Hybrid orbitals, representation

Hybrid Orbital Representation of H3C Figure 3.5 sp -Hybrid Orbital Set and an Example of an sp2-Hybridized Carbon... 

Figure Schematic representation of the two components of the ij -Hi-metal bond (a) donation from the filled (hatched) CT-H2 bonding orbital into a vacant hybrid orbital on M (b) jr-back donation from a filled d orbital (or hybrid) on M into the vacant a antibonding orbital of Hj.

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. 

Figure 3.15 An sp hybrid orbital, (a) left, radial functions for the 2s and 2p atomic orbitals right, radial function for the sp hybrid orbital (b) left, the shapes of the 2s and 2p atomic orbitals as indicated by a single contour value right, the shape of the sp hybrid orbital as indicated by the same contour, (c) The shape of a surface of constant electron density for the sp hybrid orbital, (d) Simplified representation of (c). (Reproduced with permission from R. J. Gillespie, D. A. Humphreys, N. C. Baird, and E. A. Robinson, Chemistry, 2nd Ed., 1989, Allyn and Bacon, Boston.)...

Figure 3.7. Common representations of the s, p, d and atomic orbitals, sp3 hybridized orbitals, and some representations of how they overlap to form bonds between atoms.

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... 

Problem 7.13 (a) Give an orbital representation for an S 2 reaction with (S)-RCHDX and Nu, if in the transition state the C on which displacement occurs uses sp hybrid orbitals, (b) How does this representation explain (i) inversion, (ii) the order of reactivity 3 >2°> 1° ... 

Thus we see that the MO configuration, a2, has a precise VB equivalent. What is more, any MO representation may be converted to its VB analogue, and vice versa, by simply describing the MO configuration in terms of the atomic or hybrid orbitals from which it is composed. It follows, therefore, that just as R- -X is a poor VB representation of the R—X bond because it does not take into account ionic contributions, a2 as represented by (41), is also seen to be an unsatisfactory MO wave-function since it places an excessive emphasis on ionic configurations where the two electrons lie in the same hybrid orbital, and hence repel each other. 

A necessary prelude to determining the combinations of AOs which give a hybrid orbital of correct symmetry is the classification of the AOs of the central atom A in terms of the irreducible representations of the point group to which the molecule belongs. This is discussed in 11-2. In 11-4 we consider 77--bonding systems and in the final section we discuss the relationship between localized and non-localized MO theory. 

First then, for methane, we must obtain I 71. To do this let us associate with each carbon hybrid orbital a vector pointing in the appropriate direction and let us label these vectors vv vs, v , v4 (see Fig. 11-3.1). All of the symmetry properties of the four hybrid orbitals will be identical to those of the four vectors. The reducible representation using these vectors (or hybrids) as a basis can be obtained from 4... 

Recalling our earlier discussion, this equation means that the four AOs which are combined to make the set of four hybrid orbitals must be chosen so as to include one orbital which belongs to the P 1 representation and a set of three orbitals which belong to the rT representation. Reference to Table 11-2.2 shows that the AOs fall into the categories below for the "d point group. So we can combine an s-orbital... 

As a second example, let us consider a molecule with the formula AB6 having the symmetry of a trigonal bipyramid Ih. The vector system is shown in Fig. 11-3.2. The set of five hybrid orbitals (or vectors) on A form a basis for a reducible representation of the point group, with the following character ... 

The six necessary hybrid orbitals on the boron atom can also be assigned vectors. If w-bonds are to be formed, these vectors must have the same orientation as the six vectors on the chlorine atoms. If we followed in the footsteps of 11-3, we would now construct the reducible representation Th7b from a consideration of how the six vectors on the boron atom change under the symmetry operations of the B point group. However, it is clear that since the six vectors on the chlorine atoms match the six on the boron atom, exactly the same representation rhyb can be found by using these vectors instead. Since it is less confusing to have three pairs of vectors separated in space than six originating from one point, we will take this latter approach. 

In this particular example we could have avoided some of the labour involved in finding the combinations of hybrid orbitals which are equal to px and ptf, by using the 8 point group (to which the molecule also belongs). For this point group, the two-dimensional representation, the cause of all the trouble, can be expressed as two complex one-dimensionl representations. The orbitals p, and py are then just as easy to obtain as the s-orbital. Any complex numbers which result are eliminated at the end of the treatment by addition and subtraction of the orbitals formed. This is the technique which was used in 10-7 to find the 7T-molecular orbitals of the trivinylmethyl radical. It is, however, of no avail when dealing with point groups which have three-dimensional irreducible representations as in our next example, CH4. 

To determine how to form a set of trigonally directed hybrid orbitals, we begin in exactly the same way as we did in the MO treatment. We use the three a bonds as a basis for a representation, reduce this representation and obtain the results on page 219. However, we now employ these results differently. We conclude that the s orbital may be combined with two of the p orbitals to form three equivalent lobes projecting from the central atom A toward the B atoms. We find the algebraic expressions for those combinations by the following procedure. 

Fig. 5.4 Other ways of representing hybrid orbitals (a) orbital shape shown by a single contour, (b) clcnd representation, (c) simplified r resemaiicr. The small back lobes have been omitted and the shape streamlined to make it easier to draw molecules containing several hybrid orbitals.

The atomic orbitals suitable for combination into hybrid orbitals in a given molecule or ion will he those that meet certain symmetry criteria. The relevant symmetry properties of orbitals can be extracted from character tables by simple inspection. We have already pointed out (page 60) that the p, orbital transforms in a particular point group in the same manner as an x vector. In other words, a px orbital can serve as a basis function for any irreducible representation that has "x" listed among its basis functions in a character table. Likewise, the pr and p. orbitals transform as y and vectors. The d orbitals—d d dy, d >, t, and d ,—transform as the binary products xy, xz, yr, x2 — y2, and z2, respectively. Recall that degenerate groups of vectors, orbitals, etc, are denoted in character tables by inclusion within parentheses. 

In carrying out the procedure for a tetrahedral species, it is convenient to let four vectors on the central atom represent the hybrid orbitals we wish to construct (Fig. 3.26). Derivation of the reducible representation for these vectors involves performing on them, in turn, one symmetry operation from each class in the Td point group. As in the analysis of vibrational modes presented earlier, only those vectors that do not move will contribute to the representation. Thus we can determine the character for each symmetry operation we apply by simply counting the number of vectors that remain stationary. The result for AB4 is the reducible representation, I",. 

No matter whether calculated within the perfect pairing VB approach or by the spin-coupled VB approach, in both cases the CC hybrid orbital extends outside the three-membered ring as expected by the schematic representations in Figure 7, but also inside... 

FIGURE 17. Schematic representation of the symmetry-unique spin-coupling patterns in cyclopropane (above) and benzene (below). In the case of cyclopropane, carbon hybrid orbitals and, in the case of benzene, carbon p n orbitals are shown. For each structure, Gallup-Norbeck occupation numbers as determined by spin-coupled valence bond theory are given. All data from Reference 51... 

FIGURE 1. Schematic representation of the interactions between sp3(Si) hybrid orbitals in linear oligosilanes. (a) Reprinted by permission of The Royal Society of Chemistry, from Reference 33 (b) Reprinted with permission from Reference 17. Copyright 1970 American Chemical Society... 

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