Intermolecular pair potential

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

${\mathrm{\Phi }}_{12}={\mathrm{\Phi }}_{12}(r,{\theta }_1,{\varphi }_1,{\theta }_2,{\varphi }_2)$

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

${\mathrm{\Phi }}_{12}=4\pi {\sum }_{L_1{L}_{2}m}{L}_1{L}_{2}m(r)Y_{L_1}^m({\theta }_1,{\varphi }_1)Y_{L_2}^m*({\theta }_2,{\varphi }_2)$,

where $Y_{Lm}(\theta ,\varphi )$ are the spherical harmonics.

References

1. J. A. Pople "The Statistical Mechanics of Assemblies of Axially Symmetric Molecules. I. General Theory", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 221 pp. 498-507 (1954)