Fuzzy electronic distribution

The formal "bodies" of molecules do not have boundaries and the actual shape of molecules is determined by the fuzzy electron distribution. Realistic models describing molecular shapes and chemical bonding must reflect this natural fuzziness [27]. 

Further, ions are not hard, billiard ball like spheres. Since the wave functions that describe the electronic distribution in an atom or ion do not suddenly drop to zero amplitude at some particular radius, we must consider the surfaces of our supposedly spherical ions to be somewhat fuzzy. A more subtle complication is that the apparent radius of an ion increases (typically by some 6 pm for each increment) whenever the coordination number increases. Shannon10 has compiled a comprehensive set of ionic radii that take this into account. Selected Shannon-type ionic radii are given in Appendix F these are based on a radius for O2- of 140 pm for six coordination, which is close to the traditionally accepted value, whereas Shannon takes the reference value as 126 pm on the grounds that it gives more realistic ionic sizes. For most purposes, this distinction does not mat-... 

The above fuzzy electron density membership functions reflect the relative contributions of the fuzzy, three-dimensional charge clouds of the various molecular electron density distributions to the total electronic density of molecular family L. 

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. 

All aspects of molecular shape and size are fully reflected by the molecular electron density distribution. A molecule is an arrangement of atomic nuclei surrounded by a fuzzy electron density cloud. Within the Born-Oppenheimer approximation, the location of the maxima of the density function, the actual local maximum values, and the shape of the electronic density distribution near these maxima are fully sufficient to deduce the type and relative arrangement of the nuclei within the molecule. Consequently, the electronic density itself contains all information about the molecule. As follows from the fundamental relationships of quantum mechanics, the electronic density and, in a less spectacular way, the nuclear distribution are both subject to the Heisenberg uncertainty relationship. The profound influence of quantum-mechanical uncertainty at the molecular level raises important questions concerning the legitimacy of using macroscopic analogies and concepts for the description of molecular properties. ... 

The three-dimensional shape of this fuzzy body of the electronic distribution has many important features not revealed by the simple, skeletal ball and stick model. One of the most important tasks of topological shape analysis of molecules is the precise analysis and concise description of the three-dimensional electronic charge distributions, such as that illustrated by the selected MIDCO s of allyl alcohol in Figure 1.2. Various methods and computational techniques of such topological shape analyses are discussed in detail in this book. 

However, real molecules are quantum mechanical objects and they do not have a finite body defined in precise geometrical terms and a finite boundary surface that contains all the electron density of the molecule. The peripheral regions of a molecule can be better represented by a continuous, 3D electronic charge density function that approaches zero value at large distances from the nuclei of the molecule. This density function changes rapidly with distance within a certain range, but the change is continuous. The fuzzy, cloud-like electronic distribution of a molecule is very different from a macroscopic body [251], and no precise, finite distance can be specified that could indicate where the molecule ends. No true molecular surface exists in the classical, macroscopic sense. 

This contrast is in part due to the difficulty of invoking fuzzy three-dimensional electron density clouds in a mechanistic interpretation of reactions, which is also hindered by the perceived classical nature of the concept of localization. Classical objects, often used in analogies when modeling molecular structures, usually exhibit localized features, yet the very concept of localization apparently conflicts with the delocalized nature of molecular electron distribution and the Heisenberg uncertainty relation. 

When a molecule interacts with its surroundings, or when it takes part in a chemical reaction, it is the properties on the molecular level which determine its chemical behaviour. Explanations of this are usually expressed in terms of intrinsic properties which relate to the electronic distribution over the molecular framework as well as to conformational and steric effects. However, such intrinsic properties cannot be measured directly. What can be measured are macroscopic, observable, manifestations of the intrinsic properties. These observed properties are then related back to the molecular level by physical-chemical models. For instance, the rather fuzzy concept of nucleophilocity is usually understood as the ability of a molecule to transfer electron density to an electron deficient site. A number of observable properties can be related to this ability, e.g. ionization potential, basicity as measured... 

