Doubly degenerate deformations

The elastic line, which appears at an energy transfer of zero for zero momentum transfer, is now displaced by 422 cm to higher energies. A similar effect is seen on all the internal transitions. Thus, for example, the doubly degenerate deformation mode of methane, V4, appearing at 1534 cm in the infrared, would be centred at 1956 cm (=1534 + 422) in the INS (if measured at the momentum transfer value of 9 A ). The standard deviation of the Gaussian, F (cm ), is given by [12] ... 

In-plane doubly degenerate deformation (w), cm b. Out-of-plane deformations are usually lower than in-plane ones. Note that the last two are inverted from this order. 

As a first approximation to the influence of structure on the vibrational frequencies, we shall concentrate on the M-H-M tri-atomic array. For a linear M-H-M unit, the normal modes are very simple. There is a Raman-active symmetric M-H-M stretch (Structure 9) that only involves motion of the massive metal and therefore occurs at much lower frequencies than the B-B and M-B stretching modes discussed in a previous section. In an intermediate frequency region, a doubly degenerate M-H deformation mode occurs that involves hydrogen motion (Structure 10), and at high frequencies, an asymmetric ir-active stretching vibration (Structure 11) should be observed. 

Syy, E = exx-Eyy 2i8xy, and e+z = 8xz iEyz. In order to help the derivation of the deformation potentials for WZ structure, we investigate the strain effect on the eigenstates at the F point. At the T point, assuming that e = ew and Exy = Eyz = En = 0, we can analytically solve the 6x6 strain Hamiltonian for valence bands and obtain the three doubly degenerate eigenstates ... 

Whenever the dependence r(T) is reconstructed, we can simulate it and fitting will give us the magnitudes of the potential barrier, To, the tunnelling splitting, rF, the vibrational frequency, vo, the deformation potential, b, and the energy of inevitable strain, mq. For a doubly degenerate states one can use the expressions (38)-(40). 

Asymmetric deformation IR active (530cm ) Doubly degenerate mode... 

Figure 3.7 Vibrational frequencies (cm ) for CH4 CH3 + H versus the reaction path. In descending order the modes at Thc 3.5 A are the doubly degenerate CH stretch, symmetric CH stretch, doubly degenerate CH3 deformation, out of plane CH3 bend, and...

Figure 4 Six out-of-plane vibrational coordinates of a Z)4h metalloporphyrin used in normal structural decompositions to reveal compositions of equilibrium out-of-plane heme deformations. Mulliken S5nnbols are of the I>4h point group and the abbreviations sad, ruf, dom, wav, and pro indicate saddled, ruffled, domed, wave, and propeller distortions, respectively. (Note that wav and waVy derive from a doubly degenerate pair of Eg vibrational modes of the porph5rin.) The dark and light circles indicate atoms on opposite sides of the mean heme plane. Adapted from ref. 11.

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. 

Inorganic carbonate salts have one or two very strong broad bands at 1520-1320 cm involving the out-of-phase CO3 stretch. This has two components which are doubly degenerate near 1450 cm" if the three-fold symmetry is retained around the CO3 ion, but otherwise the component bands separate. There is a medium-weak sharp band at 890-800 cm which is caused by the out-of-plane deformation of the CO3 ion, and a weak sometimes multiple band at 730-670 cm which involves CO3 in-plane deformation (2 components). The in-phase CO3 stretch is normally IR inactive if the three-fold symmetry is retained, but otherwise shows up as one or two weak bands at 1100-1030 cm" . This is a very strong band in the Raman spectra. 

The last species of vibrations that can occur for these two molecules is classified as a type E, This type of vibration is doubly degenerate, that is, two vibrations exist of the exact same energy. The E type vibration is also antisymmetrical with respect to rotational axes that exist in the molecule For the two molecules CH3—CF3 and CH3—CCI3, the vibrations that can be called rocking of the CCI3, CH3, and CF3 groups and C—H, C—F, and C—Cl stretches are E vibrations. In addition, some of the deformations of the C—H, C—F, and C—Cl groups are E type vibrations. 

Figure 1 is also illustrative of a number of theoretical concepts of infrared spectroscopy. For example, the E vibrations in CH3—CF3 are all doubly degenerate, but they split into symmetrical and asymmetrical vibrations as one fluorine atom is replaced by a chlorine atom. This occurs for all E vibrations except the CH3 stretching and deformation frequencies, where the symmetry of the CH3 group does not allow this. 

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