Doubly degenerate vibrations

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]... 

Figure 5. Normal modes for vibration of tetrahedral [Cr04] (chromate). There are four distinct vibrational frequencies, including one doubly-degenerate vibration (E symmetry) and two triply-degenerate vibrations (F2 symmetry), for a total of nine vibrational modes. Arrows show the characteristic motions of each atom during vibration, and the length of each arrow is proportional to the magnitude of atomic motion. Only F2 modes involve motion of the central chromium atom, and as a result their vibrational frequencies are affected by Cr-isotope substitution. The normal modes shown here were calculated with an ab initio quantum mechanical model, using hybrid Hartree-Fock/Density Functional Theory (B3LYP) and the 6-31G(d) basis set—other ab initio and empirical force-field models give very similar results.

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.

Quantum mechanically, there is also the possibility of nuclear vibrational angular momentum for these degenerate modes. We denote the normal coordinates for the doubly degenerate vibrations by Qx and Qy. From Fig. 6.2, we have... 

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. 

For the linear XYZ molecule, with two non-degenerate and one doubly degenerate vibrations, one Bg value and three a values must be determined ... 

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

Selection mles for overtones of doubly degenerate vibrations E species) are determined by... 

Thus, the first overtone of the doubly degenerate vibration is IR- and Raman-active. The characters of overtones for triply degenerate vibrations are given by... 

The selection rule changes as shown in Table 1-11. In Civ and the i, vibration, which is forbidden in the free ion, becomes infrared active and each of the doubly degenerate vibrations, and splits into two bands. Although the number of infrared-active fundamentals is the same for Civ and C, the splitting of the degenerate vibrations is larger in the bidentate than in the... 

On account of the larger mass of chlorine and the weaker bond with carbon (as compared with hydrogen), substitution by chlorine causes these vibrations to become increasingly C—C stretching. In perchlorobenzene, the doubly degenerate vibration at 1340 cm is overwhelmingly stretching (Fig. 34). 

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]... 

PH3, PD3, PT3. The six normal modes of the PH3 (PD3) molecule form two totally symmetrical vibrations, Vi(Ai) and V2(Ai), and two doubly degenerate vibrations, V3(E) and V4(E). Force field calculations (see p. 172) showed that approximately the higher frequency mode in each of the two symmetry species is a bond stretch and the lower an angle deformation. 

In parallel with the Vi activation, also the splitting of the V3 vibration occurs upon coordination. In the free CCb " ion, the V3 vibration is doubly degenerate (Fig. 9, bottom-left quadrant). Doubly degenerate vibrations occur only in molecules possessing an axis higher than twofold, which is the case of the Dsh symmetry, having a three-fold rotational axis (see... 

Fig. 9, upper-left quadrant C3 - three-fold axis) (Nakamoto, 1997). The lowering of the symmetiy of the carbonate ion from Dsh to either C2 or Cs, which means loss of the equivalence of the three C-O bonds in the CQ32- and, therefore, loss of the three-fold rotational axis, leads to the sep>aration (sphtting) of the doubly degenerate vibrations (Fig. 9, bottom-right quadrant). 

This representation can be reduced using eqn [73] to give 3Ai + 3E, Thus, CICH3 (3N—6 = 9) has three totally symmetric vibrations and three doubly degenerate vibrations. 

The presumably most widely known example of vibronic coupling is the Jahn-Teller effect of a doubly degenerate electronic state, that is, the coupling of the two components of the degenerate state by a doubly degenerate vibrational mode. The symmetry selection rule for this type of vibronic coupling, the so-called E x E Jahn-Teller effect, is °... 

Let us consider a system with a doubly degenerate electronic state and a threefold principal rotation axis. Then there are always doubly degenerate vibrational modes that are (linearly) JT-active, that is, the derivatives dVaa I dQi do not vanish for their (Cartesian) displacement components Qx and Qy. By elementary symmetry considerations the corresponding 2x2 JT matrix Hamiltonian in first order is found to... 

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