Editing Fully anisotropic rigid molecules

The fivefold dependence of the pair functions, <math>\Phi(12)=\Phi(r_{12},\theta_1, \theta_2, \phi_{12}, \chi_1, \chi_2)</math>, 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, 

:<math>\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) *</math>

where the orientations <math>\omega=(\phi,\theta,\chi)</math>, the [[Euler angles]] with respect
to the axial line <math>{\mathbf r}_{12}</math> between molecular centers, <math>Y_{mn}^l (\omega)</math>
is a [[Spherical harmonics | generalized spherical harmonic]] and <math>\overline{m}=-m</math>.
Inversion of this expression provides the coefficients

:<math>\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</math>

Note that by setting <math>n_1 = n_2= 0</math>, one has the coefficients
<math>\Phi_{l_1 l_2 m}^{00}(r_{12})</math> for linear molecules.
==References==
#[http://dx.doi.org/10.1063/1.469615 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)]
[[category: integral equations]]