Dynamic shape descriptors

Consequently, an accurate description of molecular shape should include a smearing effect resulting from nuclear motions and uncertainties. A number of solutions are proposed in the literature, including the use of nuclear wavefunctions, >2 o open sets of nuclear configurations,and a fuzzy set approach to molecular shape. We shall use the term dynamical shape for a description that takes into account nuclear flexibility. Such a characterization is performed with dynamic shape descriptors. The term static shape descriptors is reserved to those defined at a frozen nuclear geometry. 

Within the simplest formulation of a dynamic shape analysis method of the first type, the invariance of topological descriptors within domains of the dynamic shape... 

It is possible to design shape descriptors that incorporate nuclear vibrations. However, the simpler approach to shape dynamics is to apply static descriptors to entire domains of nuclear configurations. We then observe the interrelation between molecular shape and molecular motions (e.g., vibrations, reaction paths).32,33 in this case, we find a range of values in static parameters. The span of such a range (e.g., the fluctuations in a static descriptor) can characterize shape dynamics.3 ... 

The next sections focus mostly on the properties of absolute shape descriptors. Special attention is devoted to those that are used to study static conformations and dynamics. Among the myriad of shape descriptors in the literature, we deal with a subset of those that are conceptually distinct and serve as examples for the construction of many others. 

Let us denote the average of a general shape descriptor P( r, ) over all accessible molecular configurations by (P). If the analysis is restricted to conformational changes, then atomic connectivity is maintained. For a given type of connectivity (e.g., linear chains), the dependence of (P) with the number of atoms n relates to shape features, especially shape flexibility. Later in the chapter, we discuss shape dynamics in a broader context. Here, we can illustrate this relation with a simple example. 

Relative shape descriptors are defined in terms of a reference structure, which may be an experimental X-ray or NMR structure. Similarly, we can use a molecular mechanics conformational energy minimum as the initial structure for a dynamics simulation. The reference can also be a target conformation with desired shape features—for example, being maximally compact i -aie or... 

Throughout the realm of molecular modeling, the concept of molecular shape arises over and over in one form or another. Just what do scientists mean by a molecule s shape, and how can one use three-dimensional shape in modeling. In Chapter 5, Professor Gustavo A. Arteca examines these issues and delineates the hierarchical levels of molecular shape and shape descriptors. He explains molecular shape in terms of mathematical descriptors of nuclear geometry, connectivity, and molecular surfaces. Of special note are his comments on shape dynamics of flexible molecules and descriptors of relative shape. 

Conseqnently, the resulting RDF descriptors — here called the amplified and attennated RDF — exhibit quite different shapes. The attenuated RDF descriptors typically exhibit more, or significant, peaks than the amplified variant when dynamic properties are used. In particular, dynamic properties of carbon atoms are often small in comparison with one of the heteroatoms. Thus, peaks related to carbon atoms are also small, whereas the peaks related to the heteroatoms are intensified. 

Using differential geometry, a curve can be reconstructed from the knowledge of its curvature and torsional functions (the Frenet-Serret formulas). Therefore, these two functions give a concise shape description, provided one associates an everywhere-differentiable curve to the molecular skeleton. This approach has been employed in the analysis of protein shape. Moreover, this technique allows one to represent the dynamics of molecular chains or loops in terms of the deformation of elastic bodies. We return to this point when we discuss a number of purely topological descriptors of molecular curves. 

Molecular shape has a fundamental influence on both the static and dynamic properties of molecules. For example, the shapes of nuclear and electronic distributions determine the molecular dipole moments as well as the likely sites of approach by a nucleophilic reagent. The evolution of concepts and models used by chemists for the description of the molecular shape closely mirrors the advances made in our understanding of molecular behavior. Whereas most of the early models focused on the nuclear arrangements, the more advanced recent approaches have placed increasingly more emphasis on the electronic distribution. The nuclear position descriptors, however, is still very important, since the nuclei in molecules are more particle-like than electrons and therefore much easier to be described. For the description of the shape of complex molecules, the concept of similarity appears to be very useful. 

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