Reducible representations hybrid orbitals

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... 

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. 

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. 

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",. 

The atomic orbitals that fit the irreducible representations are those used in the hybrid orbitals. Using the symmetry operations of the D f, group, we find that the reducible representation... 

The p and s orbitals of the borons collectively have the same symmetry (which reduces to the irreducible representations A g + Eg - Ti an analysis of the orbitals in terms of symmetry is left as an exercise in Problem 15-17 at the end of this chapter) and, therefore, may be considered to form sp hybrid orbitals. These hybrid orbitals, two on each boron, point out toward the hydrogen atoms and in toward the center of the cluster, as shown in Figme 15-12. 

This representation reduces to A/ + , which best corresponds with the 2s, 2p and Ipy orbitals. Thus, the hybridization of the B atom in BH3 is sp, with the molecule lying in the xy-plane. The empty Ip orbital is not used in the hybridization. Application of the projection operator yields the same results as those obtained in Chapter 9 for the symmetry coordinates of the B-H stretching vibrations in the isomorphous BCL3 molecule. These hybrid orbitals take the following mathematical forms (after normalization and substitution of AOs for the basis functions) ... 

FIGURE 13.26 Operation of the symmetry classes of T on the sp orbitals. The a, b, c, and ti labels are used only to keep track of the individual hybrid orbitals. The nrnnber of hybrid orbitals that do not move when a symmetry operation occurs is listed in the final coliunn. This set of mrmbers is the reducible representation F of the sp orbitals. The great orthogonality theorem is used to reduce F into its irreducible representation labels. 

Next, one moves to the ligands s p orbitals analysis within the Oj symmetry the possible hybridizations are those of o(s-s s-p) and n (p-p). Figure 2.41 (left-bottom). With the aid of the Table of characters of the group 0, these hybrid orbitals will be characterized by the symmetries with the reducible representation and of Table 2.14, with the projections in... 

TABLE 2.14 The Table of Characters and the Reducible Representations and Xp of the Ligands Hybrid Orbitals for the Symmetry After (Lancashire, ... 

Therefore the reducible representation of an eight-vertex polyhedron with an inversion center contains equal numbers of even and odd irreducible representations. This corresponds to a hybridization using four symmetrical and four antisym-metrical atomic orbitals. Since only three orbitals of the sp d manifold are antisymmetrical (namely the three p orbitals), an eight-vertex polyhedron with an inversion center cannot be formed using only s, p, and d orbitals. Similarly in the case of the 3,3-bicapped trigonal prism the primary involution is the horizontal reflection, Oh, which also leaves no vertices fixed and thus has a character of zero so that the same arguments apply. 

These hybridizations are then reduced to the irreducible representations of the coordination symmetry group and the terms that correspond to the bond orbitals and the anti-bond orbitals of the ligands toward the central ion are therefore formulated. 

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