The characterization of the interrelations between chemical bonding and molecular shape requires a detailed analysis of the electronic density of molecules. Chemical bonding is a quantum mechanical phenomenon, and the shorthand notations of formal single, double, triple, and aromatic bonds used by chemists are a useful but rather severe oversimplification of reality. Similarly, the classical concepts of body and surface , the usual tools for the shape characterization of macroscopic objects, can be applied to molecules only indirectly. The quantum mechanical uncertainty of both electronic and nuclear positions within a molecule implies that valid descriptions of both chemical bonding and molecular shape must be based on the fuzzy, delocalize properties of electronic density distributions. These electron distributions are dominated by the nuclear arrangements and hence quantum mechanical uncertainly affects electrons on two levels by the lesser positional uncertainty of the more massive nuclei, and by the more prominent positional uncertainty of the electrons themselves. These two factors play important roles in chemistry and affect both chemical bonding and molecular shape. 

Some of the fundamental relations of fuzzy set theory and the actual formulation of fuzzy set methods [59-63] appear ideally suitable for the description of the fuzzy, low-density electron distributions [64. A molecular electron density exhibits a natural fuzziness, in part due to the quantum mechanical uncertainty relation. Evidently, a molecular electron distribution, when considered as a formal molecular body, is inherently fuzzy without any well-dehned boundaries. A molecule does not end abruptly, since the electron density is gradually decreasing with distance from the nearest nucleus, and becomes zero, in the strict sense, at infinity. In fact, the same consideration has been the motivation for the compactihcation technique described in the proof of the holographic electron density theorem. For a correct description of molecules, the models used for electron densities must exhibit the natural fuzziness of quantum chemical electron distributions [64. ... 

One of the important properties of an atom or ion is its size. We often think of atoms and ions as hard, spherical objects. According to the quantum mechanical model, however, atoms and ions do not have sharply defined boundaries at which the electron distribution becomes zero. (Section 6.5) The edges of atoms and ions are therefore a bit "fuzzy." Nevertheless, we can define their sizes in several different ways based on the distances between atoms in various situations. 

The infimum /(A, Aff, , . ./>) = inf/. /(A,Aff,uf, ,/>) taken over all the allowed positionings and partitionings P gives another symmetry deficiency measure. These symmetry deficiency measures are equally applicable to discrete sets, crisp continuum sets, and fuzzy sets, including nuclear distributions and fuzzy electron density distributions of molecules, molecular fragments, and functional groups. 

The fuzzy membership function defined above reflects the actual electronic charge distribution of functional group within the given molecule X, without directly involving any other density contributions from other functional groups of the molecule. 

For those of you who didn t read the first chapter in this book or the first chemistry book in this series, or if you just plain forgot, fuzzy orbitals represent probability distributions, which are representations of the probability of finding an electron in any given portion of space. The darker the shading in a probability distribution, the more likely you ll find an electron in that place. 

Whereas the molecular center of mass is of importance in both dynamics and spectroscopy, a formal center of the electronic density distribution has direct significance in shape characterization. A suitable definition of this latter center may differ from the molecular center of mass. The fuzzy set model of electron densities is represented by the... 

Following the principles of the ZPA approach, these symmetry deficiency measures are generalizations of the folding-unfolding approach, equally applicable to crisp continuum sets and fuzzy sets, for example, to entire electron density distributions of molecules and various molecular fragments representing fuzzy functional groups. 

Figure 1.2 The three-dimensional, fuzzy "body" of the charge density distribution of allyl alcohol can be represented by a series of "nested" molecular isodensity contours (MIDCO s). Along each MIDCO the electronic density is a constant value. Three such MIDCO s are shown for the constant electron density values of 0.2, 0.1, and 0.01 (in atomic units), respectively. A contour surface of lower density encloses surfaces of higher density. These MIDCO s are analogous to a series of Russian wooden dolls, each larger doll enclosing a smaller one. These ab initio MIDCO s have been calculated for the minimum energy conformation of allyl alcohol using a 6-31C basis set.

Because the boundary of an atomic orbital is fuzzy, the orbital does not have an exactly defined size. To overcome the inherent uncertainty about the electron s location, chemists arbitrarily draw an orbital s surface to contain 90% of the electron s total probability distribution. In other words, the electron spends 90% of the time within the volume defined by the surface, and 10% of the time somewhere outside the surface. The spherical surface shown in Figure 5-13b encloses 90% of the lowest-energy orbital of hydrogen. 

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