Fully anisotropic rigid molecules

The fivefold dependence of the pair functions, 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)=\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 . 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 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 \omega=(\phi,\theta,\chi)} , the Euler angles with respect to the axial line 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 {\mathbf r}_{12}} between molecular centers, 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_{mn}^l (\omega)} is a generalized spherical harmonic 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 \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 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 n_1 = n_2= 0} , one has the coefficients 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_{l_1 l_2 m}^{00}(r_{12})} for linear molecules.

References

Related reading

• R. Ishizuka and N. Yoshida "Application of efficient algorithm for solving six-dimensional molecular Ornstein-Zernike equation", Journal of Chemical Physics 136 114106 (2012)