Valence structure index

The Xs is the valence structure index introduced by Sabljic and is defined as... 

Individual formal valence structures of conjugated hydrocarbons are excellent substrates for research in chemical graph theory, whereby many of the concepts of discrete mathematics and combinatorics may be applied to chemical problems. The lecture note published by Cyvin and Gutman (Cy-vin, Gutman 1988)) outlines the main features of this type of research mostly from enumeration viewpoint. In addition to their combinatorial properties, chemists were also interested in relative importance of Kekule valence-bond structures of benzenoid hydrocarbons. In fact, as early as 1973, Graovac et al. (1973) published their Kekule index, which seems to be one of the earliest results on the ordering of Kekule structures These authors used ideas from molecular orbital theory to calculate their indices... 

Valence path connectivity index of order h = 0-6 Valence cluster connectivity index of order h = 3-6 Valence chain connectivity index of order h = 3-6 Valence path-cluster connectivity index of order h = 4-6 Total structure index... 

One can construct other indices for benzene character of benzenoid hydrocarbons, which we will designate as B in order to differentiate them from the index B. For example, one can consider the Kekule index, which assigns to individual Kekule valence structures an index derived from local properties of molecular orbitals, and take the average over all Kekule structures. Even though for many Kekule structures this index is bigger than the Kekule index of benzene, the average Kekule index appears smaller for polycyclic benzenoid hydrocarbons than it is for benzene. However, for an index to reflect benzene character, one expects certain trends among structurally related benzenoids to be satisfied, such as... 

Nevertheless, the Kekule index, which assigns to individual Kekule valence structures a numerical value, is of some interest, as it reflects the relative importance of individual Kekule valence structures, a topic which has received some but apparently not sufficient attention in the literature. 

Compounds are generally classified according to their fully unsaturated parent compound (but see below). Thus substituted, partially saturated, and fully saturated derivatives of, for example, pyrrole are all indexed under pyrrole. Benzo and similar derivatives are included under the most unsaturated parent system (e.g., quinoline, thienofuran, etc.). For any given heterocyclic parent only one indicated hydrogen isomer appears in the text, typically the most stable or the lowest numbered form thus all instances of pyran, whether of the 2H- or 4H-form, are indexed under 2H-pyran. The charges and additional valences for any heterocyclic parent structure are not indicated. 

The concept of the molecular connectivity index (originally called branching index) was introduced by Randic [266]. The information used in the calculation of molecular connectivity indices is the number and type of atoms and bonds as well as the numbers of total and valence electrons [176,178,181,267,268]. These data are readily available for all compounds, synthetic or hypothetical, from their structural formulas. All molecular connectivity indices are calculated only for the non-hydrogen part of the molecule [269-271]. Each non-hydrogen atom is described by its atomic 6 value, which is equal to the number of adjacent nonhydrogen atoms. For example, the first-order Oy) molecular connectivity index is calculated from the atomic S values using Eq. (38) ... 

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. 

There are two chair forms and two types of valences (axial and equatorial). The conversion of one chair form to the other interconverts the axial and equatorial valences (i.e., a valence which is axial in one chan form is equatorial in the other chair form and vice versa). In the structures below one of the carbons is indexed with a star ( ) to help keep track of it. 

Various reactivity indices have been derived for benzenoid hydrocarbons from the following purely topological approaches the Huckel model (HMO), first-order perturbation theory (PMO), the free electron MO model (FEMO), and valence-bond structure resonance theory (VBSRT). Since many of the indices that have been known for a long time (index of free valence Fr, self-atom polarizability ir , superdelocalizability Sr, Brown s index Z, cation localization energy Lr+, Dewar reactivity number Nt, Brown s para-localization energy Lp) have been described in detail by Streitwieser in his well-known volume [23] we will refer here only to some more recent developments. 

Yokono et al. [85] have suggested that the results obtained by Lewis and Edstrom [84] can be understood in terms of the maximum value of the index of free valence as calculated by the HMO method. However, as Herndon [30] has shown, some discrepancies occur when the free valence approach is applied to the experimental findings. He found that the structure count ratio for the single position in each compound that would give rise to the most highly resonance stabilized radical is a reliable reactivity index to correlate and predict the qualitative aspects of the thermal behaviour of benzenoid hydrocarbons. 

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