Normal modes bending mode

Normal Mode Shear Mode extension torsion bending shear warping warping... 

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. 

The CO2 laser is a near-infrared gas laser capable of very high power and with an efficiency of about 20 per cent. CO2 has three normal modes of vibration Vj, the symmetric stretch, V2, the bending vibration, and V3, the antisymmetric stretch, with symmetry species (t+, ti , and (7+, and fundamental vibration wavenumbers of 1354, 673, and 2396 cm, respectively. Figure 9.16 shows some of the vibrational levels, the numbering of which is explained in footnote 4 of Chapter 4 (page 93), which are involved in the laser action. This occurs principally in the 3q22 transition, at about 10.6 pm, but may also be induced in the 3oli transition, at about 9.6 pm. 

A2) In spite of the high individual frequencies, bond length and bond angle vibrations participate in quasi-classical low frequency collective normal modes. Bond angle bending is necessary for the flexibility of five-membered rings, which plays a key role in the polymorphism of nucleic acids. 

Figure 4 DynDom [67] analysis of the first two normal modes of human lysozyme. Dark grey and white indicate the two dynamic domains, separated by the black hinge bending region. The vertical line represents a hinge axis that produces a closure motion in the first normal mode. The horizontal line represents a hinge axis that produces a twisting motion in the second normal mode. (Adapted from Ref. 68.) The DynDom program is available from the Internet at http //md. chem.rug.nl/ steve/dyndom.html.

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... 

A nonlinear molecule consisting of N atoms can vibrate in 3N — 6 different ways, and a linear molecule can vibrate in 3N — 5 different ways. The number of ways in which a molecule can vibrate increases rapidly with the number of atoms a water molecule, with N = 3, can vibrate in 3 ways, but a benzene molecule, with N = 12, can vibrate in 30 different ways. Some of the vibrations of benzene correspond to expansion and contraction of the ring, others to its elongation, and still others to flexing and bending. Each way in which a molecule can vibrate is called a normal mode, and so we say that benzene has 30 normal modes of vibration. Each normal mode has a frequency that depends in a complicated way on the masses of the atoms that move during the vibration and the force constants associated with the motions involved (Fig. 2). 

Fig. 5.15 Schematic representation of the normal modes of the Fe(ni)-azide complex with the largest iron composition factors. The individual displacements of the Fe nucleus are depicted by a blue arrow. All vibrations except for V4 are characterized by a significant involvement of bond stretching and bending coordinates (red arrows and archlines), hi such a case, the length of the arrows and archlines roughly indicate the relative amplitude of bond stretching and bending, respectively. Internal coordinates vibrating in antiphase are denoted by inward and outward arrows respectively (taken from [63])...

Integration over the PDOS in Fig. 9.35a yields much smaller composition factors for the resonances at Vi, V2, and V3. This finding suggests that Vj, V2 and V3 are not pure stretching modes but contain considerable contributions from bending modes [89]. Normal mode analysis confirms this qualitative assignment [91]. 

It is apparent from Fig. 4 that the normal modes of vibration of the water molecule, as calculated from the eigenvectors, can be described approximately as a symmetrical stretching vibration (Mj) and a symmetrical bending vibration... 

The SQ method extracts resonance states for the J = 25 dynamics by using the centrifugally-shifted Hamiltonian. In Fig. 20, the SQ wavefunc-tion for a trapped state at Ec = 1.2 eV is shown. The wavefunction has been sliced perpendicular to the minimum energy path and is plotted in the symmetric stretch and bend normal mode coordinates. As anticipated, the wavefunction shows a combination of one quanta of symmetric stretch excitation and two quanta of bend excitation. The extracted state is barrier state (or quantum bottleneck state) and not a Feshbach resonance. 

Figure 8.13 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 normal to the surface changes the electron wave is reflected by the surface into the specular direction. Impact scattering also excites 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 highly 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.

The two stretching modes are called V and v3 here in order to conform with standard notation (Herzberg, 1950 v2 is the bending mode). Several other cases have been analyzed. Typical root-mean-square deviations for the lowest-order Hamiltonian of Eq. (4.28) are < 5 cm-1 up to the sixth overtone. For example, the calculation of water of Table 4.1 has a root-mean-square deviation of 4.0 cm. In addition to providing a calculation of stretching overtones, one is also able to determine, in a simple way, the nature of the spectrum. If one compares, for example, water, H20, with sulfur dioxide, S02, one observes the situation of Table 4.2. Thus S02 is much closer to the normal limit than H20. We shall... 

This ethylidyne intermediate has recently been subject to two IRS investigations, by Chesters and McCash and in a more detailed study by Malik et alP Their spectra show three absorption peaks above 800cm which in light of the EELS work could be assigned to the C—C stretch mode at 1120cm , the symmetric CH3 bend mode at 1340cm and the symmetric CH3 stretch mode at 2885 cm In contrast to the surface methoxy discussed in the previous section, there is no peak associated with the asymmetric CH3 stretch mode at 2950cm" This shows, in line with the discussion above, that the CCHj is oriented with the C—C axis normal to the surface (as indicated in Fig. 19) and verifies at the same time the validity of the infrared surface selection rule. 

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