Topological description

Group additivity methods must be derived as a consistent set. It is not correct to combine fragments from different group additivity techniques, even for the same property. This additivity approximation essentially ignores effects due to the location of one functional group relative to another. Some of these methods have a series of corrections for various classes of compounds to correct for this. Other methods use some sort of topological description. 

Figure 3 Mutation of a ligand Asp into Asn in solution and bound to a protein, (a) Thermodynamic cycle, (b) Dual topology description a hybrid ligand with two side chains. Blocks are used to define the hybrid energy function [Eq. (14)]. Only the ligand is shown the environment is either solvent or the solvated protein, (c) Single-topology description.

This structural similarity is also reflected in the amino acid sequences of the domains, which show 40% identity. They are thus clearly homologous to each other. The motif structures within the domains superpose equally well but their sequence homology is less, being around 30% between motifs 1 and 2 and 20 Xi between 3 and 4. This study, however, clearly shows that the topological description in terms of four Greek key motifs is also valid at the structural and amino acid sequence levels. 

Fig. 3. Topological description of the Li6P6Si2 rhombododecahedron face-capped Pe octahedron.

One further theoretical method that merits consideration at this point is the topological theory of molecular structure exemplified by Bader (1985, 1990). In this method a topological description of the total electron density in the molecule is used. A major advantage of this method is that it allows the total interaction between various centres to be probed. Cremer et al. (1983) used the Bader method to examine the homotropylium cation [12] and concluded that it was indeed homoaromatic. 

The aim ofthe present study is double i) to show that BET can be used as a tool for analyzing the adiabatic PESs and localizing the diabatic crossings which govern the overall electron changes ii) to provide a topological description of the three-electron bonds. 

The reader s attention is drawn to the discussion in Sections 14.3 and 14.4 which shows that all chemical bond models are equivalent because they all reduce to this same topological description. The derivation here is based on the ionic model because it is the simplest and most convincing. 

An improved version of the MTD approach would be of real interest as a mono-parametric Free-Wilson-type method (due to their meaning, the MTD and Free-Wilson parameters belong to the same class). The topological description of the molecular structure assures the easy to use character of the MTD method, and the hypermolecule concept allows to study widely differing structures within the data basis. 

Of course, the fitting of the data can give rise to some compensation effects and the accuracy of the determination of the wetted areas (related to the topological description of the liquid flow on the catalyst) is limited by the knowledge of all the physico-chemical properties (vapour pressure, viscosities etc.) and of the correlated parameters (mass transfer coefficient etc...) involved in the model. 

Figure 2.2 Selected families (DDj(aj)) of density domains of the water molecule, as calculated with the GAUSSIAN 90 ab initio program [253] and the GSHAPE 90 molecular shape analysis program [254], using a 6-3IG basis set. There are only two topologically different types of families of density domains either a single density domain, or a family of three density domains. The sequence of topologically distinct cases provides a topological description of chemical bonding.

Convexity and curvature properties. In the above discussion and examples we have already used the concepts of convexity and locally convex domains in an intuitive manner. Whereas our goal is to provide a topological shape characterization for molecules, we shall often use geometrical tools at intermediate steps toward a topological description. These steps often involve the concepts of convexity, curvature, and a characterization of critical points of functions. 

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