Bending modes

The analogous coupling between the antisyimnetric stretch and bend is forbidden in the H2O Hamiltonian because of syimnetry.) The 2 1 resonance is known as a Femii resonance after its introduction [ ] in molecular spectroscopy. The 2 1 resonance is often very prominent in spectra, especially between stretch and bend modes, which often have approximate 2 1 frequency ratios. The 2 1 couplmg leaves unchanged as a poly ad number the sum ... 

Figure Al.2.11. Resonant collective inodes of the 2 1 Fenni resonance system of a coupled stretch and bend with an approximate 2 1 frequency ratio. Shown is one end of a syimnetric triatomic such as H2O. The nomial stretch and bend modes are superseded by the horseshoe-shaped modes shown in (a) and (b). These two modes have different frequency, as further illustrated in figure Al.2.12.

Figure A3.13.12. Evolution of the probability for a right-handed ehiral stmetnre (fiill eiirve, see ( equation (A3,13.69))) of the CH eliromophore in CHD2T (a) and CHDT2 ( ) after preparation of ehiral stnietures with multiphoton laser exeitation, as diseussed in the text (see also [154]). For eomparison, the time evolution of aeeording to a one-dimensional model ineluding only the bending mode (dashed enrve) is also shown. The left-hand side insert shows the time evolution of within the one-dimensional ealeulations for a longer time interval the right-hand insert shows the time evolution within the tln-ee-dimensional ealeulation for the same time interval (see text).

Figure Bl.6.10 Energy-loss spectrum of 3.5 eV electrons specularly reflected from benzene absorbed on the rheniiun(l 11) surface [H]. Excitation of C-H vibrational modes appears at 100, 140 and 372 meV. Only modes with a changing electric dipole perpendicular to the surface are allowed for excitation in specular reflection. The great intensity of the out-of-plane C-H bending mode at 100 meV confimis that the plane of the molecule is parallel to the metal surface. Transitions at 43, 68 and 176 meV are associated with Rli-C and C-C vibrations.

Figure Bl.25.12 illustrates the two scattering modes for a hypothetical adsorption system consisting of an atom on a metal [3]. The stretch vibration of the atom perpendicular to the surface is accompanied by a change m dipole moment the bending mode parallel to the surface is not. As explained above, the EELS spectrum of electrons scattered in the specular direction detects only the dipole-active vibration. The more isotropically scattered electrons, however, undergo impact scattering and excite both vibrational modes. Note that the comparison of EELS spectra recorded in specular and off-specular direction yields infomiation about the orientation of an adsorbed molecule.

Figure Bl.25.12. Excitation mechanisms in electron energy loss spectroscopy for a simple adsorbate system Dipole scattering excites only the vibration perpendicular to the surface (v ) in which a dipole moment nonnal to the surface changes the electron wave is reflected by the surface into the specular direction. Impact scattering excites also the bending mode v- in which the atom moves parallel to the surface electrons are scattered over a wide range of angles. The EELS spectra show the higlily intense elastic peak and the relatively weak loss peaks. Off-specular loss peaks are in general one to two orders of magnitude weaker than specular loss peaks.

One reason that the symmetric stretch is favored over the asymmetric one might be the overall process, which is electron transfer. This means that most of the END trajectories show a nonvanishing probability for electron transfer and as a result the dominant forces try to open the bond angle during the collision toward a linear structure of HjO. In this way, the totally symmetric bending mode is dynamically promoted, which couples to the symmetric stretch, but not to the asymmetric one. 

B. R. Brooks and M. Karplus. Normal modes for specific motions of macromolecules Application to the hinge-bending mode of lysozyme. Proc. Natl. Acad. Sci. USA, 82 4995-4999, 1985. 

So, of the 9 eartesian displaeements, 3 are of ai symmetry, 3 of b2,2 of bi, and 1 of a2- Of these, there are three translations (ai, b2, and b i) and three rotations (b2, b i, and a2). This leaves two vibrations of ai and one of b2 symmetry. For the H2O example treated here, the three non zero eigenvalues of the mass-weighted Hessian are therefore of ai b2, and ai symmetry. They deseribe the symmetrie and asymmetrie streteh vibrations and the bending mode, respeetively as illustrated below. 

The first or free-free rotor bending mode frequency (based on the required margin for operation below the critical speed)... 

We will neglect the bending mode, which may not always be acceptable. 

Application of similar reasoning to the case of the bending mode of COj would indicate that the vibration is Raman inactive, infrared active. 

The vibrational energy levels associated with a single normal mode have a degeneracy of one. However, molecules with high symmetry may have several normal modes with the same frequency. For example, COi has two bending modes, with the motion of one perpendicular to the motion of the other. Such modes are often referred to as degenerate modes, but there is a subtle difference... 

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