Normal mode compositions

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... 

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. 

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. 

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]. 

The wavefunctions of the normal scattering modes are the standing waves produced by a linear combination of incoming coulomb wavefunctions which is reflected from the ionic core with only a phase shift. The composition of the linear combination is not altered by scattering from the ionic core. These normal modes are usually called the a channels, and have wavefunctions in the region rc[Pg.418]

Fullerenes. A resonance Raman study of C6o in its first allowed electronic excited state shows that the ground state hg(l) mode splits into two components, 265, 281 cm-1. The data are consistent with D5d symmetry for the excited state.175 Raman spectroscopy was used to characterise C60 units in a tantalum oxide lithium fulleride composite.176 A group theoretical analysis has been made of the vibrational normal modes for the azafullerene C4gNi2.177... 

Detailed analyses of the vibrational spectra of raacromolecules, however, have provided a deeper understanding of structure and interactions in these systems (Krimm, 1960). An important advance in this direction for proteins came with the determination of the normal modes of vibration of the peptide group in A -methylacetamide (Miyazawa et al., 1958), and the characterization of several specific amide vibrations in polypeptide systems (Miyazawa, 1962, 1967). Extensive use has been made of spectra-structure correlations based on some of these amide modes, including attempts to determine secondary structure composition in proteins (see, for example, Pezolet et al., 1976 Lippert et al., 1976 Williams and Dunker, 1981 Williams, 1983). 

Clearly, the assets of a useful, in itself noncontradictory, and physically based CNM analysis are the internal vibrational motions and their properties as well as the amplitudes that relate internal modes to normal modes. As shown in the previous section, the adiabatic internal modes an are the appropriate candidates for internal modes. Adiabatic modes are based on a dynamic principle, they are calculated by solving the Euler-Lagrange equations, they are independent of the composition of the set of internal coordinates to describe a molecule, and they are unique in so far as they provide a strict separation of electronic and mass effects [18,19]. Therefore, they fulfil the first requirement for a physically based CNM analysis. 

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. 

The optimization procedure depends on identifying the function of solvent composition and temperature that will provide a precise value for the capacity ratios of the two solutes. It will then be possible, using a relatively simple computer program, to identify the various operating parameters of the chromatographic system that will provide the separation in the minimum time. In the example given, a separation system operating in the normal mode will be used. 

Most of the Pirkle phases, and in particular the Whelk-01 phase, are stable to all types of solvent and can be used either in the reversed phase mode, or the normal mode, which again, adds to its universal applicability. In the normal phase mode of development, hexane/IPA would be a good mobile phase from which to start. A mixture, hexane/IPA 80/20 v/v, would be a practical scouting composition to assess the possible level of retention and chiral selectivity. Again,... 

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