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 . 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

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)