Degenerate states in optical rotation and time-reversal invariance.

Abstract

The consequence of time-reversal symmetry in a general second-order optical process between degenerate initial and final states is elucidated. The principle is applied to the pseudoscalar natural-optical-rotation operator (${R}_{L}$\ensuremath{\cdot}${M}_{L}$), which involves the interference of an electric-multipole (${R}_{L}$) transition and a magnetic-multipole (${M}_{L}$) transition. In addition to deriving point-group selection rules, time-reversal effects on matrix elements between degenerate states of a chiral versus an achiral molecule are compared and contrasted. Other non--time-reversal-degenerate states in chiral versus achiral molecules (${D}_{3}$ versus ${C}_{3v}$ and ${D}_{4}$ versus ${D}_{2d}$) are also compared.The analogy to the Jahn-Teller effect, which also involves (albeit first-order vibronic) matrix elements over degenerate states, is mentioned, especially with regard to the formal ``splitting'' of the degenerate levels by the pseudoscalar optical-rotation operator. A comprehensive search among the point groups for degenerate-state self-products that will give rise to the pseudoscalar function is conducted. Detailed results for the coupling of (degenerate) irreducible representations for important point groups (and their subgroups) are worked out for ${T}_{d}$ (T), O, ${D}_{6d}$ (${D}_{6}$, ${D}_{3}$, ${C}_{6v}$, ${C}_{3v}$, ${C}_{6}$, ${C}_{3}$, ${D}_{2d}$), and ${D}_{4d}$. The couplings to give dipolar (L=1), quadrupolar (L=2), and octopolar (L=3) pseudoscalars are shown explicitly.Attention is called to the differences in the transformation of a magnetic-multipole operator and an electric-multipole operator, even though they both belong to the same irreducible representation. The resulting differences in coupling coefficients for degenerate states are given to help determine nonvanishing optical-rotation matrix elements. A summary and a discussion of the conservation of parity and time symmetry in optical rotation are given.