Symmetry hybrid orbitals

The total electron density contributed by all the electrons in any molecule is a property that can be visualized and it is possible to imagine an experiment in which it could be observed. It is when we try to break down this electron density into a contribution from each electron that problems arise. The methods employing hybrid orbitals or equivalent orbitals are useful in certain circumsfances such as in rationalizing properties of a localized part of fhe molecule. Flowever, fhe promotion of an electron from one orbifal fo anofher, in an electronic transition, or the complete removal of it, in an ionization process, both obey symmetry selection mles. For this reason the orbitals used to describe the difference befween eifher fwo electronic states of the molecule or an electronic state of the molecule and an electronic state of the positive ion must be MOs which belong to symmetry species of the point group to which the molecule belongs. Such orbitals are called symmetry orbitals and are the only type we shall consider here. 

Figure 6.12 Formation of a bonding 3-centre B-H-B orbital j1/ from an sp hybrid orbital on each of B(l), B(2) and the H Is orbital, (KH). The 3 AOs have similar energy and appreciable spatial overlap, but only the combination Vr(Bl)-l-Vr(B2) has the correct symmetry to combine linearly with i/r(H).

At that time I was handicapped by my remembering a misinterpretation that I had made of some results obtained in 1932 by one of my students, Ralph Hultgren (6). He had begun to make a thorough study of sets of equivalent spd hybrid bond orbitals, and soon found that he could not handle the computational problem in those precomputer days. I pointed out that the best hybrid orbitals have cylindrical symmetry about... 

The hybrid orbital has cylindrical symmetry, and accordingly the introduction of d character and f character in the axial bond itself does not lead to an interaction dependent on the relative azimuthal orientation of the two groups. 

The period 1930-1980s may be the golden age for the growth of qualitative theories and conceptual models. As is well known, the frontier molecular orbital theory [1-3], Woodward-Hoffmann rules [4, 5], and the resonance theory [6] have equipped chemists well for rationalizing and predicting pericyclic reaction mechanisms or molecular properties with fundamental concepts such as orbital symmetry and hybridization. Remarkable advances in aeative synthesis and fine characterization during recent years appeal for new conceptual models. 

Double and triple bonds, particularly those in carbon molecules, are often described in MO terms. Thus a double bond is described as consisting of a a bond formed by the overlap of an sp hybrid orbital on each carbon atom and a tt bond formed by the sideways overlap of either the 2p or 2p orbitals (Figure 3.18). A cr orbital has cylindrical symmetry like an atomic s orbital whereas a ir orbital, like an atomic p orbital, has a planar node passing through the nucleus of each of the bonded atoms. A triple bond is similarly described as consisting of a cr orbital and two ir orbitals formed from both the 2p and 2pv orbitals on... 

The most common—and perhaps most important—hybrid orbitals are the tetrahdral ones formed by adding one s-, and three p- type orbitals. These can be arranged to form various crystal structures diamond, zincblende, and wurtzite. Combinations of the s-, p-, and d- orbitals allow 48 possible symmetries (Kimball, 1940). 

We therefore conclude that, for a combination of model, numerical and conceptual reasons the OHAO basis is well-adapted to a theory of valence. The hybrid orbital basis (for simple molecules) has a distinctive symmetry property it carries a permutation representation of the molecular symmetry group the equivalent orbitals are always sent into each other, never into linear combinations of each other. This simple fact enables the hybrid orbital basis to be studied in a way which is physically more transparent than the conventional AO basis. 

There exists no uniformity as regards the relations between localized orbitals and molecular symmetry. Consider for example an atomic system consisting of two electrons in an (s) orbital and two electrons in a (2px) orbital, both of which are self-consistent-field orbitals. Since they belong to irreducible representations of the atomic symmetry group, they are in fact the canonical orbitals of this system. Let these two self-consistent-field orbitals be denoted by Cs) and (2p), and let (ft+) and (ft ) denote the two digonal hybrid orbitals defined by... 

In molecular orbital (MO) terminology, the MO formed by overlap between the carbon hybrid orbital and the metal hybrid orbital is called a sigma orbital. An atomic 3 orbital is perfectly spherical and hence has the highest possible symmetry. It has the same sign everywhere, by... 

Hybrid orbitals may be considered as perfectioned AOs, adopted in the calculation of localized MOs in polyatomic molecules, with the LCAO method (cf. section 1.17.1). In the case of hybrid orbitals sp, the four linear combinations of s and p orbitals (Tbi, Tc2, Te, Te ) that lead to tetrahedral symmetry are... 

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