Molecular Simulation/Dipole-Dipole Interactions

Dipole-dipole interactions occur between two molecules when each of the molecules has a permanent dipole moment. These interactions can be attractive or repulsive depending on the variable orientation of the two dipoles.

A permanent dipole moment occurs when a molecule exhibits uneven charge distribution. This is caused by atoms of different electronegativities being bonded together. The electron density of the molecule will be polarized towards the more electronegative atoms. This excess of negative charge in one part of the molecule and the deficit of charge in another area of the molecule results in the molecule having a permanent dipole moment.

The orientation of the two interacting dipoles can be described by three angles (θ1, θ2, Φ). Angle Φ is the dihedral angle between the two dipoles, which corresponds to the rotation. It is this angle which accounts for the greatest change in potential energy of the electrostatic interaction. Attractive dipole-dipole interactions occur when oppositely charged ends of the dipole moments of two molecules are in closer proximity than the likely charged ends. When the opposite and like charged ends of the two dipoles are equidistant, there is a net-zero electrostatic interaction. This net-zero interaction occurs when the dihedral angle Φ is at 90°, i.e. the two dipoles are perpendicular to each other.



Derivation
The potential energy of a dipole-dipole interaction can be derived by applying a variation of Coulomb's law for point charges. By treating the two dipoles as a series of point charges, the interaction can be described by a sum of all charge-charge interactions, where the distance between charges is described using vector geometry.
 * $$\mathcal{V}(r) = \frac{q_{d1}q_{d2}}{4\pi\epsilon_0}\frac{1}{\left\vert AC\right\vert} - \frac{q_{d1}q_{d2}}{4\pi\epsilon_0}\frac{1}{\left\vert AD\right\vert} - \frac{q_{d1}q_{d2}}{4\pi\epsilon_0}\frac{1}{\left\vert BC\right\vert} + \frac{q_{d1}q_{d2}}{4\pi\epsilon_0}\frac{1}{\left\vert BD\right\vert} $$

Similarly to the derivation of a charge-dipole interaction, the derivation for dipole-dipole interactions begins by defining the distance between the centre of the two dipoles as r, and the length of each dipole as l. Using a method similar to the derivation of a charge-dipole interaction, trigonometry can be used to determine the distance of each vector corresponding to an interaction: $$\left\vert AC\right\vert$$, $$\left\vert AD\right\vert$$, $$\left\vert BC\right\vert$$, and $$\left\vert BD \right\vert$$.

This equation can be simplified so that the potential energy can be determined as a function of four variables:
 * $$\mathcal{V}(r, \theta_1, \theta_2, \phi) = -\frac{2\mu_1\mu_2}{4\pi\epsilon_0r^3} \left (\cos \theta_1 \cos \theta_2 - \frac{1}{2}\sin \theta_1 \sin\theta_2 \cos\phi \right) $$

Where μ1 and μ2 are the magnitude of the two dipole moments, r is the distance between the two dipoles and θ1, θ2, and Φ describe the orientation of the two dipoles. This potential energy decays at a rate of $$r^3$$.

The ideal attractive potential occurs when the two dipoles are aligned such that Φ = θ1 = θ2 = 0. The interaction potential energy can be rotationally averaged to provide the interaction energy thermally-averaged over all dipole orientations.