Bond-order summations

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. 

For multiple edges (unsaturated or aromatic systems) fractional distances are introduced in the distance matrix so that fractional distance sums result on summation if the bond order between vertices i, j is b, then 1/b is the entry in the row/... 

The Natural Bond Orbital analysis of Weinhold [Foster and Weinhold, 1980 Reed, Weinstock etal., 1985 Reed, Curtiss etal., 1988] generates, departing from canonical MOs, a set of localized one center (core, lone pairs) and two center (jt and a bonds) strongly occupied orbitals, and a set of one center (Rydberg) and two center (a, Jt ) weakly occupied orbitals the NBOs. The Natural Bond Orbitals (NBOs) are obtained by a sequence of transformations from the input basis to give, first, the Natural Atomic Orbitals (NAOs), then the Natural Hybrid Orbitals (NHOs), and finally the Natural Bond Orbitals (NBOs). For NAOs, atomic charges can be calculated as a summation of contributions given by orbitals localized on each atom moreover, from NBOs, bond order can be also calculated. 

This implementation requires simple modifications in the property part of the CRYSTAL program [23]. A Lowdin population analysis is introduced for self-consistent DM and the bond-order sums are calculated for atoms of the crystal. The lattice summation in (6.76) is made over the same part of the lattice that has been used in the integrals calculation for the self-consistent procedure (the lattice summation field is defined by the most severe tolerance used in the two-electron exchange integrals calculation). ... 

For the small-size clusters (L = 4,8) the imbalance of the cychcity region and the interaction radius causes a huge difference between and IV as the sum of bond orders inside the interaction radius includes the neighboring cyclic clusters around the central cyclic cluster. The PBCs force atoms outside the central cyclic cluster to be equivalent to the corresponding atoms inside the cluster. At the same time the summation of bond orders inside a cyclicity region gives values that are largely model-size independent, as the DM is calculated for the cyclic cluster under consideration. 

The resulting overlap integrals can be used in eqn (5)-(7) and (10) for the evaluation of delocalization indices, Cioslowski-Mixon bond orders and DAFH analysis respectively. The summation over orbitals in all these equations has to be performed over all bands and all the fc-points from the whole Brillouin zone. 

Here, p is Coulson s bond order and pp is bond order for para-bonds. Both summations assume for indices that r > 5. This is an interesting and much simplified approach to MO ring indices however, the indices reported by Hosoya and co-workers are not quite identical to those of Polansky and Derflinger. Recently, Professor Hosoya informed me that the difference occurs because his group based their work on SCF MO rather than on HMO calculations. 

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