Degenerate vibrational modes

If the states are degenerate rather than of different symmetry, the model Hamiltonian becomes the Jahn-Teller model Hamiltonian. For example, in many point groups D and so a doubly degenerate electronic state can interact with a doubly degenerate vibrational mode. In this, the x e Jahn-Teller effect the first-order Hamiltonian is then [65]... 

As we shall discuss in Chapters 4-6, this neglect is not quite reasonable if there are degenerate vibrational modes. 

Fig. 6JI Angular momentum arising from excitation of the doubly degenerate vibrational modes of a linear molecule. The plusses mark the equilibrium positions. Note how the motion resembles the rotation of a slightly bent molecule.

The formula (6.56), which includes anharmonicity corrections, is applicable only to molecules with no degenerate vibrational modes. Degenerate... 

Spherical tops and nonlinear symmetric tops also have degenerate vibrational modes. For a linear molecule, there is no rotational angular momentum about the symmetry axis, only vibrational angular... 

Asymmetric tops have no degenerate vibrational modes (this will be shown in Chapter 9) hence we need not worry about vibrational angular momentum for them. 

The presence of degenerate vibrational modes affects the rotational energies. In Section 4.11, we saw the effect of electronic angular momentum on the rotational energies of a diatomic molecule. In this section, we shall assume that there is no electronic angular momentum, only nuclear vibrational and rotational angular momentum. If both electronic and... 

As noted earlier, point groups with no threefold or higher proper or improper axis have only one-dimensional representations hence a necessary condition for a molecule to have degenerate vibrational modes is that it possess a Cn or an S axis with n> 3. Asymmetric tops have no degenerate vibrational modes. 

Spherical harmonics, 29, 38 Spherical polar coordinates, 21 Spherical top definition of, 199 degeneracy for, 210 degenerate vibrational modes in, 276 energies of, 209-210 microwave spectrum of, 217,225 polarizability, 202 wave functions of, 209, 211... 

The frequency (vp) of the two degenerate vibrational modes parallel to the surface of an adatom confined to an adsorption site is given to a first approximation by the relation... 

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. 

Because there is a close coincidence between certain inversion levels in NH3 and the excited vibrational levels pertaining to the doubly degenerate vibrational modes (Fig. 11), and these levels interact by a Coriolis coupling effect, a special numerical treatment is required in this case (Section 5.4). 

The E X E ]T effect, where a doubly degenerate vibrational mode lifts the degeneracy of a doubly degenerate electronic state, is presumably the most extensively investigated vibronic-coupling problem in molecular and solid-state spectroscopy, see [26-28] for reviews. 

The vibrational displacements are described in terms of dimensionless normal coordinates Qx, Qy of a degenerate vibrational mode of E symmetry. The electrostatic Hamiltonian is expanded at the reference geometry in powers of Qx, up to second order... 

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. 

There are 2 doubly degenerate vibrational modes which interact 2... 

To illustrate the gauge invariant reference section for MAB, let us revisit the linear + quadratic E< e Jahn-Teller effect, which is known to exhibit a nontrivial MAB structure. There, the symmetry induced degeneracy of two electronic states (E) is lifted by their interaction with a doubly degenerate vibrational mode (e). In the vicinity of the degeneracy point at the symmetric nuclear configuration, this may be modeled by the vibronic Hamiltonian [39]... 

The failure of the Bom-Oppenheimer approximation for degenerate vibrational modes which can lift electronic degeneracies can thus account for the poor results obtained in frequency calculations which assume the validity of the Bom-Oppenheimer approximation in the energy hypersurface calculation. A case in point is Grein s comparison (749) of the results on BH4, CH4 and NHj (see also above, as well as (44—48, 50, 52—57)) which shows the discrepancies between the experimental and calculated values of co2(E), W8(Fa) and U4(p2) of the XY4 molecules the agreement with wi(Ai) is far better. 

Example 9-1. Given the wavenumbers and vibrational modes shown below for the BCI3 molecule in the two common isotopes of B, determine the symmetries of each vibrational mode determine whether the mode is IR-active, Raman-active, or both and rationalize the observed isotope effect. For the degenerate vibrational modes, only a single stretching frequency is observed. 

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