Fully anisotropic rigid molecules

The fivefold dependence of the pair functions, $\Phi (12)=\Phi (r_{12},\theta _{1},\theta _{2},\phi _{12},\chi _{1},\chi _{2})$, for liquids of rigid, fully anisotropic molecules makes these equations excessively complex for numerical work (see Ref. 1). The first and essential ingredient for their reduction is a spherical harmonic expansion of the correlation functions,

$\Phi (12)=\sum _{l_{1}l_{2}mn_{1}n_{2}}[(2l_{1}+1)(2l_{2}+1)]^{1/2}\Phi _{l_{1}l_{2}m}^{n_{1}n_{2}}(r_{12})Y_{mn_{1}}^{l_{1}}(\omega _{1})*Y_{{\overline {m}}n_{2}}^{l_{2}}(\omega _{2})*$

where the orientations $\omega =(\phi ,\theta ,\chi )$, the Euler angles with respect to the axial line ${\mathbf {r} }_{12}$ between molecular centers, $Y_{mn}^{l}(\omega )$ is a generalized spherical harmonic and ${\overline {m}}=-m$. Inversion of this expression provides the coefficients

$\Phi _{l_{1}l_{2}m}^{n_{1}n_{2}}(r_{12})={\frac {[(2l_{1}+1)(2l_{2}+1)]^{1/2}}{64\pi ^{4}}}\int \Phi (12)Y_{mn_{1}}^{l_{1}}(\omega _{1})Y_{{\overline {m}}n_{2}}^{l_{2}}(\omega _{2})~{\rm {d}}\omega _{1}{\rm {d}}\omega _{2}$

Note that by setting $n_{1}=n_{2}=0$, one has the coefficients $\Phi _{l_{1}l_{2}m}^{00}(r_{12})$ for linear molecules.

References

1. F. Lado, E. Lomba and M. Lombardero "Integral equation algorithm for fluids of fully anisotropic molecules", Journal of Chemical Physics 103 pp. 481-484 (1995)