Fully anisotropic rigid molecules

The fivefold dependence of the pair functions, $\mathrm{\Phi }(12)=\mathrm{\Phi }(r_{12},{\theta }_1,{\theta }_2,{\varphi }_{12},{\chi }_1,{\chi }_2)$, 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,

$\mathrm{\Phi }(12)={\sum }_{l_1{l}_{2}m{n}_1{n}_2}[(2l_1+1)(2l_2+1){]}^{1/2}\underset{l_1{l}_{2}m}{\overset{n_1{n}_2}{\mathrm{\Phi }}}(r_{12})Y_{m{n}_1}^{l_1}({\omega }_1)*Y_{\overline{m}{n}_2}^{l_2}({\omega }_2)*$

where the orientations $\omega =(\varphi ,\theta ,\chi )$, 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

$\underset{l_1{l}_{2}m}{\overset{n_1{n}_2}{\mathrm{\Phi }}}(r_{12})=\frac{[(2l_1+1)(2l_2+1){]}^{1/2}}{64{\pi }^4}\int \mathrm{\Phi }(12)Y_{m{n}_1}^{l_1}({\omega }_1)Y_{\overline{m}{n}_2}^{l_2}({\omega }_2)\mathrm{d}{\omega }_1\mathrm{d}{\omega }_2$

Note that by setting $n_1=n_2=0$, one has the coefficients $\underset{l_1{l}_{2}m}{\overset{00}{\mathrm{\Phi }}}(r_{12})$ for linear molecules.

References

1. F. Lado, E. Lomba and M. Lombardero "Integral equation algorithm for fluids of fully anisotropic molecules", Journal of Chemical Physics 103 pp. 481-484 (1995)