Atomic properties additivity

When additivity of atomic properties is valid then the skeleton S disappears and Eq. (1) can be rewritten as Eq. (2). 

For any molecule, additivity of atomic properties requires as many variables as there are different atom types contained in the molecule. For example, for acetic add, C2H+O2, three different atomic increments are needed, one each for a carbon, a hydrogen, and an oxygen atom. 

Clearly, simple additivity of atom properties will no longer suffice, as the contribution of an atom will diminish the hirther it is away from the atom whose property has to be estimated. In the following, we present two methods of accounting for the influence of one atom on another, attenuated over the distance between the two atoms. 

Molecules are usually represented as 2D formulas or 3D molecular models. WhOe the 3D coordinates of atoms in a molecule are sufficient to describe the spatial arrangement of atoms, they exhibit two major disadvantages as molecular descriptors they depend on the size of a molecule and they do not describe additional properties (e.g., atomic properties). The first feature is most important for computational analysis of data. Even a simple statistical function, e.g., a correlation, requires the information to be represented in equally sized vectors of a fixed dimension. The solution to this problem is a mathematical transformation of the Cartesian coordinates of a molecule into a vector of fixed length. The second point can... 

Equation 1-5 was written for a sample containing a single element upon which monochromatic x-rays are incident. In so far as x-ray absorption is an atomic property, the mass absorption coefficients for other samples are additive functions of the weight-fractions of the elements, free or combined, that are present that is,... 

The similarity of the results obtained for finite elusters and the infinite slab allows to eonclude in favour of the validity of the eluster model of adequate size (6 or 8 molybdenum atoms). In addition to the chemisorption of organic molecules on solid surfaces which is generally considered as a localized phenomenon, the interaction between molybdenum oxide and an adsorbate can also be represented by a loeal eomplex formed by a finite eluster and the adsorbed molecule. Indeed, the study of the evolution of the electronic properties as a funetion of the cluster size shows that, for a eluster eontaining 6 or 8 molybdenum atoms, most of the electronic properties converge towards limit values. This eonvergence is sensitive to the direction of the cluster growth. On the other hand, the electronic properties of the (001), (010) and (100) faces are not identieal, the type of surface atoms being different these results allow to predict that the characteristics of the chemisorption step will depend on the particular face on which it takes place. 

An important advantage of the finite atoms defined by AIM is that they do not overlap, which is not generally true for orbital-defined atoms. Each atom has a sharp and well-defined boundary inside the molecule, given by its interatomic surfaces. The atoms fit exactly into each other, leaving no gaps. In other words, the shape and the volume of the atoms are additive. This is true also for other physical properties of an atom, such as the electron population and the charge, as seen in Table 6.2 and as indeed has been shown to be true for all other properties. (Bader 1990, Popelier 1999). 

In contrast, the AIM theory provides a clear definition of an atom in a molecule as a space-filling object, from which all its properties can be obtained. The properties of these atoms are additive to give the corresponding molecular property. 

The atomic properties satisfy the necessary physical requirement of paralleling the transferability of their charge distributions - atoms that look the same in two molecules contribute identical amounts to all properties in both molecules, including field-induced properties. Thus the atoms of theory recover the experimentally measurable contributions to the volume, heats of formation, electric polarizability, and magnetic susceptibility in those cases where the group contributions are found to be transferable, as well as additive additive [4], The additivity of the atomic properties coupled with the observation that their transferability parallels the transferability of the atom s physical form are unique to QTAIM and are essential for a theory of atoms in molecules that purports to explain the observations of experimental chemistry. 

Pettifor s structure maps additional remarks. We have seen that in a phenomenological approach to the systematics of the crystal structures (and of other phase properties) several types of coordinates, derived from physical atomic properties, have been used for the preparation of (two-, three-dimensional) stability maps. Differences, sums, ratios of properties such as electronegativities, atomic radii and valence-electron numbers have been used. These variables, however, as stressed, for instance, by Villars et al. (1989) do not always clearly differentiate between chemically different atoms. 

The electron density is a continuous function that is experimentally observable, hence uniquely defined, at all points in space. Its topology can be described in terms of the distribution of its critical points, i.e. the points at which the electron density has a zero gradient in all directions. There are four kinds of critical point which include maxima (A) usually found near the centres of atoms, and minima (D) found in the cavities or cages that lie between the atoms. In addition there are two types of saddle point. The first (B) represents a saddle point that is a maximum in two directions and a minimum in the third, the second (C) represents a saddle point that is a minimum in two direction and a maximum in the third. One can draw lines of steepest descent connecting the maxima (A) to the minima (D), lines whose direction indicates the direction in which the electron density falls off most rapidly. Of the infinite number of lines of steepest descent that can be drawn there exists a unique set that has the property that, in passing from the maximum to the minimum, each line passes successively through a B and a C critical point. This set forms a network whose nodes are the critical points and whose links are the lines of steepest descent connecting them. 

Most spectacular exceptions to the property additivity scheme come from nitrogen atoms that can change their basicity, their polar surface area, their number of donors, etc. If this becomes an issue for a library, the reagents can be initially split according to each case and a different offset used. 

A very good picture of some of the properties of hydrogen compounds can be obtained with the aid of an electrostatic model, but we must be careful not to conclude that all the hydrogen compounds are therefore ionic in character. In addition to NHS, there are two other nitrogen compounds, hydrazine H2NNH2 and hydroxylamine NH2OH, which have properties not fundamentally dissimilar to those of ammonia. It is not possible to devise a plausible electrostatic model for these compounds because of the bond between like atoms. In addition to water there is also the compound hydrogen... 

As noted above, the hardness or softness of an acidic or basic site is not an inherent property of the particular atom at that site, but can be influenced by the substituent atoms The addition of soft, polarizable substituents can soften an otherwise hard center and the presence of electron-withdrawing substituents can reduce the softness of a site. The acidic boron atom is borderline between hard and soft. Addition of three hard, electronegative fluorine atoms hardens the boron and makes it a hard Lewis acid. Conversely, addition of three soft, electropositive hydrogens54 softens the boron and makes it a soft Lewis acid. Examples of the difference in hardness of these two boron acids are... 

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