The effect of phase conventions on vibration-rotation matrix elements

Abstract

The transformation properties of vibrational and rotational basis operators and functions under symmetry operations and time reversal are investigated, with emphasis on their dependence on normal coordinate orientation and phase conventions. The effect of phase conventions on the values of the off-diagonal vibration-rotation matrix elements is examined, and it is shown that if the molecular symmetry allows for an operation denoted R′ , consisting of either a reflection through a plane containing the angular momentum quantization z-axis or a rotation about a binary axis normal to z, all vibration-rotation matrix elements of axially symmetric and asymmetric rotor molecules can be made real by appropriate vibrational and rotational phase conventions, which are discussed and recommended. Therefore vibration-rotation matrix elements of these molecules can be made all real for all molecular symmetry groups, with the exception of the groups containing separably degenerate E-species, C i , C 1 (no symmetry), and C 2, C s , and C 2 h if the binary rotation axis is oriented along z and σ h = σ xy. It is shown that, with the same conventions which render all vibration-rotation matrix elements real, the matrix elements to be used in the calculation of electric dipole vibration-rotation intensities, when the M-degeneracy is not removed, are taken all real if R′ is a reflection plane and all imaginary if R′ is a binary rotation axis. Relative phases of rotational wavefunctions differing by the value of M have to be defined in order to determine the values of matrix elements with Δ M = ±1.