Topological bond order

FIGURE 14. CC bond orders n and bond ellipticities i of cyclopropyl homoconjugated molecules (a) norcaradiene, (b) bicyclo[2.1.0]pentene, (c) bicyclo[3.1. OJhexenyl cation. On the right, the preferred mode of electron delocalization is indicated by dashed lines. Also given is the number of delocalized electrons as calculated from topological bond orders. See text... 

Hosoya, H., and Murakami, M., Topological index as applied to p-electronic systems. II. Topological bond order. Bull. Chem, Soc. J, 48(12), 3512-3517 (1975),... 

Analogous to the topological atomic charge definition is that of topological bond orders, defined as the limit of... 

The topological bond order thus defined can be interpreted as the one-electron distribution over the bonds in a molecule, analogous to the one-electron distribution over all atoms given by the topological atomic charges ... 

One way of getting rid of distortions and basis set dependence could be that one switches to the formalism developed by Bader [12] according to which the three-dimensional physical space can be partitioned into domains belonging to individual atoms (called atomic basins). In the definition of bond order and valence indices according to this scheme, the summation over atomic orbitals will be replaced by integration over atomic domains [13]. This topological scheme can be called physical space analysis. Table 22.3 shows some examples of bond order indices obtained with this method. Experience shows that the bond order indices obtained via Hilbert space and physical space analysis are reasonably close, and also that the basis set dependence is not removed by the physical space analysis. 

Bond Order Indices Calculated Using the Topological Definition... 

The topological difference between cycl[3,3,3]azine (2) and cycl-[4,3,2]azine (85) follows from a comparison of the isoelectronic anions (86) and (87) using HMO calculations.108 Since the HOMO s of 86 and 87 are nonbonding, anions, cations, and radicals are of comparable stability. Charge density and bond orders of the cation and anion of 86 depend on the symmetry of the molecule the (positive or negative) charges are located in the starred positions (1,3. ..). There are three different bond orders, as shown in formula 86. 

The perturbation of the PMD symmetry is accompanied by a decrease in the charge alternation and by the appearance of bond alternation from one end group to another. The bond alternation amplitude has been revealed to be proportional to the asymmetry degree, which can be calculated as the difference of topological indexes A 3>12 = 3>01 — 02 The effect is maximum if 01 >45° and 02 < 45°. If A<1>12 = 90°, the ideal polyene state is reached. In the excited state of the asymmetrical PMD, bond orders are essentially equalized. 

FIGURE 13. Topological CC bond orders n of homocyclopropenium (44) and homotropenylium cation (45) calculated from the MP2 electron density distribution p(r) at the bond critical points. Note that n values for C1,C3 of 44 and C1,C7 of 45 correspond to interaction indices. MP2 bond lengths (in A) are also given44,56... 

Several alternatives to the Mulliken population have been presented that attempt to provide more rigorous estimates of the charges on atoms in molecules or clusters although not all have been applied in chemisorption and catalysis. We quote the Natural Bond Order analysis, and the elegant topological analysis of the electron density or of the electron localization function, ELF, introduced by Becke and Edgecombe. " ELF analysis has... 

The bond order depends on the topology of a molecule, and since the bond order correlates with the bond length, the geometry of the molecule is to some extent defined by its topology. 

The value of the charge density at a bond critical point can be used to define a bond order (Bader et al. 1983 Cremer and Kraka 1984). The molecular graphs for ethane, ethylene, and acetylene are shown in Fig. 2.8. In each case the unique pair of trajectories associated with a single (3, — 1) critical point is found to link the carbon nuclei to one another. Multiple bonds do not appear as such in the topology of the charge density. Instead, one finds that the extent of charge accumulation between the nuclei increases with the assumed number of electron pair bonds and this increase is faithfully monitored by the value of p at the bond critical point, a value labelled p, . For carbon-carbon bonds, one can define a bond order n in terms of the values of Ph using a relationship of the form... 

---