Carlo di Lauro

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, Ci, C1 (no symmetry), and C2, Cs, and C2h if the binary rotation axis is oriented along z and σh = σxy. 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 ΔM = ±1.

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

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 relativ...

Spin-allowed and forbidden rotational intensities in doublet-doublet transitions in asymmetric top molecules

Abstract Spin-orbit coupling allows anomalous transitions in spectra involving states of the same degenerate multiplicity, due to the relaxation of the spin selection rule ΔΣ = 0. This effect is discussed for doublet-doublet transitions in asymmetric top molecules. All the nonvanishing and independent transition moments, including those responsible for the anomalous transitions, are determined by symmetry and time reversal arguments. Rotational intensities are given for transitions between case ( b ) near-symmetric top eigenstates.

Coupling of vibration, rotation, and electron spin in multiplet states

Abstract The effective Hamiltonian of Van Vleck ( 1 ) for energy calculations of rotating molecules in multiplet states, is extended to include the effect of vibrational angular momentum, within a given nondegenerate electronic state. H C and H S operators are defined, to account for vibration-rotation (Coriolis) and spin-vibration interactions, respectively. Such interactions are discussed for degenerate E vibrational states of symmetric tops, as well as for the occurrence of vibrational angular momentum involving different vibrational states, with accidentally low energy separation. Following the classification of Hunds case (b) for the spin-rotation wave function (coupled representation), we find that H C operators do not mix different F components, whereas H S operators give also matrix elements between F components differing by one unit in the quantum number N , for given J . Selection rules for single--triplet transitions in symmetric tops, using Hougens G quantum numbers, are briefly outlined.

Spin-vibronic transition moments and rotational intensities in doublet-quartet transitions in asymmetric top molecules☆

Abstract Symmetry and time reversal arguments allow the rigorous determination of the nonvanishing and independent spin-vibronic transition moments for doublet-quartet transitions in asymmetric top molecules. Less strict additional relations, leading to a smaller number of independent parameters, are derived by considering an insight of the intensity borrowing mechanism in the case of weak spin-orbit interactions. In this case it is shown that there are either two or three independent transition moments in orthorhombic molecules, depending on the symmetry species of the vibronic wavefunctions involved in the transition. Case ( b ) selection rules are derived, and it is shown that rotational intensities in orthorhombic molecules can be obtained in all cases by squaring certain sums of either all real or all imaginary terms. A diagrammatical method is devised to identify the form and polarization of the nonvanishing transition moments as well as the symmetry species of the products of those vibronic pairs whose spin-orbit mixing contributes to a given transition moment.

High resolution difference bands of ethane C2H6 from torsionally excited lower states: rotation-torsion structure of the ν 2, ν 11 and ν 4 + ν 11 vibrational states

A high resolution Fourier transform infrared spectrum of C2H6, measured at a pressure of 173.3 Pa and an optical path of 153.2 m, was analysed between 1050 and 1295 cm−1. Extensive absorption due to the difference bands ν 11–ν 4, and several rotation–torsion lines of the difference band ν 2–ν 4, in the region of the x, y-Coriolis resonance of ν 2 and ν 11, were observed. This allowed a detailed rotation–torsion analysis of the upper states ν 11 and ν 2. The anomalous torsional structure, found in the non-degenerate vibrational state ν 2, can be explained as the effect of an Hamiltonian term accounting for a strong dependence of the torsional barrier height on the normal vibrational coordinate q 2. The value of the barrier height derivative is estimated to be 127 ± 10 cm−1. Also detected and assigned were ‘hot’ difference transitions belonging to the (ν 4 + ν 11)–2ν 4 band, yielding information on the upper state ν 4 + ν 11. It is believed that transitions from 3ν 4 to 2ν 4 + ν 11 are also detectable in the investigated region.

On the Physical Reasons for the Extension of Symmetry Groups in Molecular Spectroscopy

Several situations of general interest, in which the symmetry groups usually applied to spectroscopy problems need to be extended, are reviewed. It is emphasized that any symmetry group of geometrical operations to be used in Molecular Spectroscopy should be extended for completeness by considering the time reversal operator, as far as the Hamiltonian is invariant with respect to the inversion of the direction of motion. This can explain the degeneracy of pairs of vibrational and rotational states spanning the so-called separably degenerate irreducible representations, in symmetric tops of low symmetry, and Kramers degeneracy in odd electron molecules in the absence of magnetic fields. An extension with account of time reversal is also useful to determine relative phase conventions on vibration-rotation wavefunctions, which render all vibration-rotation matrix elements real. An extension of a molecular symmetry group may be required for molecules which can attain different geometries by large amplitude periodical motions, if such motions are hindered and are not completely free. Special cases involving the internal rotation are discussed in detail. It is observed that the symmetry classification of vibrational modes involving displacements normal to the internal rotation axis is not univocal, but can be done in several ways, which actually correspond to different conventions on the separation of vibration and internal rotation in the adopted basis functions. The symmetry species of the separate vibrational and torsional factors of these functions depend on the adopted convention.