Crystal field theory potential energy term

LFT originated as a purely electrostatic model - crystal-field theory (CFT) [18], in which d-electronic multiplets of transition metals are perturbed by ligands as point charges or point dipoles. The CF operator (Equation 1) acts within the space of Slater determinants (SD) composed of purely d-spin-orbitals in which two-electron energies are taken into account with the Coulomb operator and one-electron energies with a crystal-field potential (vcp), the first and second terms in Equation 1, respectively. 

In this chapter we consider the very simplest approach to the molecular theory of liquid crystals. We shall approach the theory phenomenologically, treating the problem of the existence of the nematic phase as an order-disorder phenomenon. Using the observed symmetry of the nematic phase we shall identify an order parameter and then attempt to find an expression for the orientational potential energy of a molecule in the nematic liquid in terms of this order parameter. Such an expression is easily found in the mean field approximation. Once this is accomplished, expressions for the orientational molecular distribution function are derived and the thermodynamic functions simply calculated. The character of the transformation from nematic liquid crystal to isotropic fluid is then revealed by the theory, and the nature of the fluctuations near the transition temperature can be explored. 

The van der Waals approximation discussed in Section LA applies the mean field approximation in the solid phase in the same way as in the fluid phase. Baus and co-workers [150,151] have recently presented an alternative formulation in which the localization of the molecules in the solid phase is taken into account. They have applied this to the understanding of trends in the phase diagrams of systems of hard spheres with attractive tails as the range of attractions is changed. For the mean field term in the solid phase they use the static lattice energy for the given interaction potential and crystal lattice. A similar approach was used earlier to incorporate quad-rupole-quadrupole interactions into a van der Waals theory calculation of the phase diagram of carbon dioxide [152]. 

The first rheological theory for Uquid crystals was developed by LesUe and Ericksen, building on Ericksen s earUer transversely isotropic fluid. The theory is formulated in terms of a director field n, and it is similar to the fiber theory in the preceding section, except that it includes a contribution to the free energy from an interactive potential that causes the molecules to align at rest. The usual form of the free energy F resulting from distortions of the director field is... 

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