Overlap matrix, hybrid orbitals

An extended Hiickel MO calculation supports the assumptions made in the above analysis in that the three t2g orbitals are indeed close together in energy and remain nearly nonbonding metal-based d-orbitals. The detailed agreement is less satisfactory the SOMO is predicted to be primarily dx2 y2 with a small dxz admixture (hybrid 6 of Table 4.13), a result that can be ruled out from our analysis of the ESR results. The EHMO overlap matrix based on the X-ray structure suggests that the molecule is much closer to C2 symmetry than to Cs. If we accept that conclusion, then dxzjdxy hybridization is less likely than dyjdxy, as we tacitly assumed above. 

The hybridization principle of atomic orbitals used from the thirties up to the present days as a prelude to the formation of chemical bonds is surveyed. The expose is centered on the ab-initio procedure of overlap-matrix localization suggested by G. Del Re in 1963 (Theoretica Chimica Acta 1, pp 188-197), and its successive extensions and various applications in Quantum Chemistry are described. Some conceptual aspects of hybridization are discussed. 

To obtain eqs. (7.2.4) and (7.2.5) from eq. (7.2.6), we only need to set angle a to be 90° and 0°, respectively. It is not difficult to show that the five hybrids form an orthonormal set of wavefunctions. The parameter a in the coefficient matrix in eq. (7.2.6) may be determined in a number of ways, such as by the maximization of overlap between the hybrids and the ligand orbitals, or by the minimization of the energy of the system. In any event, such procedures are clearly beyond the scope of this chapter (or this book) and we will not deal with them any further. 

With much more powerful quantum mechanical computations available (i.e., Gaussian 98), the method was applied to a variety of photochemical reactions (note Scheme 1.12). The expression in Equation 1.12 for the delta-density matrix elements includes overlap integrals to take care of basis set definitions. Weinhold NHOs (i.e., hybrids) were used in order to permit easy analysis in terms of basis orbital pair bonds comprising orbital pairs. Note A refers to a reactant and B refers to the corresponding excited state in this study. 

Self-consistent paramagnetic total energy calculations are performed to study the ground state properties. In this case the 4f-electrons are treated as localized core states. Since the local density approximation is known to overestimate the extent of the 4f-orbitals, and hence the effects of interatomic hybridization and overlap, these terms in the matrix elements are neglected the f-states are now atomic-like and in that respect can be called core-like. The fee structure for the lanthanides are assumed, except for bcc Eu, because that is the simplest closed packed structure which approximates the real structure. 

FIGURE 12.1 Schematics showing the interaction between two identical molecular orbitals with energy e. The interaction introduces repulsion on each level that is proportional to the overlap 5, and the subsequent hybridization produces bonding and antibonding states separated by two times the coupling matrix element V. 

Sodamide also reacts with bromoarenes by elimination of HBr (Figure 10.37) to give benzyne, a very reactive and unstable intermediate. Mechanistically, the elimination may be E2 or ElcB or somewhere in between. Although it is common, and quite conventional, to write the structure of benzyne with a triple bond, this is not a triple bond of the same type as we find in alkynes. The extra bond is formed by weak overlap of two sp orbitals, each with one electron in it (10.20). It is probably more correct to think of it as a resonance hybrid of the triply bonded form, diradicals, and charge-separated species. The IR spectrum of isolated benzyne in an inert gas matrix at 8 K has been studied although assignments have been a matter of discussion, the value of 1846 cm for the C-C triple bond stretch is at least approximately correct. This suggests that the triple bond is considerably weaker than that in a conventional alkyne where a frequency of 2200 cm would be typical. 

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