Bond order, definition

The calculations of local properties of metal-oxide electronic structure [571,581-583] were made in the cychc-cluster model, in the CNDO approximation. As in the CNDO approximation AOs are supposed to be orthogonalized by the Lowdin procedure (see Chap. 6), the definitions of local properties given in Sect. 9.1.1 for nonorthog-onal basis, have to be modified. In particular, the overlap population (9.9) becomes zero in the CNDO approximation, so that the electronic population is defined only by diagonal density matrix elements In (9.6) and in the bond-order definition... 

The most well-known and at the same time the earliest computer model for a molecular structure representation is a wire frame model (Figure 2-123a). This model is also known under other names such as line model or Drciding model [199]. It shows the individual bonds and the angles formed between these bonds. The bonds of a molecule are represented by colored vector lines and the color is derived from the atom type definition. This simple method does not display atoms, but atom positions can be derived from the end and branching points of the wire frame model. In addition, the bond orders between two atoms can be expressed by the number of lines. 

P is the total spinless density matrix (P = P + P ) and P is the spin density matrix (P = p" + P ). For a closed-shell system Mayer s definition of the bond order reduces to ... 

In this chapter, we use the definitions of bond order and valence indices provided by Mayer [4-6] (for a historical account, see Ref. [6a] and for other types of bond indices, see Ref. [6b]). In terms of electronic structure theory, they represent an extension to Mulliken s population analysis. The bond order is defined as... 

Tables 22.1 and 22.2 show how the general principles sketched above are manifested in real systems. The C-C and C-H bond order indices and the C and H valence indices were calculated for ethane, ethene, ethyne, and benzene at the HF/6-31G geometry with various basis sets. The bond order of the C-H bond is close to unity in all cases. The carbon-carbon bonds have bond orders close to one, two and three in ethane, ethene, and ethyne, respectively. In benzene, all C-C bonds have the same bond order, which is close to 1.5. Note that definition (Equation 22.1) yields nonzero bond orders between nonbonded atoms also, and in certain cases,...

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... 

Here Zg is the number of tt electrons provided by atom is essentially an ionization potential for an electron extracted from in the presence of the part of the framework associated with atom r alone (a somewhat hypothetical quantity), is a framework resonance integral, and is the coulomb interaction between electrons in orbitals < >, and <(>,. The essential parameters, in the semi-empirical form of the theory, are cug, and and from their definition these quantities are expected to be characteristic of atom r or bond r—s, not of the particular molecule in which they occur (for a discussion see McWeeny, 1964). In the SCF calculation, solution of (95) leads to MO s from which charges and bond orders are calculated using (97) these are used in setting up a revised Hamiltonian according to (98) and (99) and this is put back into (95) which is solved again to get new MO s, the process being continued until self-consistency is achieved. It is now clear that prediction of the variation of the self-consistent E with respect to the parameters is a matter of considerable difficulty. 

It is not necessary to restrict ourselves to bonds that are described by Morse potentials. We can regard eqn. (56) as a quadratic equation in x, use any form of the potential energy V(R) with the usual shape (i.e., a minimum, a repulsive barrier at short distances, and a monotonical increase at large distances), and determine x to get another definition of the bond order. This is called the unity bond index quadratic exponential potential (UBI QEP) method by Shustor-ovich and Sellers. ... 

Chemists very successfully use many concepts in a more intuitive manner, although they generally tend not to define their concepts clearly (Uke the bond order), and to ignore definitions if they happen not to be convenient. 

Cremer and coworkers have shown that the analysis of p(r) provides a basis for a rigorous definition of homoaromaticity27 44. Utilizing the definitions of covalent bonding, bond order, -character and Tt-delocalization (Section III.E), they translated Winstein s definition of homoaromaticity14 (Section III. A) into density language27 44 ... 

Based on values of vco, it would appear that triply bridging carbonyls among the more common modes of CO coordination most closely approach the notion of activation by maximal CO bond order reduction. However the reactivity of the /i3-carbonyl ligand has not been demonstrated in a systematic way, and the lack of many triply bridging carbonyl species cannot be taken as a definite indication of enhanced reactivity. 

---