# 9-6 Lennard-Jones potential

The 9-6 Lennard-Jones potential (also known as the 6-9 potential) is a variant the more well known Lennard-Jones model. It is used for computing non-bonded interactions. The potential is given by [1] :

$\Phi_{12}(r) = \epsilon \left[ 2\left(\frac{r_m}{r} \right)^{9} - 3\left( \frac{r_m}{r}\right)^6 \right]$

or in terms of $\sigma$ (the value of $r$ at which $\Phi_{12}(r)=0$) one has:

$\Phi_{12}(r) = 6.75 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{9} - \left( \frac{\sigma}{r}\right)^6 \right]$

where

• $r := |\mathbf{r}_1 - \mathbf{r}_2|$
• $\Phi_{12}(r)$ is the intermolecular pair potential between two particles or sites
• $r_m$ is the distance, $r$, at which $\Phi_{12}(r)$ is a minimum, which corresponds to $r_m = 1.5^{1/3} \sigma$.
• $\epsilon$ is the well depth (energy)

It is worth noting that the inclusion of an odd power (here the 9) adds an additional computational overhead, and the 8-6 Lennard-Jones potential has been suggested as a viable alternative.