Calculation of orientation distribution functions in hard ellipsoid and Gay- Berne liquid crystals from order parameters

The most fundamental characteristic of liquid crystals، at least from a microscopic point of view، is the presence of long-range orientational order while the positional order is limited or absent. We introduce distribution functions which can help us in calculating order parameters and other relevant averages. The singlet and pair distributions are discussed. Equations for the relevant thermodynamic observables in terms of these distributions are given. Also، we introduce order parameters by expanding the singlet distribution in a complete basis set. The effect of molecular symmetry on the orientational order parameters is examined. Using the known order parameters، the orientation distribution function of hard ellipsoid and the Gay- Berne liquid crystals are calculated. The sixth rank order parameters of these liquid crystals are compared.