Phased degenerate vibration

Geometric phase effect (GPE) (Continued) adiabatic states, conical intersections invariant operators, 735-737 Jahn-Teller theorem, 733-735 antilinear operator properties, 721-723 degenerate/near-degenerate vibration levels, 728-733... 

Longuet-Higgins phase-based treatment, two-dimensional two-surface system, scattering calculation, 154-155 three-state molecular system, 134-137 two-state molecular system, single conical intersection solution, 98-101 permutational symmetry, degenerate/near-degenerate vibrational levels, 730-733 Polyene molecules ... 

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). 

It is well known that the v, band of liquid acetonitrile is significantly asymmetric due to an overlap of hot band transitions in the low frequency side. A study of gas phase rotation-vibration spectrum [19] showed that the hot band transition from the first exited state of the degenerated C-C = N bending v8 mode, v hl = v + v8 - vs, has its center at 4.944 cm 1 lower than that of the fundamental transition, v,. Also the presence of v,h2 = v, + 2v8 - 2v8 transition is expected. The careful study on the v band of liquid acetonitrile by Hashimoto et al [20] provided the reorientational and vibrational relaxation times of liquid acetonitrile molecule. They corrected the contribution by the hot band transition using the Boltzmann population law and approximated the v , v,hl, 2h2, and v, + v4 bands by Lorentzian curves. 

It is well known that the v, band of liquid acetonitrile is significantly asymmetric due to overlap of hot band transitions in the low frequency side. From a study of gas phase rotation-vibration spectrum [16], the hot band transition from the first exited state of the degenerate C-C=N bending Vj, mode, v, = V, + Vg - Vg, has its center at 4.944 cm ... 

The high symmetry of some of the sulphur structures (Fig. 1) and the presence of a number of twofold degenerate vibrations mean that the infrared and Raman spectra are quite different, so that it is often possible to identify these species, even in mixtures with other molecules of this type in the gas and liquid phases [11]. The correlations found between the vibration frequencies and structural properties (e.g., the relationship in a ring structure between the length of a bond and the dihedral angle) have been very useful in making structural predictions in cases where X-ray diffraction data are presently unavailable. 

The six normal modes of the NF3 molecule form two totally symmetrical vibrations, vi(ai) and V2(ai), and two doubly degenerate vibrations, V3(e) and V4(e). Force-field calculations, discussed on p. 194, show that to a good approximation the higher frequency mode in each of the two symmetry species is a bond stretch and the lower an angle deformation. Fundamental frequencies for various phases of NF3 are compiled in Table 3, p. 192. 

Equation (6.63) gives the first of two degenerate vibrations to obtain the second we need a new generating vector that is likely to lead to an orthogonal result. In < i(Ei ), M2 and M3 occur with opposite sign, so that H2 and H3 move out of phase with one another. Accordingly, a possible choice of generating vector is one which forces these two atoms to move in phase with one another, the simplest form of which is... 

The VCD spectrum of (S)-(-)-epoxypropane [10] in the liquid and in the gaseous phase shows the splitting of the degenerate vibrational modes of the methyl group. Its analysis verified the VCD theory of the perturbed vibrational degenerate modes. Using a resolution of 1 cm i, the CD in the rotational-vibrational spectrum of (R)-(-h)-methyloxirane has been measured with the result that the Q branch in some bands has the opposite sign to the R and P branches. This can be explained if methyloxirane (in spite of its chirality) is an approximate symmetrical top. 

Figure 29, The effect of the phase-preserving component of the degenerate 2 distorting mode, It may be regarded as a major component of the reaction coordinate that leads to the A] structure (going left, one phase of the mode). Going right, the other phase of the same vibration, the B2 state is formed. (A type-V structure is also obtained along the same coordinate).

We now turn to a more complex instance of the Davydov-split spectral lines corresponding to the bending vibrations of C02 molecules in a monolayer adsorbed on the NaCl(100) surface. A molecule C02 in gaseous phase exhibits two degenerate bending vibrations in the plane perpendicular to the long molecular axis. 

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