Absolute shape descriptors

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. 

Other methods can be used to quantify shape fluctuations and flexibility. The alternatives listed briefly here employ absolute shape descriptors ... 

The STI values vary within the interval 0 < 5 < 4. When calculated from the relation of both global curvatures [each of which can be either concave ( + ), flat (0), or convex (-)], where c, > C2, the STI gives an expression for the regional shape of every surface point. The shape varies continuously among five basic shape descriptors, namely, bag ( + / + ), cleft (+ /O), saddle ( + /-), ridge (0/ - ), knob (-/-), and as a special case, plateau (0/0). However, information about the absolute curvature is lost during the process of STI calculation. 

A relative shape descriptor is a d-dimensional function associated with a pair of molecules (in their p-dimensional models). These functions measure relative molecular similarity instead of the absolute (i.e., nonrelative) shape of the corresponding model. In this category we find, among others, root-mean-square (rms) deviations, °-2 quantum similarity measures, and some polymer compactness measures. 

The discussion in this section has dealt with descriptors of absolute shape, which are very discriminating functions. There are also numerous descriptors of relative shape (e.g., deviations between conformations), which characterize flexibility. We deal briefly with these descriptors in the next section. 

The development and application of molecular shape descriptors is an active area in computational chemistry and biology. The main goal of our work is to develop mathematical descriptors that can determine whether two molecules have comparable shapes. In this chapter we present a series of molecular shape descriptors developed oti the basis of molecular vdW space. The molecules are treated in the hard sphere approximation, as a body composed from a collection of atomic fused spheres. Each sphere is centered in the corresponding nucleus and it is characterized by its Cartesian coordinates and by its vdW radius, r. These molecular vdW shape descriptors depend only on the internal structure of the molecule, being invariants to any translation and rotation movement. Consequently, they may inform us that two molecules have comparable shapes, but since they carry no information about the absolute orientation or position of the molecule, they are not useful for computing molecular superposition. 

To take into account the absolute contribution that a single double-bond makes to the whole size and shape of alkene molecules, second-grade structural parameters were derived from a - molecular graph [Zhang et ai, 1997]. The topological descriptors representing the size w and the shape related to the presence of a double bond are, respectively ... 

As noted above, adoption of the CIS method necessitates use of the FIXDM keyword and rules out -dependent NBO options such as second-order perturbative analysis or Fock matrix deletions. However, numerous NBO descriptors remain useful for analyzing the CIS wave function, including NBO Lewis structure, orbital shapes and occupancies, and NRT weightings and bond orders. In the following, we illustrate the use of these descriptors for the three lowest CIS excited states. Our emphasis is not on the absolute accuracy of the CIS wave functions themselves but on how the NBO/NRT descriptors provide a chemist s picture of the wave functions, useful for predicting structural rearrangements... 

Relative descriptors, which usually are less discriminating than the absolute descriptors, are somewhat limited in their usefulness. For example, a given value of rms deviation between two structures may be the result of a number of completely unrelated conformations, sizes, and shape features. A detailed characterization of shape must employ absolute descriptors or at least the simultaneous use of two or more independent relative descriptors. 

Relative and absolute descriptors also differ from each other at a more fundamental level. If we compare two molecules (either their nuclear geometries or their electron densities), the result will normally depend on how they are oriented relative to each other. In contrast to absolute descriptors, relative descriptors are not invariant if one of the molecules compared is rigidly translated or rotated. Therefore, the proper use of relative descriptors must be accompanied by some sort of optimization in the superposition between two structures. Yet, a maximum superposition (e.g., by minimizing the rms deviation of paired atoms) may not produce a relative orientation that is most relevant in a given comparison of molecular shapes. This is a problem with no unique solution and is still under much research. [See discussions in Ref. 27.]... 

Ultrafast shape recognition (USR) [19] is a recent and unusually rapid descriptor-based shape similarity technique. USR is based on the observation that the shape of a molecule is determined by the relative positions of its atoms. This 3D spatial arrangement of atoms is accurately described by a set of distributions of interatomic distances measured from four strategically located reference points, which are in turn characterized by its first three statistical moments. The shape similarity of two molecules is Anally calculated through an inverse of the sum of least absolute differences in their respective descriptors (full details about this recent technique along with applications can be found in a recent review [20]). 

The short, medium and long axis of the enclosing box gives S, M and L, respectively (Figure 5.2). The ratios of these axes are representative of the shape of the molecule. A low SjL ratio corresponds to a flat molecule, a low MjL ratio to a rod-shaped molecule. These axis ratios showed much stronger correlation than the absolute values of S, M and L. Analogous descriptors are referred to as shadow descriptors in the QSAR literature. 

It should be noted that there are ways for compensating the limitation of PGA as a unique descriptor of a seismic event, either on the hazard definition - relying on notions such as the intensity or cumulative absolute velocity (CAV refer to EPRI (2006), Campbell and Bozorgnia (2011)) in the hazard curve development - or on the structural response (—>spectral shape factor in the fragility definition below). 

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