Skeletal bond orders, calculated

Figure 3.8. B-C and B-B bond orders calculated and CH and BH group charges for closo boranes and closo carboranes [B H ]2 [1-CB iHJ" and 1. //-(CB LI, show the skeletal electron distributions in these systems.

The average edge bond order for a particular closo species can be calculated by dividing the total number of skeletal bond pairs ( 4- 1) by the number of edges, which, for exclusively triangular-faced poly-hedra with n vertices, is (3 — 6). This leads directly to the formal boron-boron bond order for those borane anions (ITU) and... 

For deriving the aromaticity index AN proposed by Pozharskii (85KGS867) the sum of absolute values of all differences between the bond orders of n skeletal bonds including those with equal values of the orders is calculated and it is normalized with respect to the number of those differences equaling that of the dual combinations of n... 

A method is developed for calculating even moments of the end-to-end distance r of polymeric chains, on the basis of the RIS approximation for rotations about skeletal bonds. Expressions are obtained in a form which is applicable in principle to arbitrary k, but practical applications are limited by a tremendous increase in the order of the matrices to be treated, with increasing k. An application is made to the PE chain by using the familiar three-state model. Approximate values of the distribution function Wn (r) of the end-to-end vector r, Wn (0), and , are calculated from these even moments. 

The skeletal configuration of polyphenyl molecules, that is, biphenyl, para-terphenyl, pora-quaterphenyl, and me/a-terphenyl, and the numbering of atoms in those molecules are shown in Fig. 1. In our previous papers we calculated the mobile bond orders and discussed the related phenomena for... 

Calculation of the Chains Unperturbed Dimensions. Inspection of the conformational energy maps (Figures 2 and 3) reveals that adoption of the familiar three-state (i.e., T, G", G") scheme is sufficient for application of the RIS theory to these chains.The corresponding statistical weight matrix U, inclusive of both first-order and second-order interactions (those interactions depending on, respectively, one and two skeletal-bond rotations), is represented by ... 

The conformational energies of the lower members of POM, 2,4-dioxapentane and 2,4,6-trioxaheptane are estimated by the empirical force field method. The gauche states of the Internal rotation around the skeletal C—0 bonds are successfully predicted to be of lower energies in both molecules. In order to calculate the unperturbed dimension and dipole moment of POM, RIS approximations are made by using the results obtained from the force field calculations on 2,4,6-trioxaheptane. Although these parameters are significantly different from those estimated earlier, they reproduce the observed values fairly well. 

The polyethylene molecule may be represented skeletally in a planar zigzag form as shown in Fig. 2.12, where I = 0.154 nm and 9 = 109.5°. In order to perform the calculation, the number n of backbone bonds is required. It can be obtained... 

To calculate the p.e.c. the valence electrons of all the skeletal atoms are summed, together with the number of electrons contributed by the ligands (p. 192). This sum is characteristic of the structure of the cluster (Table 11.1). In order to meet cluster bonding requirements, transition metal atoms require ten more valence electrons (to fill the (n — l)d orbitals) than Main Group atoms. 

The Wiener index [86] can be expressed in terms of the distance matrix [87] and equals the half-sum of all distance matrix entries. Randi(5 [88] and Kier and Hall indices of order 0-3 [89] are calculated from coordination numbers of atoms or from values of atomic connectivity. The Kier shape index (order 1-3) [90] depends on the number of skeletal atoms, molecular branching, and the ratio of the atomic radius and the radius of the carbon atom in the sp hybridization state. The Kier flexibility index [90] is derived from the Kier shape index. The Balaban index depends on the row sums of the entries of the distance matrix and the cyclomatic number [92,93]. The information content index and its derivatives (order 0-2) are based on the Shannon information theory [95]. Modifications of the information content index are structural information content, complementary information content, and bond information content [96],... 

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