Generalized Similarity Indices

As can be seen from this Table, the original insufficiency of the index r is indeed remedied by the index g p. This result is very interesting since if we realize that the primary source of the increased information content of the index g p is the partial inclusion of electron correlation, then the discrimination between the allowed and forbidden reactions in these cases seems to suggest that certain delicateness of cycloadditions and sigmatropic reactions, which both belong to the class of the so-called multibond reactions [99], can be apparently related to the greater sensitivity of these reactions to the effects of electron correlation. This conclusion, together with the systematic analysis of the role of correlation effects in pericyclic reactivity will be discussed in more details in chapter 8. 

The philosophy of introducing the generalized and integral similarity indices arises from the simple idea not to base the description of the reaction on only two, even if 

however, is not very important since the formalism is entirely general and the same procedure can be applied also to other types of pericyclic processes so that the possibilities of the eventual violations can be systematically analyzed. One of the potential candidates where such a violation seems especially probable are the so-called valence isomerizations. An example in this respect may be the cyclization of 2-substituted butadienes to corresponding [1,1,0] bicyclobutanes [106]. In this connection it would be, of course, interesting to confront our theoretical predictions with experiment, but because of lack of convenient data this question has so far to remain open. 

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The stmcture and composition of DLC may vary considerably and, as a result, so do some of its properties. This is not necessarily a disadvantage since it is often possible to control and tailor these properties to fit specific applications (for instance, the index of refraction). Its properties are generally similar to those of diamond, such as high hardness and chemical inertness, but different in some key areas. As opposed to diamond, DLC has a variable index of refraction and variable electrical conductivity, both a function of hydrogen content. 

Most chemicals used in the procedure will appear in the index. Thus, there will generally be entries for all starting materials, reagents, intermediates, important by-products, and final products. Most products shown in the Tables in the discus.sion sections of this volume are included unless the compounds are quite similar in which case a general descriptive name was entered. Chemicals generally nut indexed included coiimion solvents, standard inorganic acids and bases, reactants shown in the Tables, and compounds cited in the discussion section in connection with other methods of preparation. 

The technology of proximity indices has been available and in use for some time. There are two general types of proximity indices (Jain and Dubes, 1988) that can be distinguished based on how changes in similarity are reflected. The more closely two patterns resemble each other, the larger their similarity index (e.g., correlation coefficient) and the smaller their dissimilarity index (e.g., Euclidean distance). A proximity index between the ith and th patterns is denoted by D(i, j) and obeys the following three relations ... 

Programme algorithms are considered in detail elsewhere e.g. [54] but in general a similarity index or correlation coefficient has to be calculated for each fit due to variations in spectral data. The latter arise from the differences in spectra on different instruments or under different conditions, from additional components in unresolved GC-peaks and from discrepancies due to concentration changes. A yes/no answer can be more closely approached when all spectra are unique and completely reproducible. The best fits within specified correlations are normally printed out and further criteria have to be applied to these to obtain the final answer. This would be straightforward assuming the correct spectrum was on file. In cases where this is not so, some indication of the class and type of compound may be suggested from the list of best fits. 

The application of the Lorentz-Lorenz equation gives a convincing demonstration of the general similarity of the linear response in gas and liquid but its application in the liquid introduces an approximation which has not yet been quantified. A more precise objective for the theory would be to calculate the frequency dependent susceptibility or refractive index directly. For a continuum model this may lead to a polarizability rigorously defined through the Lorentz-Lorenz equation as shown in treatments of the Ewald-Oseen theorem (see, for example Born and Wolf, plOO),59 but the polarizability defined in this way need not refer to one molecule and would not be precisely related to the gas parameters. 

Molecular Similarity and QSAR. - In a first contribution on the design of a practical, fast and reliable molecular similarity index Popelier107 proposed a measure operating in an abstract space spanned by properties evaluated at BCPs, called BCP space. Molecules are believed to be represented compactly and reliably in BCP space, as this space extracts the relevant information from the molecular ab initio wave functions. Typical problems of continuous quantum similarity measures are hereby avoided. The practical use of this novel method is adequately illustrated via the Hammett equation for para- and me/a-substituted benzoic acids. On the basis of the author s definition of distances between molecules in BCP space, the experimental sequence of acidities determined by the well-known a constant of a set of substituted congeners is reproduced. Moreover, the approach points out where the common reactive centre of the molecules is. The generality and feasibility of this method will enable predictions in medically related Quantitative Structure Activity Relationships (QSAR). This contribution combines the historically disparate fields of molecular similarity and QSAR. 

The Carbo index Rab was originally proposed as a method of comparing molecules in terms of their electron density q [1066]. In a more general version (eq. 212) this similarity index can be applied to compare any structural properties and Pb of two molecules A and B. 

Isomorphism is similarity of crystal shape, unit-cell dimensions. and structure between substances of similar chemical composition. Ideally, the substances are so closely similar that they can generally form a continuous series of solid solutions. The degree of similarity between crystals can be calculated using Kalman s parameters the unit cell similarity index, IT. and isostructurality index /,( ) ... 

The C-NMR Reproducibility-based Retrieval ( C13RR )system uses only chemical shifts as features as these appear to contain sufficient characteristic information. Peak intensities are not useful since they exhibit a very poor reproducibility, and mxiltiphcities cannot be used because the general concept requires that the features be of a continuous nature. The similarity index was developed on the basis of a reproducibihty model of chemical shifts using 200 pairs of repUcate C-NMR spectra. The database contained 6000 spectra originating from the Netherlands Organisation for Applied Scientific Research (TNO). 

This generalized cosine index is often called the Carbo index. Naturally the Carbo index is limited to the range (0,1), where Cab = 1 means perfect similarity. Still more (dis)similarity measures have been introduced and will be discussed further in this chapter. The range of the Carbo index naturally agrees with the Schwartz integral inequality " ... 

In the simplest case of pair density matrices this procedure leads to the introduction of the so-called second order similarity index g p. This index, introduced in the direct analogy with the first order index r is generally defined by eq. (73), where Pp(1,2) and pp(1,2) are the pair density matrices of the molecules R and P respectively. 

Although the correct description of forbidden reaction generally requires the variation of the reaction coordinate within the range (0, -x/2), the second order similarity index (116) is independent of the direction of the reaction path and both the allowed and forbidden reactions can be described by cp varying within (0, tc/2)... 

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