Doubly-degenerate excitations

Typically, three to four resonantly enhanced modes are observed in the 250-400-cm 1 region, which presumably derive from M—S stretching modes, of which one is polarized and more intense than the rest. The expected depolarization value (p) for a totally symmetric mode in resonance with a doubly degenerate excited state is 0.125, and the polarized totally symmetric mode for these complexes typically fall in the 0.14-0.17 range. Thus, since only one totally symmetric M—S stretching mode is anticipated for these complexes,... 

Qise A. Consider the HT coupling between two doubly-degenerate excited states, le> and s>, as for example between two distinct states in a molecule of D4h symmetry. Transitions to each state are allowed, from an. <4 xg ground state, with both X mdy polarisation. In addition to the zero-order transition moments x , y , the first-order borrowed moments x, y, must be considered. In order to demonstrate the origin of the antisymmetry of the tensor in this case ( xy — yx) the non-zero off-diagonal elements are written as follows ... 

The first examples are doubly-degenerate excitations. The simplest ones are doubly-degenerate n-jt excitations in diatomic molecules such as N2(Dooh) and CO (Coov)-The doubly-degenerate highest-occupied MOs (HOMOs), as denoted by n, belong to the irreducible representations of jtu in Dooh and it in Coov, while the doubly-degenerate lowest-unoccupied MOs (LUMOs), as denoted by jt, belong to those of Kg in Dooh and k in Coov, respectively. The direct products between Ku and Kg and between k and k can be decomposed as follows ... 

An example that is closely related to organic photochemishy is the x e case [70]. A doubly degenerate E term is the ground or excited state of any polyatomic system that has at least one axis of symmetry of not less than third order. It may be shown [70] that if the quadratic tenn in Eq, (17) is neglected, the potential surface becomes a moat around the degeneracy, sometimes called Mexican hat, The polar coordinates p and <(>, shown in Figure 20, can be used to write an expression for the energy ... 

The 21 23 excitation has an equally large coefficient these are the other halves of the two pairs of doubly degenerate orbitals. 

On the other hand, in the anion radical of fulvalene and the cation radical of heptafulvalene, the energy gaps between the ground and lowest excited state (which is in both cases doubly degenerate in the Hiickel approximation (Fig. 4)) are predicted to be reasonably large (1.4 and 1.7 eV, respectively), so that these radicals would not suffer a symmetry reduction. 

The effect of electric quadrupole interaction for Fe is exemplified in Fig. 4.6. The ground state remains unsplit because of the lack of quadrupole moment for 7 = 1/2. The excited state with 7 = 3/2 splits into two doubly degenerate substates 3/2, 3/2) and 3/2, 1/2) due to the w/ dependence of the quadrupole energies ... 

The ground and first excited electronic energy levels in the H atom are doubly degenerate (i.e., go = gi = 2). The first excited electronic state is 82,258 cm-1 above the ground state. At T = 5000 K, exponential term in Eq. 8.73 is c-23-670, and... 

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.

There are several important effects associated with degenerate vibrational levels. We begin by considering the doubly degenerate v2a and v2b vibrational modes of a linear triatomic molecule. In these modes, the three atoms each vibrate in a plane perpendicular to the molecular axis, the vibrations being either in the xz or the yz plane, where the z axis is the molecular axis (Fig. 6.2). Classically, if both these modes are excited, the vibrations may give rise to vibrational angular momentum about the internuclear axis for example, if v2a and v2b are of equal amplitude and differ in phase by 90°, then the resultant motion of each nucleus is a circle about the molecular axis, as shown in Fig. 6.8 (see Problem 6.18). 

In the actual case of the CO molecule, which provides an excellent example, the three fundamental transitions have frequencies of 667, 1300, and 2350 cm-1. The first overtone of the 667-cm-1 vibration, which is doubly degenerate, has a frequency of 1334 cm 1, which is quite close to that of the 1300-cm"1 fundamental. Now it can be shown that the excited state for the 1300-cmfundamental and one component of the representation generated by the excited state corresponding to the first overtone of the 667-cm"1 vibration do belong to the same representation of the group ),, and hence a Fermi resonance occurs. Thus, in the Raman spectrum of CO two strong bands at 1285 and 1388 cm 1 are observed, instead of just one at —1300 cm-1. 

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