Stockmayer potential

From SklogWiki

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

$\Phi_{12}(r, \theta_1, \theta_2, \phi) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] - \frac{\mu_1 \mu_2}{4\pi \epsilon_0 r^3} \left(2 \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 \cos \phi\right)$

where:

$r := |\mathbf{r}_1 - \mathbf{r}_2|$
$Φ(r)$ is the intermolecular pair potential between two particles at a distance r;
$σ$ is the diameter (length), i.e. the value of $r$ at $Φ(r) = 0$ ;
$ε$ : well depth (energy)
$ε 0$ is the permittivity of the vacuum
$μ$ is the dipole moment
$θ 1,θ 2$ is the inclination of the two dipole axes with respect to the intermolecular axis.
$φ$ is the azimuth angle between the two dipole moments

If one defines the reduced dipole moment, $μ *$

$\mu^* := \sqrt{\frac{\mu^2}{4\pi\epsilon_0\epsilon \sigma^3}}$

one can rewrite the expression as

$\Phi(r, \theta_1, \theta_2, \phi) = \epsilon \left\{4\left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] - \mu^{*2} \left(2 \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 \cos \phi\right) \left(\frac{\sigma}{r} \right)^{3} \right\}$