Stockmayer potential

The Stockmayer potential consists of the Lennard-Jones model with an embedded point dipole. Thus the Stockmayer potential becomes:

$\Phi (r,\theta _{1},\theta _{2},\phi )=4\epsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right]-{\frac {\mu ^{2}}{4\pi \epsilon _{0}r^{3}}}\left(2\cos \theta _{1}\cos \theta _{2}-\sin \theta _{1}\sin \theta _{2}\cos \phi \right)$

where:

• $\Phi (r)$ is the intermolecular pair potential between two particles at a distance r;
• $\sigma$ is the diameter (length), i.e. the value of $r$ at $\Phi (r)=0$ ;
• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \epsilon }  : well depth (energy)
• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \epsilon _{0}} is the permittivity of the vacuum
• $\mu$ is the dipole moment
• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \theta _{1},\theta _{2}} is the inclination of the two dipole axes with respect to the intermolecular axis.

If one defines the reduced dipole moment, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mu ^{*}}

$\mu ^{*}:={\sqrt {\frac {\mu ^{2}}{4\pi \epsilon _{0}\epsilon \sigma ^{3}}}}$

one can rewrite the expression as

$\Phi (r,\theta _{1},\theta _{2},\phi )=\epsilon \left\{4\left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right]-\mu ^{*2}\left(2\cos \theta _{1}\cos \theta _{2}-\sin \theta _{1}\sin \theta _{2}\cos \phi \right)\left({\frac {\sigma }{r}}\right)^{3}\right\}$

For this reason the potential is sometimes known as the Stockmayer 12-6-3 potential.

References

1. M. E. Van Leeuwe "Deviation from corresponding-states behaviour for polar fluids", Molecular Physics 82 pp. 383-392 (1994)