Fully anisotropic rigid molecules
The fivefold dependence of the pair functions, , 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,
![{\displaystyle \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})*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e36354dc3bf8a1ef0278b90cc53063435bc5248)
where the orientations , the Euler angles with respect to the axial line Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathbf {r} }_{12}} between molecular centers, is a generalized spherical harmonic and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\overline {m}}=-m} . Inversion of this expression provides the coefficients
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n_{1}=n_{2}=0} , one has the coefficients Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Phi _{l_{1}l_{2}m}^{00}(r_{12})} for linear molecules.