Vibrational averaging effects

The similarity in the behaviour of coupling constants as a function of e in both radicals allows to discuss vibrational averaging effects simply in terms of the potential governing the out-of-plane motion. 

Often the original structure determination will have involved some uncorrected vibrational averaging effects it may be an r0 or an r, structure.28 However, once /2, or some approximation to /3, has been determined it is possible to correct r to re and obtain an improved equilibrium structure (in most cases this correction can be made directly from the cubic anharmonic calculation, but in some cases the calculation will allow unobserved a. values to be determined, perhaps for other isotopic species, etc.). Similarly, it is often true that the harmonic field /2 is calculated from the observed fundamentals (the v values) rather than the harmonic vibration wavenumbers (the to values), for want of information on the corrections. However, once /4, or some approximation to/4, has been determined, it may be used to calculate a complete set of x values and hence to calculate all the corrections to obtain the co values. Thus the calculation of re and may be improved from a knowledge of /3 and /4. 

As we explained earlier in this chapter, one of the great merits of the effective Hamiltonian is that it allows the two tasks of fitting experimental data and interpreting the resultant parameters to be separated. In this section we discuss the latter aspect and explain how the quantities obtained from a fit of experimental data can be interpreted in terms of the geometric and electronic structure of the molecule concerned. We have seen how the process of averaging the parameters over the vibrational motion of the molecule leads to additional terms which describe the vibrational dependence of the parameters. We shall assume in what follows that all such vibrational averaging effects have been properly taken into account and that we are left to deal with the equilibrium value of the parameter, Pe, in equation (7.180). 

In most cases these differ significantly from the theoretical value of 6.514 an explanation is not given in the original papers, but the differences may reflect subtle vibrational averaging effects or breakdown of the Born-Oppenheimer approximation. 

In the following section, we will see that vibrational averaging plays a significant role in the computation of reliable hcc values for a number of interesting systems. Here, we just discuss vibrational averaging effects related to inversion at the radical center of typical n (methyl) and o (vinyl) radicals. 

From another point of view biological molecules are often very flexible, so that a realistic computation of their properties cannot neglect vibrational averaging effects from large amplitude motions. This aspect is examined in the second part of this work. 

The simplest way to combine electronic stnicture calculations with nuclear dynamics is to use harmonic analysis to estimate both vibrational averaging effects on physico-chemical observables and reaction rates in terms of conventional transition state theory, possibly extended to incorporate tunneling corrections. This requires, at least, the knowledge of the structures, energetics, and harmonic force fields of the relevant stationary points (i.e. energy minima and first order saddle points connecting pairs of minima). Small anq)litude vibrations around stationary points are expressed in terms of normal modes Q, which are linearly related to cartesian coordinates x... 

The behavior of coupling constants as a function of x is qualitatively very similar for the different radicals a(C) is always positive and increases with x due to the progressive contribution of carbon s orbitals to the singly occupied molecular orbital (SOMO). The effect is similar for a(F) and a(H), but since a(H) is negative for the planar conformation (due to first order spin polarization) the absolute value of a(H) decreases up to x 10° and next increases. This allows to discuss vibrational averaging effects simply in terms of the potential governing the out of plane motion. 

Thus, the present results show that PCM is a very powerfiil tool for describing the behavior in solution of compounds able to form intramolecular hydrogen bonds. Direct computation of vicinal NMR coupling constants including vibrational averaging effects confirms this conclusion further and paves the route for more reliable quantum mechanical studies of bioactive systems. 

The failure to remove all of the vibrational averaging effects is not too upsetting. The ability to determine a potential function with a barrier accurate to within a few cm-1 is rather remarkable. The fact that the data can reveal even a very small barrier - in the case of trimethylene oxide, smaller than the zero point energy -shows that they are a very sensitive probe of the molecular dynamics. 

In order to take vibrational averaging effects into proper account, we resorted to classical treatment with a purposely tailored force field using a new accurate force field obtained by extending the ff99SB force field [88] with a reliable... 

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