Global shape property

A practical implementation of the above approach is the following a global shape property of the molecule is assigned to each point of the sphere S, followed by the determination of those domains of S where this shape property is invariant. A pair of examples is shown in Figure 5.9, where the shape globe invariance domains of a MIDCO surface for two relative convexity shape domain partitionings (P) with respect to two reference curvatures, b = 0, and b < 0, are given. As... 

One should note that within each shape globe map an entire family of topological descriptors Fj(s) = I(i), i=l,...k is assigned to each point s of the sphere S, providing information on a global shape property of the enclosed molecule. 

As an example of absolute shape criteria, the local curvature properties of a MIDCO can be used for defining absolute shape domains on it [156], and for a subsequent global shape characterization. In Figure 5.1 a MIDCO G(a) is shown as an illustration of some of the concepts discussed. The simplest method [155] is based on comparisons to a reference of a tangent plane what leads to the identification of locally convex, concave, and saddle-type domains, as mentioned previously, although much finer characterizations are also possible [156,199]. 

In many chemical problems the comparisons of local molecular regions are more important than global comparisons. The presence of functional groups or other molecular moieties with specified shape properties often imply similar chemical behavior even if the molecules compared have very different global shapes. For this reason, local molecular shape descriptors and local shape codes are of major importance. 

In nearly all chemically important problems, shape complementarity refers to local shape properties. Most of the typical molecular interactions where shape complementarity is relevant involve only some local moieties of the molecules. Global shape complementarity is more difficult to achieve and seldom plays a role. 

Keinan et al. found that the quantitative degree of chirality of substrates correlates with their efficiency of reaction with active sites [7]. The degree of chirality, a global shape descriptor, was determined by the use of the continuous chirality measure (CCM) methodology developed previously, which treats chirahty as a continuous structural property and not as a binary quality (chiral/not chiral) [8]. 

Mechanical properties of polymer nanocomposites can be predicted by using analytical models and numerical simulations at a wide range of time- and length scales, for example, from molecular scale (e.g., MD) to microscale (e.g., Halpin-Tsai), to macroscale (e.g., FEM), and their combinations. MD simulations can study the local load transfers, interface properties, or failure modes at the nanoscale. Micromechanical models and continuum models may provide a simple and rapid way to predict the global mechanical properties of nanocomposites and correlate them with the key factors (e.g., particle volume fraction, particle geometry and orientation, and property ratio between particle and matrix). Recently, some of these models have been applied to polymer nanocomposites to predict their thermal-mechanical properties. Young s modulus, and reinforcement efficiency and to examine the effects of the nature of individual nanopartides (e.g., aspect ratio, shape, orientation, clustering, and the modulus ratio of nanopartide to polymer matrix). 

In Fig. 16.2(A-D) we present the spectra as a function of energy for all four cases. In Fig. 16.2A we show the exact 24-mode result and for the sake of comparison the remaining Fig. 16.2(B-D) also display this curve. It was already mentioned in the theory section that the 3(QVC)-mode model reproduces the zeroth- to second-order cumulants of the molecule, connected to observable properties of the spectrum in such experiments. These are the total intensity, the position of the maximum and the with of the spectrum. Due to the absence of theoretical proof concerning the conservation of the third cumulant, which is related to the major asymmetry of the spectra, we can not discuss this feature with certainty. As expected the spectra obtained by the 6-mode model (see Fig. 16.2B) is most similar to the exact one. However the global shape of the exact spectrum is more or less reproduced by our 3(QVC)-mode formulas too. Some oscillations appear in particular within the (0-0.3 eV) interval, but the picture is definitely closer to the exact one than that obtained by the 3(LVC)-mode approach. Apart from these oscillations one can see that the position of the maximum, the width and even the main asymmetry of the spectrum are satisfactorily reproduced by our present method. 

In this section, the adiabatic picture will be extended to include the non-adiabatic terais that couple the states. After this has been done, a diabatic picture will be developed that enables the basic topology of the coupled surfaces to be investigated. Of particular interest are the intersection regions, which may form what are called conical intersections. These are a multimode phenomena, that is, they do not occur in ID systems, and the name comes from their shape— in a special 2D space it has the fomi of a double cone. Finally, a model Flamiltonian will be introduced that can describe the coupled surfaces. This enables a global description of the surfaces, and gives both insight and predictive power to the fomration of conical intersections. More detailed review on conical intersections and their properties can be found in [1,14,65,176-178]. 

Also important to the correlation between global conformation and optoelectronic properties are the investigations of single molecules of polysilanes by AFM, by which molecules could be directly imaged, and their contour lengths, diameters, and general shapes studied. These studies are dealt with in Section 3.11.5.3. 

ADRIANA.code Global physicochemical descriptors, size and shape descriptors, atom property-weighted 2D and 3D autocorrelations and RDF, surface property-weighted autocorrelations 1,244... 

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