G 3 Molecular fluids

The purpose of this section is to reformulate the RSOZ equations in A -space given in Eqs. (7.41a)-(7.42) in an appropriate form for a molecular fluid in a molecular matrix such that the resulting expressions are still numerically tractable. We specialize to the case of linear molecules, such as spheres with dipole moments or ellipsoidal particles whose orient,ation can be specified by two angles u = (0, p)- The correlation functions fy k,uJi, LJ2) (where f = h or c, 7 = mm, mf, fm, ff , b or c) then depend on seven variables altogether. They describe the orientations of the pair of particles, that of the wavevector, and the wavenumber k = jfcj. 

To handle these variables, we expand each correlation function in an angle-dependent basis set of rotational invariants [258]. Taking advantage of the fact that, in an globally isotropic system, the direction of the wavevector k does not matter, we choose k to be parallel to the -axis of the space-fixed coordinate system. The resulting fc-frame expansion is then defined by [258] 

Inserting the fc-frame expansit)us (G.8) of the correlation functions into the RSOZ Eqs. (7.41a)-(7.42), the angular integral buried in the products /j fs can be easily performed by using the orthogonality of the spherical harmonics [258] [cf. Eq. (F.39)]. Introducing, for compactness, matrices with elements 

We note that the RSOZ equations in Eqs. (G.lla)-(G.lld) are decoupled with respect to the wavenumber k and the angular index x- practice, the matrices and have finite dimensions due to a truncation of the rotationally invariant expansion in Eq. (G.8) at appropriate values of Zj and Z2. The RSOZ Eqs. (G.lla)-(G.lld) can thus be solved by standard matrix inversion techniques. 

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