Applying the Reduction Formula

To see how the reduction process works we can return to the x, y, z basis in the T 4h example, taking the tth representation to be each of the irreducible representations for the point group in turn. Table 5.3 shows all the terms in the summation for each irreducible representation and demonstrates that the summation in Equation (5.19) gives zero for every one 

For any application of the reduction formula we will always find that the number of objects in the irreducible set of representations is equal to the number used in the definition of the reducible representation, i.e. the number of basis functions. 

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These representations can be reduced by applying the reduction formula. 

This observation is confirmed straightforwardly, by applying the reduction formula and the result is summarized for the example of the regular character of Csv in the table... 

These representations can be reduced by applying the reduction formula. First, < >, ... 

Before being able to apply the reduction formula, it is therefore necessary to determine the characters of the reducible representation being studied. 

We shall now apply the reduction formula (6.5) to some of the reducible representations that we have already studied. Table 6.9 contains the character table for the Czv point group and the characters that we have obtained for the Fh basis that is formed by the two Uh, and IShz orbitals in the H2O molecule (Table 6.7). 

In Chapter 1 it was noted that the number of vibrational modes of a molecule can be calculated by counting the degrees of freedom of the atoms (three per atom for X,Y and Z movement) and subtracting the degrees of freedom for motion of the molecule as a whole, three for its translation and (for nonlinear molecules) three for rotation. This was used in Section 5.2 to arrive at a reducible representation for the basis of nine atomic degrees of freedom for H2O, the classic C2V molecule. The characters for this representation were given in Table 5.1. We can now apply the reduction formula to identify the irreducible representations for the three vibrations of HjO. 

Taking the values of Xi(Q for each of the irreducible representations from the standard >31, character table in Appendix 12, we can now apply the reduction formula to this problem. The values of the individual triple products required in the summation are written out in Table 5.7, which shows that the reducible representation has the composition... 

In the following examples, the point group of a variety of complexes is used to derive the symmetry labels for the atomic orbitals (AOs) of the central atom. The central atom orbitals are at the intersection of all the symmetry elements in the point groups considered and so are never moved through space by an operation. However, they may be reorientated, and so we will work out the characters for each AO set (p, d) and then apply the reduction formula to find the appropriate irreducible representation labels. These results will be used in Chapter 7 when assembling MOs for some of the complexes, and there we will use the fact that the standard character tables have the p and d functions written in the rightmost columns. For the central atom, this means that we can simply read off the symmetry label from the table. 

The results can then be used to deduce the functional form of each d-orbital after the transformation and so find the required character set for the d-orbitals. In general, the p- and d-orbitals will give reducible representations to which we can apply the reduction formula to find the irreducible representations for the point group. 

For the d-orbitals we will apply the reduction formula. To make the job easier note that these functions are not changed by the inversion centre, since they all contain only even products of the x, y and z basis. This means that the d-orbitals have gerade symmetry, and so we only include irreducible representations with the g subscript in the reduction. The application of the reduction formula is laid out in Table 5.16, which shows that... 

Problem 6.6 The structure of 1,2-difiuorobenzene is shown in Figure 6.15. This isomer belongs to the C2v point group. Using a basis of the four C—H bonds, demonstrate that the reducible representation in this case is that shown in Table 6.7 and apply the reduction formula to show that... 

For the four H(ls) orbitals we can obtain the irreducible representations by applying the reduction formula to the reducible representation. Table 7.7 shows that this process results in... 

Apply the reduction formula (Equation (7.33)) to the reducible representation found in part (2) and list the irreducible representations for the 7r-orbitals. 

In this expression, N is the number of times a particular irreducible representation appears in the representation being reduced, h is the total number of operations in the group, is the character for a particular class of operation, jc, in the reducible representation, is the character of x in the irreducible representation, m is the number of operations in the class, and the summation is taken over all classes. The derivation of reducible representations will be covered in the next section. For now, we can illustrate use of the reduction formula by applying it to the following reducible representation, I-, for the motional degrees of freedom (translation, rotation, and vibration) in the water molecule ... 

The reduction formula can only be applied to finite point groups. For the infinite point groups, D h and C h, the usual practice is to reduce the representations by inspection of the character table. 

The reduction formula cannot be applied to the infinite point groups (Chapter 4). Here inspection of the character table may help. Since 2 cosd> at appears with the I u irreducible representation, it is worth a try to subtract this one from rvib ... 

Table 5.7 The reduction formula is applied to the basis of five C=0 bonds in the complex [Fe(CO)sl, shown In Figure 5.10.

In Table 5.10, the reduction formula is applied to F(d) in the >41, point group, with the result that... 

We have grouped the pt and pty cases together because they form a degenerate pair within the E representation. The first direct product is set out in Table 6.2. Under the identity operator we have generated the character 4 since no irreducible representation contains 4 under this column, this product must be reducible. In Table 6.3, the reduction formula is applied in the normal way to obtain... 

One more obvious example illustrates strong influence of particle s sedimentation upon the sensitivity threshold. Assume that we have to ensure the detection of the cracks with the depth 10 > 2 mm in the case when the same product family indicated above is applied and h = 20 pm. The calculation using formula (1) shows that in the absence of sedimentation only the cracks with the width H > 2 pm could be detected. But when the effect of sedimentation results in the reduction of the value of developer layer thickness from h = 20 pm to h = 8 pm, then the cracks of substantially smaller width H > 0,17 pm can be revealed at the same length lo = 2 mm. Therefore we can state that due to the sedimentation of developer s particles the sensitivity threshold has changed being 12 times smaller. Similar results were obtained using formula (2) for larger particles of the developers such as kaolin powder. 

A further anomaly in nomenclature is thebainol, the name applied to the substance formed by the alkaline reduction of metethebainone, and which was at first believed to be formed by reduction of the carbonyl group, but which Gulland and Robinson proved to be a ketone. It is isomeric with dihydrothebainone referred to above, and has been re-named dihydrom tathebainone (Schopf). The interrelationships of these substances are shown by the following formulae —... 

Sauers and coworkers have applied the Paterno-Biichi reaction to engeneral formula 427 (Scheme XXXIV) Reductive cleavage of these products with lithium aluminium hydride is also regioselective and leads, following oxidation, to ketones... 

The Lanczos method is based on generating the orthonormal basis in Krylov space Ki =span c, Ac, A c by applying the Gram-Schmidt orthogonaliza-tion process, described in Appendix A. In matrix notations this approach is associated with the reduction of the symmetric matrix A to a tridiagonal matrix and also with the special properties of T/,. This reduction (called also QT decomposition) is described by the formula... 

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