# Modified Lennard-Jones model

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The modified Lennard-Jones model is given by (Eq. 2 [1]):

${\displaystyle \Phi _{12}(r)=\left\{{\begin{array}{ll}4\epsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right]+C_{1}&r\leq 2.3\sigma \\C_{2}\left({\frac {\sigma }{r}}\right)^{12}+C_{3}\left({\frac {\sigma }{r}}\right)^{6}+C_{4}\left({\frac {r}{\sigma }}\right)^{2}+C_{5}&2.3\sigma

where ${\displaystyle C_{1}=0.016132\epsilon }$, ${\displaystyle C_{2}=3136.6\epsilon }$ ${\displaystyle C_{3}=-68.069\epsilon }$ ${\displaystyle C_{4}=0.083312\epsilon }$ and ${\displaystyle C_{5}=0.74689\epsilon }$. These parametrs are chosen so that the function ${\displaystyle \Phi _{12}(r)}$, as well as the first derivative, is continuous both at ${\displaystyle r=2.3\sigma }$ and ${\displaystyle r=2.5\sigma }$. These parameters have recently been recalculated with greater precision by Asano and Fuchizaki [2], leading to ${\displaystyle C_{1}=0.0163169237\epsilon }$, ${\displaystyle C_{2}=3136.5686\epsilon }$ ${\displaystyle C_{3}=-68.069\epsilon }$ [3], ${\displaystyle C_{4}=-0.0833111261\epsilon }$ and ${\displaystyle C_{5}=0.746882273\epsilon }$.

## Virial coefficients

The virial coefficients up to the seventh order have been calculated for the range of temperatures ${\displaystyle k_{B}T/\epsilon =0.3-70}$ [4]. See also [5].