Bond order approximation, definition

Selected atom charges q, bond orders p and distances do-Mo. (A) of the M015O56H22 cluster representing MoOslOlO) surface. For a definition of the two sets of data, bulk termination and optimized cluster , see text. Results refer to DFT calculations using the RPBE approximation. 

In the LCAO MO description, the H2 molecnle in its ground state has a pair of electrons in a bonding MO, and thus a single bond (that is, its bond order is 1). Later in this chapter, as we describe more complex diatomic molecules in the LCAO approximation, bond orders greater than 1 are discussed. This quantum mechanical definition of bond order generalizes the concept first developed in the Lewis theory of chemical bonding—a shared pair of electrons corresponds to a single bond, two shared pairs to a double bond, and so forth. 

In addition to the conventional overlap energy term there is a second term that depends on the bond-order Pq and the repulsive integral between an electron in adsorbate orbital 0 and an electron in orbital I on the neighboring atom, is equal to /fjj, Ek).(2.282), and the second term in follows from the first term of (2.283a) when using definition Eq.(2.35c) for the bond-order Pi2 and the zero-differential overlap approximation Eq.(2.284). In section 2 Eq.(2.40b) related to Eq.(2.309) was derived. It was shown to lead to a reduction of the interaction energy of two electrons with different spins in the same orbital (see Eq.(2.42,II)). The result implies that Ucff is not independent of the bond-order. An improved expression for would be ... 

This arbitrariness most clearly manifests itself in going beyond the scope of the HF approximation, as evidenced by a wide variety of definitions for molecular systems available in the literature for valences and bond orders in the case of post-Hartree-Fock methods for molecular systems [570,578-580]. In post-HF methods local characteristics of molecular electronic structure are usually defined in terms of the first-order density matrix and in this sense there is no conceptual difference between HF and post-HF approaches [577]. It is convenient to introduce natural (molecular) spin orbitals (NSOs), i.e. those that diagonalise the one-particle density matrix. The first-order density matrix in the most general case represents some ensemble of one-electron states described by NSOs... 

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... 

What is new in the re-examination of covalent interactions is that the approach in terms of a four-dimensional wave structure leads to a precise definition of bond order, not achieved before. Together with the new freeatom ionization radii the parameters of interatomic distance, dissociation energy, stretching force constants and diatomic dipole moments can now be derived as simple functions of the ionization radii and the golden ratio. These results have nothing in common with the more approximate simulations described before. 

Yielding bond orders that are true observables, the definitions described above call for computationally expensive evaluations of AOMs. The high cost of such calculations has to be weighed against the superior numerical stability of the computed bond orders with respect to basis set extensions and the clarity of their interpretation. As in the case of atomic charges, electron correlation affects the values of Bab- Predictably, bond orders computed at the correlated levels of theory are lower than those obtained within the HF approximation. For example, HF/6-31G and MP2/6-31G calculations produce values of 3.037 and 2.779, respectively, for Bnn in the N2 molecule. It should be emphasized that Bab measures only the covalent component of bonding, as reflected in the HF/6-31G bond orders of 0.480, 0.201, and 0.107 computed for the HF, LiH, and LiF molecules, respectively. 

Several approximations that allow simple estimates of bond parameters are presented as a demonstration that predictions based on quantum potentials are of correct order, and not as an alternative to well-established methods of quantum chemistry. In the same spirit it is demonstrated that the fundamental thermodynamic definition of chemical equilibrium can be derived directly from known quantum potentials. The main advantage of the quantum potential route is that it offers a logical scheme in terms of which to understand the physics of chemical binding. It is only with respect to electron-density distributions in bonds that its predictions deviate from conventional interpretations in a way that can be tested experimentally. 

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