Higher-Order Connectivity Indices

As with the molecular connectivity indices, higher-order shape indices have also been defined. The kappa indices themselves do not include any information about the identity of the atoms. This is the role of the kappa-alpha indices. Tlie alpha value for each atom is a measure of its size relative to some standard (chosen to be the sp -hybridised carbon) ... 

For molecular connectivity indices with orders higher than 2, it is also necessary to specify the subclass of index. There are four subclasses of higher order indices path, cluster, path/cluster, and chain. These subclasses are defined by the type of structural subunits they are describing, a subunit over which the summation is to be taken when the respective indices are calculated. Naturally, the valence counterparts of all four subclasses of higher order indices can be easily defined by analogy, described above for the first-order valence molecular connectivity index. 

Another important consideration is that all results reported here are based on the calculation of zero- and first-order connectivity indexes only. It is possible that die inclusion of higher-order indexes, which in turn means more detailed... 

It follows for this specific choice (the TS -e1 5 e+5 with energy E0), that the Franck-Condon energy Efc E+ E equals the orbital difference s. — e1, because the two-electron parts cancel - here, the upper index t refers to quantities connected with the TS. This result is violated, if higher order terms in the expansion A. 1 are taken into account however, calculations using a LDA + GGA functional yield, that the error does usually not exceed 2%. 

To derive these equations, log P (hydrophobic parameter), MR (molar refrac-tivity index), and MV (molar volume) were calculated using software freely available on the internet (wwwlogP.com, www.daylight.com). The first-order valence molecular connectivity index of substituents was calculated as suggested by Kier and Hall [46,47]. In these equations, is cross-vahdated obtained by the leave-one-out jackknife procedure. Its value higher than 0.6 defines the good predictive ability of the equation. The different indicator variables in these equations were defined as follows. 

Kier and Hall defined [Kier and Hall, 1986 Kier and Hall, 1977b] a general scheme based on the Randic index to calculate also zero-order and higher-order descriptors, thus obtaining connectivity indices of m th order, usually known as Kier-Hall connectivity indices. They are calculated by the following ... 

The map connectivity matrices are another set of map matrices based on partitioning of the —> Randic connectivity index and the higher order —> connectivity indices into contributions arising from paths of length k [Randic and Basak, 2002]. They are defined as... 

For the butanoic acid molecule shown in Figure 8.1, the five bond connectivities are the reciprocal square roots of (1 x 3), (1 x 3), (2 x 3), (2 x 1) and (2 X 1), which gives a molecular connectivity value of 2.977. This simple connectivity index is known as the first order index because it considers only individual bonds, in other words paths of two atoms in the structure. Higher order indices may be generated by the consideration of longer paths in a molecule, and other refinements—such as valence connectivity values, path, cluster and chain connectivities—have been introduced. 

Kier and Hall demonstrated very shortly after the connectivity index was proposed that this index, in combination with structurally related higher order connectivity indices, turned out to correlate well with many physical properties of molecules, in particular, alkanes, and a few molecular properties of molecules having heteroatoms, such as amines, alcohols, etc. In the next section, we outline the construction of the connectivity index. 

Here, we would like to comment on mathematical modification, not of the connectivity index, but the higher-order connectivity indices. 

The higher order connectivity indices supplement the information that is given by the connectivity index x (also referred to as X)- The list of the higher order connectivity indices X. X. X. X, was augmented later by °X- This index can... 

The (/, y)th elements of this matrix are given by the contribution of the path /, j (of length k) to the higher order connectivity index The electrotopological matrix for 3-methylhexane is illustrated in Table 7. 

Sulfur. Thiophene and benzo[ >] thiophene are both aromatic heterocycles, as discussed earlier in this review. Isothiazole is a planar molecule with an aromaticity comparable with those of thiazole and pyrazole, and higher than those of isoxazole and oxazole,122 140 as evaluated on the basis of Bird s aromaticity index A, based upon the statistical degree of uniformity of the bond orders of the ring periphery. Theoretical calculations and experimental data in connection with the aromaticity of isothiazole have been reviewed.141 Thiazole is also viewed as an aromatic molecule, similar to thiophene. It lacks an experimental aromaticity value, but the heat of formation together with bond lengths and angles have been calculated by various computational meth-... 

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