Temperature-Dependent Optical Activity Due to Quantized Molecular Rotation and Optical Rotation by Oriented Molecules

Abstract

Classical works on optical rotation have dealt with random systems for which all molecular orientations are equally probable. We have considered here optical rotation by quantum mechanically rotating molecules. The contributions to optical rotation from the individual rotational levels of a symmetric- and a spherical-top molecule have been derived. These may be asymmetric molecules belonging to the C3, D3, or T point group. The contributions are weighed by the Boltzmann distribution over the rotational levels, thus giving rise to temperature dependence. Such dependence will be significant at low temperatures. It is shown that, the same as for classical random systems, the optical rotation by quantized rotors comes from the interference of an electric multipole of a given rank with a magnetic multipole of the same rank. However, because of the dependence on the rotational energy level K, for the rotation of infrared “light” which is close to the frequency of a rotational transition, the optical rotation may arise purely from the quantized rotation of symmetric-top molecules that are not necessarily asymmetric. The difference in behavior between a symmetric-top molecule with rotational angular momentum K and a linear molecule with electronic angular momentum Λ is discussed. In the limit of equally probable orientations and in the dipole approximation, it is shown that the results for quantized rotors reduce to the classical expression of optical rotation. Whereas for rotating molecules optical rotation normally requires asymmetry and interference between one electric and one magnetic multipole of the same rank, for oriented molecules, optical rotation may come purely from molecular arrangement requiring no asymmetry of the individual molecule, and may come from arbitrary interference mechanisms. We have considered the optical rotation by molecules uniformly oriented in solid matrices or in weakly interacting molecular crystals. We have derived the optical rotation from the following interference mechanisms: electric dipole–electric dipole (E1E1), electric dipole–magnetic dipole (E1M1), electric dipole–electric quadrupole (E1E2), magnetic dipole–magnetic dipole (M1M1). The angles of rotation have been expressed in terms of the Euler angles of orientation of the individual molecule with respect to the incident polarized light and in terms of the scattering tensor matrix elements over the molecular states. The conventional rotatory strength proportional to the matrix element of the pseudoscalar, R·M, was shown to come from the orientation-independent term in the E1M1 interference mechanism. The possible contributions, of these various mechanisms and various scattering tensors, to optical rotation by oriented molecules belonging to a few simple groups are illustrated in tabular form.