Non-degenerate vibrations

Let us take as our example the three vibrational coordinates of a non-linear triatomic molecule, illustrated in Fig. 4.3. Each of them describes an in-plane motion, so they are necessarily distributed between aj and 6i, the two irreps of C2v that are symmetric to reflection in the molecular plane. The expression for the potential energy of the vibrating XYX molecule in the harmonic approximation is  

The potential energy ([/) increases as the square of each of the the three symmetry coordinates, and also includes a cross-term between the two totally symmetric coordinates. The inclusion of this fourth quadratic term allows for the possibility that it may be easier - or more difficult - to simultaneously stretch the bonds and open the angle between them than to stretch them while closing the bond-angle. In order to get rid of the unwanted cross-term, the two coordinates are combined in such a way that the center of mass remains fixed the arrows showing that Y moves to —x as two X-atoms move to -fx (and vice versa) in the two i vibrations is an attempt to depict this. When the symmetry coordinates are redefined so as to include the relative masses of the atoms, they become normal coordinates and the potential energy of the non-linear XYX molecule becomes  

Qi and Q2 both transform as ai but do not mix, provided that the nuclear displacements are small enough that the harmonic approximation to the potential energy is adequate. The harmonic force constants Ai i = 1,2,3), are all positive, because the potential energy increases as Qi departs from its equilibrium value.Also, barring an accidental degeneracy they are all different, each 

Ai — d UIdQ], which is positive with respect to all of the normal coordinates of a stable molecule at its geometry of minimum potential energy. 

The conclusions just outlined can be summarized for the general case as follows  

Each set of this type will contribute three degrees of freedom to each symmetry species. If there are m sets they will contribute 3m degrees of freedom to each symmetry species, as indicated in Table 6.5. 

If the motions of nuclei of this type of set are to be of species then, in order to be symmetric to all operations, they can move only in the xz-plane and have two degrees of 

Degrees of freedom Degrees of freedom Number of normal 

The motions of nuelei in sets of this type are analogous to those diseussed in item (2) and their assignment to symmetry speeies is analogous. The number of sets of this type is designated and assignment of the motions to symmetry speeies is given in Table 6.5. 

However, we know that a non-linear moleeule has three rotational and three translational degrees of freedom, all of whieh ean be assigned to symmetry speeies (Seetion 4.3.1). These are indieated in Table 6.5 and subtraeted from the total number of degrees of freedom to give the total number of vibrational degrees of freedom. 

Degrees of freedom3 Degrees of freedomb Number of normal 

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In a molecule belonging to a degenerate point group, for example C, , the non-degenerate vibrations of the various sets of equivalent nuclei can be treated as in Section 6.2.2.1. 

So that there are three distinct (having different A s or y s) non-degenerate vibrational normal modes with symmetry species TAl (two) and... 

For an unsymmetrical linear molecule there are N—1 non-degenerate vibrational normal coordinates belonging to the symmetry species 2+ (see Table 7-8.1) and they are of the longitudinal type and N—2 pairs of vibrational normal coordinates (doubly degenerate), each pair belonging to the II symmetry species and they are of the transverse type. 

This term describes the rotational Zeeman effect, that is, the coupling between the external field and the magnetic moment of the rotating nuclei. We note that there is no corresponding vibrational contribution since R a k is zero. The physical reason for this lack is that it is not possible to generate vibrational angular momentum in a diatomic molecule because it possesses only one, non-degenerate, vibrational mode. 

The coupling of a doubly degenerate electronic state with a single non-degenerate vibrational mode is the simplest possible example of the Jahn-Teller effect, for which the E (gi bi Hamiltonian is. 

The rotational sub structure of electronic and rovibrational spectra of a diatomic molecule is mainly characterized by only one moment of inertia or one rotational constant in each of the two participating states. Only one (non-degenerate) vibrational degree of freedom is present. 

INTRAMOLECULAR ROTATION-VIBRATION CORRELATIONS IN DILUTED SOLUTIONS. NON-DEGENERATE VIBRATIONS. 

The first theory which has been elaborated to explain the effect of centrifugal forces on non-degenerate vibrations is the moment... 

Cederbaum LS, Domcke W, Koppel H (1978) Jahn-Teller effect induced by non-degenerate vibrational modes in cumulenes. Chem Phys 33 319... 

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