Quantum theory of chiral interactions in cholesteric liquid crystals

The effective chiral interaction between molecules arising from long-range quantum interactions between fluctuating charge moments is analyzed in terms of a simple model of chiral molecules. This model is based on the approximations that (a) the dominant excited states of a molecule form a band whose width is small compared to the average energy of excitation above the ground state and (b) biaxial orientational correlation between adjacent molecules can be neglected. Previous treatments of quantum chiral interactions have been based on a multipole expansion of the effective interaction energy within second-order perturbation theory. We consider a system consisting of elongated molecules and, although we invoke the expansion in terms of coordinates transverse to the long axis of constituent molecules, we treat the longitudinal coordinate exactly. Such an approximation is plausible for molecules in real liquid crystals. The macroscopic cholesteric wave vector $\mathbf{Q}$ $(Q=2\mathrm{}\pi /P,$ where P is the pitch) is obtained via ${Q=h/K}_2,$ where $K_2$ is the Frank elastic constant for twist and h is the torque field which we calculate from the effective chiral interaction ${\kappa }_{\mathrm{IJ}}{\mathbf{a}}_I\times {\mathbf{a}}_J\cdot {\mathbf{R}}_{\mathrm{IJ}},$ where the unit vector ${\mathbf{a}}_I$ specifies the orientation of molecule I and ${\mathbf{R}}_{\mathrm{IJ}}$ is the displacement of molecule I relative to molecule J. We identify two distinct physical limits depending on whether one or both of the interacting molecules are excited in the virtual state. When both molecules are excited, we regain the $R_{\mathrm{IJ}}^{-8}$ dependence of ${\kappa }_{\mathrm{IJ}}$ on intermolecular separation found previously by Van der Meer et al. [J. Chem. Phys. 65, 3935 (1976)]. The two-molecule, unlike the one-molecule term, can be interpreted in terms of a superposition of pairwise interactions between individual atoms (or local chiral centers) on the two molecules. Contributions to ${\kappa }_{\mathrm{IJ}}$ when one molecule is excited in the virtual state are of order $R_{\mathrm{IJ}}^{-7}$ for helical molecules which are assumed not to have a global dipole moment, but whose atoms possess a dipole moment. It is shown that for a helical molecule Q can have either the same or the opposite sign as the chiral pitch of an individual molecule, depending on the details of the anisotropy of the atomic polarizability. The one-molecule mechanism can become important when the local atomic dipoles become sizable, although biaxial correlations (ignored here) should then be taken into account. Our results suggest how the architecture of molecular dipole moments might be adjusted to significantly influence the macroscopic pitch.