Fully anisotropic rigid molecules

From SklogWiki

The fivefold dependence of the pair functions, $Φ(12) = Φ(r 12,θ 1,θ 2,φ 12,χ 1,χ 2)$ , 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,

$\Phi(12)=\sum_{l_1 l_2 m n_1 n_2} [(2l_1 +1)(2l_2 +1)]^{1/2} \Phi_{l_1 l_2 m}^{n_1 n_2}(r_{12}) Y_{mn_1}^{l_1}(\omega_1) * Y_{\overline{m}n_2}^{l_2}(\omega_2) *$

where the orientations $ω = (φ,θ,χ)$ , 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

$\Phi_{l_1 l_2 m}^{n_1 n_2}(r_{12})= \frac{[(2l_1 +1)(2l_2 +1)]^{1/2}}{64 \pi^4} \int \Phi(12) Y_{mn_1}^{l_1}(\omega_1) Y_{\overline{m}n_2}^{l_2}(\omega_2) ~{\rm d}\omega_1 {\rm d} \omega_2$