Spherical harmonics, intermolecular

Here, Yx m( j) denotes a spherical harmonic, coj represents the spherical polar angles made by the symmetry axis of molecule i in a frame containing the intermolecular vector as the z axis. The choice of the x and y axes is arbitrary because the product of the functions being averaged depends on the difference of the azimuthal angles for the two molecules which are separated by distance r. At the second rank level the independent correlation coefficients are... 

The orientations of linear molecules, relative to the global frame, can be specified by two Euler angles (oP = 8P, /> the symmetry-adapted functions G icop) that occur in the intermolecular potential [Eq. (15)] reduce to Racah spherical harmonics C (0/>, />). If the molecules possess a center of inversion such as N2 (when we disregard the occurrence of mixed isotopes l4Nl5N, the natural abundance of l5N being only 0.37%),... 

Maier and SaupeS first pointed out the importance of attractive orientational interactions such as the induced dipolar forces. Later work included both the hard core repulsion and Van der Waals attraction in a mean field approximation. The total effective pseu-dopotiential is found by expansion of the intermolecular interaction in spherical harmonics for cylindrical rods. The anisotropic part therefore includes all interactions with the symmetry of the quadrupolar Legendre polynomial P2(Cos 0). 

As said before, in practice the integral equations for molecular liquids are almost always solved using spherical harmonic expansions. This is because the basic form [8.76] of the OZ relation contains too many variables to be handled efficiently. In addition, harmonic expansions are necessarily truncated after a finite number of terms. The validity of the truncations rests on the rate of convergence of the harmonic series that depends in turn on the degree of anisotropy in the intermolecular potential. 

In this chapter we will consider a more systematic approach to the molecular theory of the nematic state. In deriving the theory we will follow the development of Humphries, James, and Luckhurst. We start with a completely general pairwise intermolecular interaction potential. After expanding in a series of appropriate spherical harmonics we will systematically average the pair-interaction potential to... 

Each element of the r-mer molecules is tied to its equilibrium position on the lattice by a spherically S5munetric harmonic field arising from intermolecular interactions. Let us note by kr the force constant and by. .. ygr the cartesian coordinates of the r elements referred to their lattice points. The intermolecular potential energy is then... 

---