Reducible representations group orbitals from

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 I 7 from a consideration of how the six vectors on the boron atom change under the symmetry operations of the 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 rh7b can be found by wring 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. 

Group Orbitals from Reducible Representations Each of the representations from Step 3 can be rednced by the procednre described in Section 4.4.2. For example, the representation F(25 ) reduces to Ag + 5i ... 

From Table 7-9,2 and using eqn (5-7,2) we can find the diagonal elements of the matrices which represent the 4h point group in the p-orbital basis and in the d-orbital basis. From these elements we get the characters of two reducible representations they are shown in Table 7-9,3, By applying eqn (7-4.2)... 

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",. 

This is a reducible representation of the D point group which reduces to ug + uu. Two molecular orbitals must be generated, one with ag and the other with antibonding orbitals which can be formed from the two 1. s atomic orbitals. 

Reduce each representation from Step 3 to its irreducible representations. This is equivalent to finding the symmetry of the group orbitals or the symmetry-adapted linear combinations (SALCs) of the orbitals. The group orbitals are then the combinations of atomic orbitals that match the symmetry of the irreducible representations. 

In electronic structure problems, we would normally be interested in bonding interactions of s, p and d-atomic orbitals from atoms sited on the vertices of a molecular stmcture and hence only Fcr, F r and F. In vibrational problems, the mechanical representation is Fcoordinates = Fff X Fxyz = Fcr + F r for an empty cluster and F + F r + Fxyz for a cluster with a centrally placed atom. For the cases of interest in molecular problems, equations 3.5 to 3.7, which follow from a knowledge of the permutation characters alone, can be used to generate, once and for all, the foil set of reducible characters for the orbits of the molecular point groups. 

Consider the basis Fh constituted by the two orbitals Ishi ISH2 on the hydrogen atoms in the water molecule. From Table 6.1, we notice that the two orbitals are transformed into themselves by the operations E and ayz (characters equal to 1 +1 = 2), whereas they are interchanged by the operations Cf and (characters equal to 0+0 = 0). The characters associated with the basis Fh, listed in Table 6.7, do not correspond to any of the irreducible representations of the Czv point group, which are all one-dimensional (Table 6.5). This is therefore a basis for a reducible representation. 

Ptcy has a square-planar geometry and a point group. The four group orbitals can be found from the reducible representation, where F = Aj + + E . 

Here z is the square root of-1. As in equation 12.9,/runs from 0, 1, 2. ... We shall see below, and, very importantly, in the next chapter, that this complex form of the wavefunction is very useful. It is interesting to see where this expression comes from. Group theory provides the answer. The molecular point group of, for example, benzene is However, the group Q is the simplest one we can use which will generate the tt orbitals of the molecule. Table 12.1 shows its character table. The reducible representation for the basis set of six n orbitals is... 

Because we have assembled the reducible representations for complete sets of orbitals, the character totals obtained are independent of the choice of symmetry elements or operations from each class in the point group. We can now proceed to using the reduction formula to find the irreducible labels for p- and d-orbitals in O. For the p-orbitals, inspection of the standard character table from Appendix 12 shows that... 

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