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
\[\Phi(12)=\sum_{l_1 l_2 m n_1 n_2} [(2l_1 +1)(2l_2 +1)]^{1/2} \Phi_{l_1 l_2 m}^{n_1 n_2}(r_{12}) Y_{mn_1}^{l_1}(\omega_1) * Y_{\overline{m}n_2}^{l_2}(\omega_2) *\]
where the orientations \(\omega=(\phi,\theta,\chi)\), 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
\[\Phi_{l_1 l_2 m}^{n_1 n_2}(r_{12})= \frac{[(2l_1 +1)(2l_2 +1)]^{1/2}}{64 \pi^4} \int \Phi(12) Y_{mn_1}^{l_1}(\omega_1) Y_{\overline{m}n_2}^{l_2}(\omega_2) ~{\rm d}\omega_1 {\rm d} \omega_2\]
Note that by setting \(n_1 = n_2= 0\), one has the coefficients \(\Phi_{l_1 l_2 m}^{00}(r_{12})\) for linear molecules.
[edit] References
- ↑ 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)
- Related reading
