Higher asymmetry in optical rotation

Abstract

This study shows that optical rotation could arise from higher-order interference effects of two-photon processes in quantum electrodynamics. In the lowest-order limit, applicable to electric- or magnetic-dipole allowed transitions in small molecules, the conventional expression of angle of rotation proportional to Im(0 | R | b) · (b | M | 0) is attributable to the interference of one photon being in an electric-dipole (R) mode which has odd parity and the other photon being in a magnetic-dipole (M) mode which has even parity. The asymmetry stems from the opposite parities of the transition operators R, M, both transforming as tensors of the first rank. For application to larger molecules and forbidden transitions, we dispense with the usual refractive index and medium polarization formalism and derive, using quantum electrodynamics, the higher-asymmetry contribution to optical activity due to the interference of higher electric and magnetic multipoles of any order. It is shown that the perpendicularity of the polarization directions of the incident and scattered photon in the forward direction requires that nonvanishing interference be between an electric-multipole and a magnetic-multipole transition. Furthermore, random molecular arrangement requires that these multipoles transform as tensors of the same rank. The asymmetry then stems from the opposite parities of the electric and magnetic multipoles of the same (L) rank, being (-)L and (-)L-1, respectively. The derivation is given in detail showing the properties of the radiation field of polarized photons and the transformation properties of the multipole operators. The multipole tensor expansion of the radiative interaction in emission is given in the Appendix, in anticipation of its future use in multiphoton processes. The angle of rotation is expressed in terms of the matrix element over the scalar contraction of the electric and magnetic tensor operators. Such contraction is shown to transform as a pseudoscalar regardless of L. In the lowest limit of L=1, the result agrees with conventional work and has the correct dimension of radians per centimeter. Specific examples of optical rotation due to the interference of electric quadrupole (even)-magnetic quadrupole (odd) and the interference of electric octopole (odd)-magnetic octopole (even) are given. The symmetry of these quadrupole and octopole operators in the C2ν and C3ν point group is derived. As asymmetric potentials inducing optical activity the pseudoscalar representations of functions in these point groups are also derived up to spherical harmonics of L=6. The relative magnitudes of these higher-order interference effects and the physical situations in which these effects may be useful are discussed.