Chirality parameter

The PECD measurement clearly takes the form of a cosine function with an amplitude given entirely in terms of the single chiral parameter, b. It therefore provides exactly the same information content as the y asymmetry factor dehned above [Eq. (8)]. Experimental advantages of examining the PECD rather than the single angular distribution /p(0) are likely to include some cancellation of purely instrumental asymmetries (e.g., varying detection efficiency in the forward-backward directions) and consequent improvements in sensitivity. 

In the more novel case of the P coefficient, only interference terms for which 7 = 1 1 arise, so now all summands contributing to contain phase shift information. This hints that the chiral parameters could be more sensitive to phase differences than traditional (3 parameters, and one would... 

It was shown in the preceding section that PECD can be anticipated to have an enhanced sensitivity (compared to the cross-section or p anisotropy parameter) to any small variations in the photoelectron scattering phase shifts. This is because the chiral parameter is structured from electric dipole operator interference terms between adjacent -waves, each of which depends on the sine of the associated channels relative phase shifts. In contrast, the cross-section has no phase dependence, and the p parameter has only a partial dependence on the cosine of the relative phase. The distinction between the sine... 

For the carbonyl carbon Ij core level ionization, excellent quantitative agreement of the b parameters is found, both between the alternative calculations and between either calculation and experiment (see Section VLB.I). Given the spherical, therefore achiral, nature of the initial orbital in these calculations, any chirality exhibited in the angular distribution must stem from the final-state photoelectron scattering off the chiral molecular ion potential. Successful prediction of any non-zero chiral parameter is clearly then dependent on a reliable potential model describing the final state. At this level, there is nothing significant to choose between the potential models of the two methods. 

In one of the very first realistic computational studies of PECD effects, performed for the amino acid alanine [51], it was noted that different results were obtained at each of three fixed geometries corresponding to low lying conformations identified in previous structure investigations [69-71]. A later combined experimental-theoretical study of PECD in 3-hydroxytetrahydrofuran [61] further looked at the influence of presumed conformation and concluded that while the predicted cross-section, a, and p parameters were mildly affected by conformation, the chiral parameters were much more strongly... 

Figure 10. The CMS-Xa predictions for cross-section, the anisotropy parameter —P/2), and the chiral parameter in the carbonyl C li photoionization of (R)-carvone (I) and its indicated derivatives.

The 2 angular-distribution parameters show somewhat more molecule-to-molecule variation, but are essentially still quite similar to one another. They also display low energy structures that readily correlate with the noted features in the cross-section curves, and the variations are mainly confined to changes in the relative intensity of these. Much bigger differences are, however, found in the calculated chiral parameters, though here as well there is a common... 

From the available evidence Stener and co-workers [53, 60] conclude that the chiral parameter is more sensitive to small asymmetries in the molecular potential than to continuum collapse effects at resonance. At present, such conclusions must be provisional as there is little direct evidence. There is also no evidence regarding likely behavior at autoionization resonances, and this too deserves attention. 

At a fundamental level, it has been shown that PECD stems from interference between electric dipole operator matrix elements of adjacent continuum f values, and that consequently the chiral parameters depend on the sine rather than the cosine of the relative scattering phases. Generally, this provides a unique probe of the photoionization dynamics in chiral species. More than that, this sine dependence invests the hj parameter with a greatly enhanced response to small changes in scattering phase, and it is believed that this accounts for an extraordinary sensitivity to small conformational changes, or indeed to molecular substitutions, that have only a minimal impact on the other photoionization parameters. 

This interaction term, which is proportional to the first pseudoscalar term in a multipole expansion of the intermolecular interaction energy, as obtained by Goossens [79], [77], [80], was sufficient to obtain a rich polymorphism of chiral liquid crystal phases given by cholesteric and several blue phases in dependence on temperature and the chirality parameter c measuring the strength of the energy of chiral interaction, both for calamitic and discotic chiral Gay-Beme molecules [77], [81]. 

The physical properties (e.g., pitch, spontaneous polarization) are independent of the chemical origin of the chirality (see Sections 4.2.1 to 4.2.10). Exact models concerning the effects of molecular chirality are difficult to find and to quantify. Further, each chiral parameter may have independent rules. The helix of the cholesteric phase is oriented perpendicular to the main axis of the molecules, whereas the helix of the phase is oriented parallel to the molecular axis. The spontaneous polarization also depends on the strength of the dipole monents connected to the chiral centers. Direct quantification of the different chiral effects from the chemical structure is a big challenge for the future. An example for molecular models of chiral properties might be the Boulder model of the spontaneous polarization [30]. 

To verify the accuracy of the code for isotropic, chiral particles, we consider a spherical particle of size parameter k a = 10. The refractive index of the particle is m-r = 1.5 and the chirality parameter is [3k = 0.1, where ki = m ks. Calculations are performed for A rank = 18 and A int = 200. Figure 3.7 compares the normalized differential scattering cross-sections computed with the... 

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