Degenerate modes of vibration

Upon coordination of an oxyanion to a metal the symmetry of the anion may change,degenerate modes of vibration of the free ion may be resolved... 

Fig. 1. The variation of the potential energy of an adatom moving parallel to the surface. Em is the barrier height measured from the zero point energy of the two degenerate modes of vibration parallel to the surface, i.e. Em = E — hvp, where is the frequency of vibration of a localised adatom parallel to the surface.

The statement of the selection rules given above also applies to degenerate modes of vibration. For example in Section 5.7 we show that the C=0 vibrational modes of the Aa complex [Fe(CO)5] have the irreducible representations ... 

Figure 3.12. CO adsorbed on a surface possesses five modes of vibration Note that the surface is not necessarily symmetric in x and y, and hence the vibrations in the x and y direction do not have to be degenerate.

The analysis of vibration spectra proceeds by the use of normal modes. For instance, the vibration of a nonlinear water molecule has three degrees of freedom, which can be represented as three normal modes. The first mode is a symmetric stretch at 3586 cm , where the O atom moves up and the two H atoms move away from the O atom the second is an asymmetric stretch at 3725 cm where one H atom draws closer to the O atom but the other H atom pulls away and the third is a bending moment at 1595 cm , where the O atom moves down and the two H atoms move up and away diagonally. The linear CO2 molecule has four normal modes of vibration. The first is a symmetric stretch, which is inactive in the infrared, where the two O atoms move away from the central C atom the second is an asymmetric stretch at 2335 cm where both O atoms move right while the C atom moves left and the third and fourth together constitute a doubly degenerate bending motion at 663 cm where both O atoms move forward and the C atom moves backward, or both O atoms move upward and the C atom moves downward. 

With p = (Qg + Ql f being the Jahn-Teller radius, FE the linear vibronic coupling constant, GE the quadratic vibronic coupling constant, KE the force constant for the Eg normal mode of vibration, and Qo, Qe the two degenerate vibrations of eg symmetry. 

The Jahn-Teller theorem was proved by showing that for all symmetry groups except and there was at least one normal mode of vibration which belonged to a non symmetric representation f,- such that the direct product of F/ with the representation Fy of the degenerate electronic state contained the representation Fy. 

Each such vibration (6.32) is called a normal mode of vibration. For each normal mode, the vibrational amplitude Aim of each atomic coordinate is constant, but the amplitudes for different coordinates are, in general, different. The nature of the normal modes depends on the molecular geometry, the nuclear masses, and the values of the force constants ujk. The eigenvalues m of U determine the vibrational frequencies the eigenvectors of U determine the relative amplitudes of the vibrations of the q, s in each normal mode, since Ajm / A im = Ijm/L- For H2° here are 9-6-3 normal modes, and the solution of (6.17) and (6.18) yields the vibrational modes shown in Fig. 6.1. For some molecules, two or more normal modes have the same vibrational frequency (corresponding to two or more equal roots of the secular equation) such modes are called degenerate. For example, a linear triatomic molecule has four normal modes, two of which have the same frequency. See Fig. 6.2. The general classical-mechanical solution (6.30) is an arbitrary superposition of the normal modes. 

Fig. 3.24 Normal modes of vibration of the XeF4 molecule. Note that both modes are doubly degenerate. [Modified from Harris, D C. Bertolucci,...

There are (3n—6) modes of vibration in a polyatomic molecule except that for a linear molecule one mode is doubly degenerate so that effectively the number is (3n-5) n is the number of atoms in the molecule. Symmetry rules may restrict the kinds of transitions which can occur during an electronic transition but often the number of possible transitions is very large. 

The vibrational modes of the molecule also show a high degree of degeneracy so that only three of the 3N — 6 = 174 modes of vibration are non-degenerate, the normal modes transforming as ... 

The problem of a triply degenerate electronic level interacting with a mode of vibration of fivefold degeneracy was actually considered long ago in different systems [2,3]. The electron-phonon coupling will now be described by ... 

The symmetry of the normal mode of vibration that can take the molecule out of the degenerate electronic state will have to be such as to satisfy Eq. (6-7). The direct product of E with itself (see Table 6-11) reduces to A + A 2 + E. The molecule has three normal modes of vibration [(3 x 3) - 6 = 3], and their symmetry species are A + E. A totally symmetric normal mode, A, does not reduce the molecular symmetry (this is the symmetric stretching mode), and thus the only possibility is a vibration of E symmetry. This matches one of the irreducible representations of the direct product E E therefore, this normal mode of vibration is capable of reducing th eZ)3/, symmetry of the H3 molecule. These types of vibrations are called Jahn-Teller active vibrations. 

The Hiickel it MOs of a square planar cyclobutadiene are well known. They are the one below two below one set shown in 81. We have a typical Jahn-Teller situation, i.e., two electrons in a degenerate orbital. (Of course, we need worry about the various states that arise from this occupation, and the Jahn-Teller theorem really applies to only one.67) The Jahn-Teller theorem says that such a situation necessitates a large interaction of vibrational and electronic motion. It states that there must be at least one normal mode of vibration that will break the degeneracy and lower the energy of the system (and, of course, lower its symmetry). It even specifies which vibrations would accomplish this. 

For. the majority of simple molecules the normal modes of vibration may be obtained by a straightforward, if tedious, procedure and the motion of the atoms relative to each other obtained. In certain cases several normal modes possess the same frequency and the system is degenerate this point will be discussed later when we consider specific molecules. 

In the absence of a JT effect, the atoms will be arranged in the high-symmetry Dsh configuration and the electron density sampled will correspond to Ex + Ey for some particular bias. A simulation of the STM image produced by this electron density is shown in Fig. 18, which also shows the doubly-degenerate, in-plane normal modes of vibration of interest... 

Fig. 18 A simulated, constant-current STM image of a hypothetical X3 molecule in its high-symmetry D31, configuration is shown in (a). The (x, y)-axes are arranged so that atom a lies on the y-axis and the centre of mass is at the origin. The corresponding degenerate normal modes of vibration are shown in (b) and (c)...

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