Transformation, unitary, hybrid orbitals

The canonical m.o.s of diamond are delocalized over the entire crystal. However, as we have seen in Chapter 8 for other systems, the occupied m.o.s can be the object of a unitary transformation leading to a set of equivalent and quasi-localized molecular orbitals . This is why the structure of diamond can (for some purposes) be described in terms of the overlap of sp hybrid orbitals, four for each C atom. As we have seen in Chapter 8, we must stress that such an alternative description cannot be used to infer information about electron energies. In particular, the localized bond description of the structure of diamond does not imply that all valence electrons have the same energy. This would be the case only if the sp -sp bonds were independent. It is because of residual interactions such as f and (3"... 

Such a transformation can be used for relocalizing a given set of delocalized molecular orbitals in conformity with the chemical formula. For instance, the occupied orbitals of methane can be transformed into orbitals very close to simple two-center MO s constructed from tetrahedral sp3 hybrid orbitals and Is hydrogen orbitals 24,25,26) a. unitary transformation can hardly modify the wave function, except for an immaterial phase factor therefore, it leads to a description which is as valid as that in terms of the canonical delocalized Hartree-Fock orbitals. Of course, the localization obtained in this way is not perfect, but it is usually much better than is often believed. In the case of methane, the best localized orbitals are uniquely determined by symmetry 27> for less symmetric molecules one needs a criterion for best localization 28 29>, a problem on which we shall not insist here. A careful inspection reveals that there are three classes of compounds ... 

In a series of papers the connection between /couplings and bond parameters was extensively revisited during the review period. The atomic hybridizations of the bonded atoms were described on the basis of MBOHOs (Maximum Bond Order Hybrid Orbitals). The MBOHO approach is based on the maximum bond order principle and the basic idea of the maximum overlap symmetry orbitals. Unitary transformations T and U of the m (orthogonal) AOs on atom A and n AOs on atom B ( > m) can be defined. 

The Lewis bonding model may be rationalized using quantum mechanics if the ordinary C2s and C2p orbitals, for example, are mixed in such a way that they point into the comers of a tetrahedron with the carbon atom at its center. Generally, AOs on the same atom mix to form directed orbitals. This is referred to as (atomic) hybridization. Hybridization is important for the interpretation of the wave function, but of little interest computationally, since it just amounts to a unitary transformation of the orbitals, and Slater determinants are invariant under unitary transformations. 

In the previous section, we showed how one could transform from a basis set of STOs to a basis set of symmetry orbitals. Since these two sets are related through a unitary transformation, they are equivalent and must lead to the same MOs when we do a linear variation calculation. However, there are an inhnite number of unitary transformations available, and so the set of symmetry orbitals is only one of an infinite number of possible equivalent bases. Of course, this set has the unique advantage of being a set of bases for representations of the symmetry group, which makes it easy to work with. Another set of equivalent basis functions are the hybrid orbitals. These have the distinction of being the functions that are concentrated along the directions of bonds in the system. Consider, for example, methane, which was discussed in detail in Chapter 10. The minimal basis set of valence STOs on carbon can be transformed to form four tetrahedrally directed hybrids ... 

A symmetrically orthogonalized directed hybrid orbital derived through unitary transformation of natural atomic orbitals centered on a particular atom. See Population Analyses for Semiempirical Methods. 

A unitary transformation between the s and p orbitals of an individual atom we shall refer to this as hybridization invariance. 

Two remarks are important in connection with the partitioning of the total moment. First, hybridization is a unitary transformation which does not change the total electron density of the orbitals if the latter are all equally occupied Therefore, the center of gravity of the whole set of hybrids is the nucleus = 0... 

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