2}}{d_{i}^{2}+\left(h_{i}-\cos{\theta_{ijk}}\right)^{2}}\end{aligned}\end{align} \]

In these formulas, \(\theta_{ijk}\) is the angle formed by three atoms \(i\), \(j\) and \(k\), atom \(i\) being the center.

  • Extended Lennard-Jones potential (ELJ). This is a generalized version of Lennard-Jones potential, although at small size its GM does not change much compared to that of a usual LJ cluster.

\[U_{\mathrm{ELJ}} = \sum_{i=1}^{N}\sum_{i < j}^{N}\sum_{k = 6}^{k_{\mathrm{max}}}\frac{c_k}{r_{ij}^{k}}\]