Intermolecular pair potential

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \Phi_{12} \right. = \Phi_{12}(r, \theta_1, \phi_1, \theta_2, \phi_2) }

The angles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_i} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_i} can be considered to be polar angles, with the intermolecular vector, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} , as the common polar axis. Since the molecules are axially symmetric, the angles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_i} do not influence the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12} } . A very powerful expansion of this pair potential is due to Pople (Ref. 1 Eq. 2.1):

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

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{L m}(\theta, \phi)} 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)