Stretching and bending terms

In the case of ethylene, because of 2-fold symmetry, odd terms drop out of the series, V3, V5,... = 0. In the case of ethane, because of 3-fold symmeti-y, even temis drop out, V2, V4,... = 0. Terms higher than three, even though permitted by symmetry, are usually quite small and force fields can often be limited to three torsional terms. Like cubic and quaitic terms modifying the basic quadratic approximation for stretching and bending, terms in the Fourier expansion of Ftors (to) beyond n = 3 have limited use in special cases, for example, in problems involving octahedrally bound complexes. In most cases we are left with the simple expression... 

Stretch and bend terms are most simply given in terms of quadratic ( Hook s law ) forms. 

The stretching and bending terms for structure 2 are the same as for structure 1, except for the contribution of the central C-C bond strictly speaking, the torsional term should be smaller, since the opposing C(CH3) groups have been moved apart. 

Several extensions of the MCY potential were soon introduced. A flexible version containing stretch and bend terms through quartic terms has been implemented with some improvement over the original MCY predictions. This also led to an improved description of the temperature dependency of the second virial coefficient. A further extension involves the introduction of many-body terms into the model. Initially, this was done by estimating an explicit form that includes three-body and four-body terms. This brute force approach to induction, which does not lead to a self-consistent treatment of the problem, resulted in a variety of computational problems. A more direct approach is discussed later. 

Rahman potential is an extension of the SPC model. The extension is the inclusion of quadratic stretch and bend terms, so that internal degrees of freedom are present. The model was used to examine the effect of the liquid environment on internal modes, and the pair functions were found to be close to those for the SPC model. In contrast. Teleman et al., ° who used the same SPC model with a harmonic intramolecular potential, reported some large effects on properties. However, Barrat and McDonald find that flexibility has a small effect on calculated properties, stating... 

A flexible, polarizable model that can also dissociate has been developed. The intramolecular stretch and bend terms are localized by a Gaussian function, making it possible for a molecule to separate into individual ions. Significantly, this model can include hydronium and hydroxyl ions in a Simula-... 

In addition to the descriptions of group vibrations as stretch and bend (or deformation) the terms rock, twist, scissors, wag, torsion, ring breathing and inversion (or umbrella) are used frequently these motions are illustrated in Figure 6.13. 

It is important to note that the behavior of polymers below the yield point is Hookean and essentially reversible for short-term service. Thus this range, which is associated with stretching and bending of covalent bonds, is called the elastic range. The area under the stress-strain curve is a measure of toughness. 

The spectra of normal alkanes (paraffins) can be interpreted in terms of four vibrations, namely, the stretching and bending of C—H and C—C bonds. Detailed analysis of the spectra of the lower members of the alkane series has made detailed assignments of the spectral positions of specific vibrational modes possible. 

This approximation shows that the vibrational frequency is inversely proportional to the mass, and directly proportional to the force constant. The force constants are defined in terms of internal coordinates of the molecules they correspond to the forces resisting stretching and bending... 

The spectra have been analyzed in terms of a local excitation of the calcium and internal van der Waals coordinates (stretching and bending). Indeed, strong resonances are observed in these spectra, which can be assigned to van der Waals modes. The excitation of these vibrations allows the exploration of an important region in the excited PES. This exploration is more extensive when the laser-excited coordinate is different from the reaction coordinate. In such a case, the longer lived the reaction intermediate is, the more of the PES will be explored. The excitation of different resonances is reflected in the different widths of the transitions—an indication of their reactivities. 

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