Case study 1: Adding noble gases to TransRot

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