Case study 1: Adding noble gases to TransRot - steventopper/TransRot GitHub Wiki
We are grateful to William Wang for completing and sharing this implementation and the following case study as a tutorial for the TransRot community.
Converting Lennard-Jones parameters to the TransRot standard form
In Part 1, we implement the OPLS-AA Lennard-Jones parameters for clusters which include one or more noble gas atoms. For rare gas atoms, the OPLS-AA model uses the famous Lennard-Jones equation $$U_{\mathrm{LJ}}(r_{ij})=4 \epsilon\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^6\right]$$ where $r_{ij}$ is the distance between any two atoms, $\sigma$ is the distance at which the potential is zero, and $\epsilon$ is the well depth (see Figure 1 below).
The ($\epsilon$, $\sigma$) values were taken from the OPLS-AA parameters document provided by the TINKER molecular modeling project. The file provides $\epsilon$ in $\mathrm{kcal/mole}$ and $\sigma$ in Å, which are the same units required by TransRot; these are provided below in Table 1. Because Lennard-Jones energies are often given in units of Kelvin in the literature, $\epsilon$ is provided in these units as well for purposes of comparison.
Table 1. TINKER OPLS-AA Noble Gas Parameters
| Element | $\sigma$ (Å) | $\epsilon$ (kcal/mole) | $\epsilon$ (Kelvin) |
|---|---|---|---|
| He | 2.5560 | 0.0200 | 10.064 |
| Ne | 2.7800 | 0.0690 | 34.722 |
| Ar | 3.4010 | 0.2399 | 120.72 |
| Kr | 3.6240 | 0.3170 | 161.89 |
| Xe | 3.9350 | 0.4330 | 217.90 |
However, as explained in the README, TransRot utilizes an alternate form of the equation: $$U_{\mathrm{LJ}}(r_{ij}) =\frac{D_{ij}}{{r_{ij}}^{12}}-\frac{C_{ij}}{{r_{ij}}^6}$$ where ($C_{ij}$, $D_{ij})$ are parameters for the system of interest (all other TransRot parameters are equal to zero here). These parameters have units of $\mathrm{(kcal\times Å^{6})/mole}$ and $\mathrm{(kcal\times Å^{12})/mole}$, respectively. Fortunately, $\epsilon$ and $\sigma$ are easily converted to the new parameters through the relationships $C_{ij}=4\epsilon\sigma^{6}$ and $D_{ij}=4\epsilon\sigma^{12}$.
Table 2. TransRot-ready TINKER OPLS-AA Noble Gas Parameters
| Element | $C_{ij}$ $\mathrm{(kcal\times Å^{6})/mole}$ | $D_{ij}$ $\mathrm{(kcal\times Å^{12})/mole}$ |
|---|---|---|
| He | 22.30771482 | 6220.426757 |
| Ne | 127.4024728 | 58809.38436 |
| Ar | 1447.871451 | 2240628.193 |
| Kr | 2872.422947 | 6506950.779 |
| Xe | 6430.078987 | 23871775.86 |
The resulting interaction potentials are shown below in Figure 1.
Figure 1. OPLS-AA Lennard-Jones interaction potentials (kcal/mole) as a function of interparticle distance (Å) for the noble gases.
Key: red=He, blue=Ne, green=Ar, purple=Kr, orange=Xe.
These parameters and atom types were added to the dbase.txt file as shown below. Note that each atom is initially placed at the origin (0,0,0) . For other particles the collision radius may have to be estimated from the structure or the potential model, but here the OPLS-AA values of $\sigma$ are conveniently provided. We note that this parameter is only used in the initial placement of cluster atoms and can be adjusted quite a bit without affecting the final outcome. Average atomic masses are used in each case, but individual isotope masses could alternately be used.
Figure 2: Addition of rare gas atom types and parameters to the dbase.txt file
1
He 2.5560
He 0 0 0 0.0000 0.0000 22.30771482 6220.426757 0 4.00260
1
Ne 2.7800
Ne 0 0 0 0.0000 0.0000 127.4024728 58809.38436 0 20.180
1
Ar 3.4010
Ar 0 0 0 0.0000 0.0000 1447.871451 2240628.193 0 39.9
1
Kr 3.6240
Kr 0 0 0 0.0000 0.0000 2872.422947 6506950.779 0 83.80
1
Xe 3.9350
Xe 0 0 0 0.0000 0.0000 6430.078987 23871775.86 0 131.29
It should be noted that recent work has led to more accurate effective two-body interaction potentials for the rare gases. In particular, the model due to Deiters and Sadus (2019) appears to be both accurate and numerically efficient. In general, the conventional wisdom in the literature supports the idea that for liquids (and nanoparticles), it is more appropriate to use an effective two-body interaction potential than one which accurately describes the interaction between two otherwise isolated rare gas atoms, as explained in Allen and Tildesley (2017).
End of Part 1
-WW and RQT