Mode composition factor

NIS provides an absolute measurement of the so-called normal mode composition factors that characterize the extent of involvement of the resonant nucleus in a given normal mode. On the basis of the analysis of experimental NIS data, one can therefore construct a partial vibrational density of states (PVDOS) that can be... 

One can hence think of (normal-mode composition factor) ej = ejaSja as the fractional involvement of atom j in normal mode a.The dimensionless vector eja also specifies the direction of the motion of atom j in the ot-th normal mode. Interestingly, the mode composition factors are also related to the magnitude of the atomic fluctuations. In a stationary state ) of a harmonic system, the mean square deviation (msd) of atom j from its equilibrium position may be expressed as a sum over modes of nonzero frequency ... 

The PVDOS directly characterizes the involvement of the probe nucleus in different normal modes and provides a graphical representation of the calculated normal mode composition factors. 

Figure 5.14 presents experimental, fitted, and purely quantum-chemically calculated NIS spectra of the ferric-azide complex. It is clear that the fitted trace perfectly describes the experimental spectra within the signal-to-noise ratio. Furthermore, the purely theoretical spectrum agrees well with the fitted spectrum. This indicates that the calculations provide highly realistic force field and normal mode composition factors for the molecule under smdy and are invaluable as a guide for least-square fittings. 

The fitted and calculated vibrational frequencies and normal mode composition factors corresponding to the 17 most important NIS bands are presented in Table 5.9. It is evident that the vibrational peaks in the calculated NIS spectrum are typically 0-30 cm lower than to the experimental values. In the calculated NIS spectra, there are two small peaks at 635 and 716 cm (Fig. 5.14b) that are not visible in the experimental spectrum. According to the normal mode calculations these are Fe-N-N and Fe-O-C deformation vibrations. Small admixtures of Fe-N and Fe-O stretching modes account for the calculated nonzero normal mode composition factors. Although the calculated relative intensities are slightly above detection limit dictated by the signal-to-noise ratio, they are determined by values of pea which are very small (0.028 and 0.026 for the peaks at 635 and 716 cm ). They must be considered to be within the uncertainties of the theoretical... 

Table 5.9 Experimental and calculated at the BP86ATZVP level frequencies and corresponding values of the iron normal mode composition factors of the most important vibrations that appear in the NIS signal of the Fe(III)-azide complex (taken from [101])...

The normal-mode analysis has shown that there are 17 vibrational modes that are characterized by significant involvement of the Fe nucleus (i.e. large values of Fea)- frequencies and normal mode composition factors corresponding to these vibrations are described in Table 5.9. 

Figure 5 presents an example of the excitation probability S (v) and the VDOS D (v) for the iron atom in the molecule Fe(TPP)(l-MeIm)(CO), as determined from measurements on a polycrystalline sample. Sharp features in both representations of the experimental data clearly identify vibrational frequencies above 100 cm, although low-frequency vibrational features are more apparent in the VDOS representation. The VDOS also provides the most convenient estimate of the mode composition factor ej, since the area of each feature directly yields the sum of values for all contributing vibrations. This avoids the need to remove the additional factors in equation (5) that contribute to the area of a feature in S (v), with the subtleties associated with determining an appropriate value for the recoilless fraction Z. However, calculation of D (v) from S (v) involves implicit assumptions that may not be valid in some situations, for example, when more than one molecular species contributes to the experimental signal or when vibrational anisotropy is significant. 

As noted above, the area of a peak in the VDOS provides a straightforward measure of the mode composition factor e j according to equation (6) (possibly summed over a number of unresolved modes). However, there are nontrivial approximations implicit in the calculation. In addition to the Debye approximation used to subtract the recoiUess contribution, the Fourier-log algorithm assumes a unique environment for the probe atom and neglects vibrational anisotropy. The resulting errors are often smaller than the experimental uncertainty, particularly for protein samples. However, there may be situations where these assumptions are questionable, for example, if the probe nucleus occupies... 

NRVS directly samples the KED because the spectral area contributed by mode a directly yields the mode composition factor for the Fe probe atom, according to equations (3 6). Using more traditional vibrational methods, frequency shifts resulting from isotope substitution of atom j can yield an estimate ... 

Here and in the remainder of the chapter, we express the mode composition factor as setting j = Fe and suppressing the mode index a. 

The coefficients eja governing the mathematical transformation from normal coordinates to atomic Cartesian coordinates rj provide a transparent description of mode character. The vector eja parallels the motion of atom j in normal mode a, while the squared magnitude describes its relative mean squared amplitude. Normalization, according to J] = 1, then ensures that the mode composition factor is equal to the fraction of mode energy associated with motion of atom j. The resulting KED, together with the directional information, facilitates model-independent comparison of experiments with each other and with computational predictions. 

---