Formula reduction

When evaluating matrix elements of operators for coupled systems, it is often convenient to make use of reduction formulas. These formulas reduce the evaluation of products of operators to matrix elements of individual operators. Two situations can occur. 

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 ... 

Readers please evaluate the triple matrix product in the reduction formula (6.120) with pencil and paper.]... 

The decomposition of the reducible representation into irreducible ones proceeds through the reduction formula... 

The application of the reduction formula is exemplified by the decomposition of the D2 representation of the group R3 (Table 65) in terms of the irreducible representations of the cubic group O having the character table according to Table 64. Now the appearances of the individual irreducible representations are evaluated according to the reduction formula of the form... 

Side Note 17.8. Diazene anions have a key role in both the Wolff-Kishner reduction and its alternative, the Reductive Cyanation semicarbazone reduction (Formula D in Figure 17.67 and Formula G in Figure 17.68, respectively) they decompose into elemental nitrogen and an organometallic compound. The italics immediately explain what happens in the second step of the reductive cyanation of a ketone shown in Figure 17.69. 

A more general and more complicated way is to use a reduction formula ... 

The reduction formula can be simplified by grouping the equivalent operations into classes,... 

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. 

With the 12-dimensional reducible representation of the Cartesian displacement vectors of HNNH, the inspection method probably does not work. However, the reduction formula can be used. The reducible representation is ... 

This is, of course, a reducible representation. Reduce it now with the reduction formula (see Chapter 4) ... 

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 ... 

This representation can now be reduced by using the reduction formula introduced in Chapter 4 ... 

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... 

We consider first the tridiagonalization process using Krylov space of dimension N Kl) =span c, Ac, A c. In this case, according to the definition, the matrix Q v is orthogonal, Therefore, the reduction formula (E.21)... 

Gram equivalent weight (as oxidatant or reductant) = formula weight (g) difference in oxidation state... 

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