Phase conventions that render all matrix elements of the vibration-rotation Hamiltonian real

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Abstract
Rotational and vibrational operators of a molecule possessing an ℜ symmetry element, consisting of either a binary rotation axis normal to the angular momentum quantization z axis or of a reflection plane containing z, are classified according to the irreducible co-representations of the group (E, ℜ′, Θ, ℜ′Θ), Θ being the time-reversal operator. Using this classification, it is shown that the matrix elements of the vibration-rotation Hamiltonian, in the usual vibration-rotation basis, can be made all real for all classes of molecules possessing the ℜ′ symmetry element, by appropriate rotational and vibrational phase conventions, which are defined and recommended. With the same phase conventions that render the matrix elements of the vibration-rotation Hamiltonian all real, the matrix elements of the electric dipole vibration-rotation transition moment can be made all real if ℜ′ is a reflection plane, and all imaginary if ℜ′ is a binary rotation axis, by appropriate additional conventions about the relative phases of |JkM> basis functions differing by the values of J and M.