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,

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)=\sum_{l_1 l_2 m n_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

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}^{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)