Normal mode composition factors

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. 

The first-principles calculation of NIS spectra has several important aspects. First of all, they greatly assist the assignment of NIS spectra. Secondly, the elucidation of the vibrational frequencies and normal mode compositions by means of quantum chemical calculations allows for the interpretation of the observed NIS patterns in terms of geometric and electronic structure and consequently provide a means of critically testing proposals for species of unknown structure. The first-principles calculation also provides an unambiguous way to perform consistent quantitative parameterization of experimental NIS data. Finally, there is another methodological aspect concerning the accuracy of the quantum chemically calculated force fields. Such calculations typically use only the experimental frequencies as reference values. However, apart from the frequencies, NIS probes the shapes of the normal modes for which the iron composition factors are a direct quantitative measure. Thus, by comparison with experimental data, one can assess the quality of the calculated normal mode compositions. 

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. 

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]. 

An extension of Q-mode factor analysis [ 1,303-306] provides a solution to the first, second and fourth problems listed above, those associated with obtaining absolute compositions of the end-members themselves. Normally when... 

Polymers generally have a low absorption coefficient for x-rays. Thus for a pure polymer sample, contrast is more likely to be a limiting factor than the instrument resolution in normal absorption mode x-ray tomography. Most published x-ray tomography on polymer samples has involved higher contrast systems. These include determination of the distribution of catalyst residue in as-polymerized particles [286], fiber composites [287] and damage in these composites [288,289], bone regrowth into biomaterial [290], and many porous structures such as foams [291] and biopolymer scaffolds [292]. The laboratory and tabletop instruments... 

Now that the e, s, a, b, and v system parameters are known for the four Chirobiotic columns and the A, B, E, S, and V solute descriptors are obtained for five enantiomeric compounds, it is possible to calculate the retention factor k of the five enantiomeric compounds using Eq. (5) and compare it with the experimental two retention factors obtained for the two enantiomers on the CSP. Figure 7 illustrates the results three situations were observed, (i) The retention factors predicted by LSER corresponded to the first eluting enantiomer (e.g., dihydrofurocoumarin. Fig. 7a) (ii) the LSER-predicted retention factors corresponded to the last eluting enantiomer (e.g., 5-methyl-5-phenyl hydantoin in the normal-phase mode. Fig. 7b) and (iii) the LSER-predicted retention factors did not correspond to a particular enantiomer for all mobile phase compositions (e.g., 5-methyl-5-phenyl hydantoin in the RPLC mode and bromacil. Fig. 7c and d). From a mechanistic point of view, it can be speculated that in case i, the chiral selector has overall attractive enan-tioselective interactions with the second enantiomer more retained than the LSER prediction in case ii, the chiral selector has overall repulsive enantioselective interactions with the first enantiomer less retained than the LSER prediction and in case iii, the chiral selector has enantioselective interactions with both enantiomers. 

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