# Stockmayer potential

The Stockmayer potential consists of the Lennard-Jones model with an embedded point dipole. Thus the Stockmayer potential becomes (Eq. 1 [1]):

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

where:

• $r:=|{\mathbf {r}}_{1}-{\mathbf {r}}_{2}|$
• $\Phi (r)$ is the intermolecular pair potential between two particles at a distance $r$
• $\sigma$ is the diameter (length), i.e. the value of $r$ at $\Phi (r)=0$
• $\epsilon$ represents the well depth (energy)
• $\epsilon _{0}$ is the permittivity of the vacuum
• $\mu$ is the dipole moment
• $\theta _{1}$ and $\theta _{2}$ are the angles associated with the inclination of the two dipole axes with respect to the intermolecular axis.
• $\phi$ is the azimuth angle between the two dipole moments

If one defines a reduced dipole moment, $\mu ^{*}$, such that:

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

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.

## Critical properties

In the range $0\leq \mu ^{*}\leq 2.45$ [2]:

$T_{c}^{*}=1.313+0.2999\mu ^{{*2}}-0.2837\ln(\mu ^{{*2}}+1)$
$\rho _{c}^{*}=0.3009-0.00785\mu ^{{*2}}-0.00198\mu ^{{*4}}$
$P_{c}^{*}=0.127+0.0023\mu ^{{*2}}$

## Bridge function

A bridge function for use in integral equations has been calculated by Puibasset and Belloni [3].