# Stockmayer potential

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

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

where:

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

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

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

one can rewrite the expression as

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

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

## Critical properties

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

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

## Bridge function

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