Molecular bodies

Semiclassical Concepts of Molecular Bodies and Functional Groups... 

Quantum Chemical Representations of Molecular Bodies and their Subdivisions Using Fragmentation Schemes... 

One approach to the approximate representation of molecular bodies is based on molecular isodensity contours, MIDCOs, defined with respect to some fixed nuclear configuration K and some electron density threshold a. A MIDCO G(a,K) is defined (in the fixed nuclear configuration approximation) as the collection of all those points r of the three-dimensional space where the electronic density is equal to the threshold a ... 

A molecule contains a nuclear distribution and an electronic distribution there is nothing else in a molecule. The nuclear arrangement is fully reflected in the electronic density distribution, consequently, the electronic density and its changes are sufficient to derive all information on all molecular properties. Molecular bodies are the fuzzy bodies of electronic charge density distributions consequently, the shape and shape changes of these fuzzy bodies potentially describe all molecular properties. Modern computational methods of quantum chemistry provide practical means to describe molecular electron distributions, and sufficiently accurate quantum chemical representations of the fuzzy molecular bodies are of importance for many reasons. A detailed analysis and understanding of "static" molecular properties such as "equilibrium" structure, and the more important dynamic properties such as vibrations, conformational changes and chemical reactions are hardly possible without a description of the molecule itself that implies a description of molecular bodies. 

In the general scheme described in subsequent sections, a functional group is regarded as a fuzzy body of electronic charge cloud, a fuzzy subset of the electronic charge density cloud of the complete molecule. In this context, a functional group is a special case of a fuzzy fragment of a molecular body, obtained by some subdivision... 

A density domain DD(a,K) represents a formal molecular body at an electronic density threshold a and nuclear configuration K. A body DD(a,K) may be a single piece or it may be a collection of several disconnected pieces, the maximum connected components DD (a,K) of DD(a,K) ... 

There is no diflBculty in the imdamped plane motion which corresponds to the wagging of a rigid dipolar group elastically connected to the rest of a molecule which rotates freely as a whole, since in this case normal modes of motion exist, and are indeed obvious. For simplicity we take the mass centre of the polar group to be fixed in position within the molecule. If the angular position of the molecular body is given by a and that of the... 

A fuzzy set B is called an R-deficient set if B has none of the point symmetry elements of family R. However, by analogy with the case of crisp sets, it takes only infinitesimal distortions to lose a given symmetry element. Consequently, unless further restrictions are applied, the total mass difference between a fuzzy set of a specified symmetry and another fuzzy set that does not have this symmetry can be infinitesimal. As a result, i -deficient fuzzy sets and fuzzy R sets can be almost identical. Nevertheless, the actual symmetry deficiencies of fuzzy continua, such as formal molecular bodies represented by fuzzy clouds of electron densities, can be defined in terms of the deviations from their maximal R subsets and minimal R supersets, defined in subsequent text. 

However, the need for a better description of the formal molecular body, the need to account for molecular volume effects, the necessity to describe finer details of changes in electron distributions during conformational changes and chemical reactions, and the requirement of a more precise evaluation of molecular similarity are the factors which have motivated chemists to move beyond the stereochemical skeletal shape concept. 

It is useful to emphasize that the above representation of a formal molecular body by a single level set F(a) and a single nuclear arrangement K involves two simplifications [109] ... 

The model refers to a specified density value a along the contour. Those points of space where the electronic density is less than this threshold value a are not regarded to belong to the formal molecular body considered. 

Level sets F(a) [as well as the closely related density domains DD(a), as we shall see in the next section] provide a representation of formal molecular bodies. A similar definition gives a useful concept of a formal molecular surface the concept molecular isodensity contour surface (MIDCO). For any formal nuclear configuration K, it is possible to define a surface by choosing a small value a for the electronic density, and by selecting all those points r in the 3D space where the density p(r) happens to be equal to this value a, that is, where equation (2.3) is fulfilled. For an appropriate small value a, this contour surface may be regarded as the surface of the essential part of the molecule and, in short, it may be referred to as the molecular surface. These surfaces, the molecular isodensity contour surfaces, or MIDCO s, are denoted by G(a) and are defined as... 

For a continuous function, such as the electronic density p(r), all points r fulfilling equation (2.3) do form a continuous surface. Consequently, the terms contour surface and isodensity surface are appropriate for G(a). For the study of the 3D shape properties of molecular bodies, represented by level sets F(a) of electronic charge densities, it is sufficient to study the shape of their boundaries these boundaries are the MIDCO s G(a). 

The Density Domain Approach (DDA) to chemical bonding has been proposed [109] as a tool that is able to describe the global properties as well as the fine details of the full, three-dimensional bonding pattern within molecular bodies. 

F(a) and G(a), are now considered as a single family, denoted by DD(a). A DD(a) set can be regarded as the molecular body at an electronic density threshold a. Indeed, if we imagine that we can actually see molecules, and the sensitivity of our eyes is adjustable to notice only electronic densities that are equal to or greater than the value a, we would then see these DD(a) sets as the molecular bodies. 

The Density Domain Approach to chemical bonding is based on the topological analysis of the dominant shape variations of the molecular body DD(a) [or, equivalently, those of the G(a) contour surface] regarded as a function of the density parameter a. 

We call such a sequence a topological sequence of families of density domains or, since each family DDj(aj) represents a formal molecular body DD(aj) at the density level a=aj, a topological sequence of molecular bodies. Below we shall discuss several examples as illustrations of the DD approach. The techniques of actual topological characterization will be described in the following chapters. 

It is worth emphasizing that there are only finitely many (actually only three) density threshold ranges of the water molecule which are distinguishable using the simplest topological criterion of identifying maximum connected components, the density domains DD (a) of the molecular bodies DD(a). 

Anyone with nostalgia for tradition may obtain some consolation from our next observation the ethyl group is a density domain functional group of the ethanol molecule. This is evident if one follows the changes in the density domains as the density threshold a is further lowered. One by one, the two remaining hydrogen density domains are linked up with the earlier DD3(a,C,H,C,H,H) density domain of a=0.291, as shown by the molecular bodies DD(a) at a=0.284 and at a=0.282. The DD family making up the molecular body DD(0.282) consists of only two density domains,... 

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