9-6 Lennard-Jones potential

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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]


  • 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.