Angle-bending vibrations

This Schrodinger equation forms the basis for our thinking about bond stretching and angle bending vibrations as well as collective phonon motions in solids... 

Table 6.3 lists a number of group vibration wavenumbers for both bond-stretching and angle-bending vibrations. 

Table 7.1. Characteristic frequencies (wave numbers, in cm ) typical for some chemical bonds (stretching vibrations) and bond angles (bending vibrations).

In this article we describe an extension of SISM to a system of molecules for which it can be assumed that both bond stretching and angle bending describe satisfactorily all vibrational motions of the molecule. The SISM presented here allows the use of an integration time step up to an order of magnitude larger than possible with other methods of the same order and complexity. 

Intensive use of cross-terms is important in force fields designed to predict vibrational spectra, whereas for the calculation of molecular structure only a limited set of cross-terms was found to be necessary. For the above-mentioned example, the coupling of bond-stretching (f and / and angle-bending (B) within a water molecule (see Figure 7-1.3, top left) can be calculated according to Eq. (30). 

Figure 7-13. Cross-terms combining internal vibrational modes such as bond stretch, angle bend, and bond torsion within a molecule.

The harmonie oseillator energies and wavefunetions eomprise the simplest reasonable model for vibrational motion. Vibrations of a polyatomie moleeule are often eharaeterized in terms of individual bond-stretehing and angle-bending motions eaeh of whieh is, in turn, approximated harmonieally. This results in a total vibrational wavefunetion that is written as a produet of funetions one for eaeh of the vibrational eoordinates. 

Even with this simple model it is clear that if one of the nuclei is given a sudden displacement it is very likely that the whole molecule will undergo a very complicated motion, a Lissajous motion, consisting of a mixture of angle-bending and bond-stretching. The Lissajous motion can always be broken down into a combination of the so-called normal vibrations of the system which, in the Lissajous motion, are superimposed in varying proportions. 

Table 6.3 Typical bond-stretching and angle-bending group vibration wavenumbers co...

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. 

Display water as a ball-and-spoke model. How many different vibrations are there Explain. One after the other, animate these vibrations. For each, record the vibrational frequency and provide a description of the atomic motions. What appears to be easier (lower frequency), motions primarily associated with bond stretching or with angle bending ... 

Repeat the analysis with deuterium oxide (D2O). Are the vibrational frequencies the same, larger or smaller than those in water Rationalize your observations. Are the changes in vibrational frequencies greatest for bond stretching or angle bending motions ... 

Molecular mechanics calculations use an empirically devised set of equations for the potential energy of molecules. These include terms for vibrational bond stretching, bond angle bending, and other interactions between atoms in a molecule. All of these are summed up as follows ... 

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