# Intermolecular pair potential

The intermolecular pair potential is a widely used approximation. Real intermolecular interactions consist of two-body interactions, three-body interactions, four-body interactions etc. However, the calculation of even three-body interactions is computationally time consuming, and the calculation of only two-body interactions is frequent. Such "effective" pair potentials often include the higher order interactions implicitly. Naturally the interaction potential between atoms or molecules remains unchanged regardless of where one is in the phase diagram, be it the low temperature solid, or a high temperature gas. However, when one only uses two-body interactions such 'transferability' is lost, and one may well need to modify the the potential or the parameters of the potential if one is studying a hot gas or a cooler high density liquid.

## Axially symmetric molecules

In general, the intermolecular pair potential for axially symmetric molecules, ${\displaystyle \Phi _{12}}$, is a function of five coordinates:

${\displaystyle \left.\Phi _{12}\right.=\Phi _{12}(r,\theta _{1},\phi _{1},\theta _{2},\phi _{2})}$

The angles ${\displaystyle \theta _{i}}$ and ${\displaystyle \phi _{i}}$ can be considered to be polar angles, with the intermolecular vector, ${\displaystyle r}$, as the common polar axis. Since the molecules are axially symmetric, the angles ${\displaystyle \psi _{i}}$ do not influence the value of ${\displaystyle \Phi _{12}}$. A very powerful expansion of this pair potential is due to Pople (Ref. 1 Eq. 2.1):

${\displaystyle \left.\Phi _{12}\right.=4\pi \sum _{L_{1}L_{2}m}L_{1}L_{2}m(r)Y_{L_{1}}^{m}(\theta _{1},\phi _{1})Y_{L_{2}}^{m}*(\theta _{2},\phi _{2})}$,

where ${\displaystyle Y_{L}^{m}(\theta ,\phi )}$ are the spherical harmonics.