Doubly 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]... 

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.

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. 

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 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... 

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.

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... 

Benzene thus has 10 nondegenerate and 20 doubly degenerate normal modes. The convention8 is to number the vibrational frequencies according to the order their symmetry species are listed in the character table modes of the same symmetry species are numbered in order of decreasing frequency. Thus the a2u frequency is called p4, the frequency of the two degenerate elg modes is called vn, the lower of the two e2u frequencies is called p20. 

In a smaller molecule (HCP), these diagnostically important changes in vibrational resonance structure are manifest in several ways (i) the onset of rapid changes in molecular constants, especially B values and second-order vibrational fine-structure parameters associated with a doubly degenerate bending mode (ii) the abrupt onset of anharmonic and Coriolis spectroscopic perturbations and (iii) the breakup of a persistent polyad structure 15]. 

Recent works by Herman et al. and Field et al. have focused on molecules of the family of acetylene, in particular C2HD [112] and C2H2 (see Refs. 122 and 123 and Field et al., Intramolecular Dynamics in the Frequency Domain, this volume). These linear molecules have three stretching modes, 1, 2, and 3, and two doubly degenerate bending modes, trans 4 and cis 5. Isotopic effects appear particularly striking in the vibrational dynamics, as shown in the comparative study of the dynamics of the above isotopomers. 

For a doubly degenerate normal mode, both components must be used together as the basis of a two-dimensional irreducible representation. For example, the operations C2 and ctv on the two normal vibrations that constitute the i>6 mode lead to the character (sum of the diagonal elements of the corresponding 2x2 matrix) of -2 and 0, respectively, as illustrated below. Working through the remaining symmetry operations, the symmetry species of can be identified as Eu. 

For the nondegenerate C state, only totally symmetric vibrations (vi and vi) can possess non-vanishing /r s. For the doubly degenerate states X and B in addition the four doubly-degenerate E2g modes (ve - vg) can contribute, recovering the well-known result that these modes are JT active in Bz+. 

Mode 1 denotes the totally symmetric C-C stretching mode of Bz+ while the modes 6a-8a, 6b-8b derive from the doubly degenerate E2g modes 6-8 of Bz+ (they are to be considered as components of these doubly degenerate modes for the parent cation, but distinct modes with similar displacement patterns for the fluoro derivatives). The similarity of the vibrational frequencies throughout the series is noted (Table 3a). The same holds for the coupling constants for the X, A states (corresponding to the Eig state of Bz+, see Table 3b) and also for the coupling constants of the E2g - derived states (see Table 3c). 

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. 

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