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] :

${\displaystyle \Phi _{12}(r)=\epsilon \left[2\left({\frac {r_{m}}{r}}\right)^{9}-3\left({\frac {r_{m}}{r}}\right)^{6}\right]}$

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

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

where

• ${\displaystyle r:=|\mathbf {r} _{1}-\mathbf {r} _{2}|}$
• ${\displaystyle \Phi _{12}(r)}$ is the intermolecular pair potential between two particles or sites
• ${\displaystyle r_{m}}$ is the distance, ${\displaystyle r}$, at which ${\displaystyle \Phi _{12}(r)}$ is a minimum, which corresponds to ${\displaystyle r_{m}=1.5^{1/3}\sigma }$.
• ${\displaystyle \epsilon }$ 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.