Construction of hybrid orbitals

As introduced in Chapter 3, for AX systems, the hybrid orbitals are the linear combinations of atomic orbitals on central atom A that point toward the X atoms. In addition, the construction of the sp hybrids was demonstrated. In this section, we will consider hybrids that have d-orbital contributions, as well as the relationship between the hybrid orbital coefficient matrix and that of the molecular orbitals, all from the viewpoint of group theory. 

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So far we have only considered which AOs are required for the construction of hybrid orbitals of the appropriate symmetry. We now will show how we can obtain explicit mathematical expressions for the hybrid orbitals which will allow us to see exactly how much each AO contributes. Though hybrid orbitals are most frequently used in qualitative discussions of bonding, they do have their quantitative use when one carries out an exact MO calculation and when one deals with coordination compounds, where it is often necessary to use hybrid orbitals for evaluating overlap integrals which are often related to bond strengths in these situations the explicit expressions are required. 

Covalent bonds can be described with a variety of models, virtually all of which involve symmetry considerations. As a means of illustrating the role of symmetry in bonding theory and laying some foundation for discussions to follow, this section will show the application of symmetry principles in the construction of hybrid orbitals. Since you will have encountered hybridization before now, but perhaps not in a symmetry context, this provides a facile introduction to the application of symmetry. You should remember that the basic procedure outlined here (combining appropriate atomic orbitals to make new orbitals) is applicable also to the derivation of molecular orbitals and ligand group orbitals, both of which will be encountered in subsequent chapters. 

These are acceptable for many purposes, but the complex number VI (= — 1) makes them difficult to visualize. Given a set of solutions to the wave equation, which we find inconvenient, it is always permissible to transform these into an equal number of new functions obtained by taking linear combinations of the original ones, provided that orthogonality is upheld, that is, the overlap integral f dr i j) is always zero. (This procedure is utilized in other familiar situations, for example, in the construction of hybrid orbitals, and of MOs as linear combinations of AOs see Molecular Orbital Theory.) Eliminating i from equations (1-5) by equations (6-9),... 

Construction of molecular orbitals Construction of hybrid orbitals Predicting the decrease of degeneracies of d orbitals under a ligand field Predicting the allowedness of chemical reactions... 

Molecules such as NHj and HjO etc. are described in terms of an inequivalent hybridization scheme based on sp in valence bond theory. The construction of hybridized orbitals in such molecules is different from that developed above. The tetrahedral molecule XAY3 (3) provides a useful starting point. Since the hyl is distinguished from hy2, hy3 and hy4, the symmetry-adapted linear combinations of these hybrids cannot be generated in terms of the spherical harmonic expansion in Eq. (1). But they can be derived as follows ... 

Both hybridization and molecular orbital theory (discussed in Section 3.5) involve the mixing of atomic orbitals however, the construction of hybrid orbitals involves mixing of atomic orbitals on the same atom, whereas the construction of molecular obitals involves atomic orbitals on different atoms. 

Any hybrid orbital is named from the atomic valence orbitals from which It Is constmcted. To match the geometry of methane, we need four orbitals that point at the comers of a tetrahedron. We construct this set from one s orbital and three p orbitals, so the hybrids are called s p hybrid orbitais. Figure 10-8a shows the detailed shape of an s p hybrid orbital. For the sake of convenience and to keep our figures as uncluttered as possible, we use the stylized view of hybrid orbitals shown in Figure 10-8Z). In this representation, we omit the small backside lobe, and we slim down the orbital in order to show several orbitals around an atom. Figure 10-8c shows a stylized view of an s p hybridized atom. This part of the figure shows that all four s p hybrids have the same shape, but each points to a different comer of a regular tetrahedron. 

STRATEGY Draw the Lewis structure and determine the electron arrangement about the central atom. The number of orbitals required for the electron arrangement determines the hybridization scheme used, as shown in Table 3.2. Construct the hybrid orbitals from atomic orbitals, using the same number of atomic orbitals as hybrid orbitals. Start with the s-orbital, then add p- and d-orbitals as needed. 

Pauling showed that the quantum mechanical wave functions for s and p atomic orbitals derived from the Schrodinger wave equation (Section 5.7) can be mathematically combined to form a new set of equivalent wave functions called hybrid atomic orbitals. When one s orbital combines with three p orbitals, as occurs in an excited-state carbon atom, four equivalent hybrid orbitals, called sp3 hybrids, result. (The superscript 3 in the name sp3 tells how many p atomic orbitals are combined to construct the hybrid orbitals, not how many electrons occupy each orbital.)... 

In Section 7.2.1, we have seen that many hybridization schemes involve d orbitals. In fact, we do not anticipate any technical difficulty in the construction of hybrids that have d orbital participation. Let us take octahedral d2sp3 hybrids, directed along Cartesian axes (Fig. 7.1.10), as an example. From Table 7.1.5,... 

The equations (59)—(63) solve the formal problem of which and how many multiplets 25+1/I arise from a given set of covalent and ionic configurations, and of how to construct the corresponding symmetry-adapted VB functions. However, one seeks to express at least some part of an expansion in many VB functions in a more compact and, hopefully, physically suggestive form. This is essentially the motivation behind the introduction of hybrid orbitals. 

The simple VB model is augmented with the concept of orbital hybridization to account for the valence of second-row atoms and the structures of their compounds. Hybrid orbitals are constructed by adding s and p orbitals with different coefficients (weights or percentage contributions) and phases. The number of hybrid orbitals produced equals the number of starting AOs there are two sp hybrid orbitals, three sp hybrid orbitals, and four sp hybrid orbitals. 

We construct the first set of hybrid orbitals from one s atomic orbital, the three p atomic orbitals and the d atomic orbital they are called dsp hybrid orbitals. The principal quantum numbers of the participating atomic orbitals depend on the particular metal atom under consideration for Co, they would be the 3d, 4s, and 4p atomic orbitals. The dsp hybrid orbitals in the most general case are written out as... 

Table 8.7 shows the variety of hybrid orbitals that can be constructed from various combinations of s, p, and d orbitals, the shapes of the molecules that result, and selected examples. 

Bent s Rule and When a set of hybrid orbitals is constructed by a linear combination of atomic orbitals,... 

The conventional representations of hybrid orbitals used in Fig. 1.18 are just as misleading as the conventional representations of the p orbitals from which they are derived. A more accurate picture of the sp3 hybrid is given by the contours of the wave function in Fig. 1.19. Because of the presence of the inner sphere in the 2s orbital (Fig. 1.11a), the nucleus is actually inside the back lobe, and a small proportion of the front lobe reaches behind the nucleus. This follows from the way a hybrid is constructed by adding one-quarter of the wave function of the s orbital (Fig. 1.11a) and three-quarters in total of the wave functions of the p orbitals (Fig. 1.12a). As usual, we draw the conventional hybrids relatively thin, and make the mental reservation that they are fatter than they are usually drawn. 

Initially it would appear that, of the four bonds formed by a carbon atom, one would involve the 2s orbital and the other three the three 2p orbitals. This, however, implies that one bond is different from the other three, whereas experimentally the CH4 molecule is perfectly symmetrical—all four bonds being identical. The solution to this dilemma was given by Pauling who suggested that we should use a linear combination of orbitals instead of the pure 2s and 2p orbitals of the carbon atom. On the basis of a simple quantum-mechanical treatment, he concluded that from the one 2s and three 2p orbitals one can construct four hybridized orbitals as follows ... 

Theoretical determination methods of hybrid orbitals 2.1 Geometrical constructions... 

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