The reduction formula

In Chapter 4 we saw how a representation could be used to mimic the symmetry properties of a molecule by describing the interaction of group operations with a particular basis. 

Any collection of basis vectors that complies with the molecular symmetry can generate a character representation of the group, but in most cases it will be a reducible one and so can be simplified. In this section we will show that the simplification of a reducible representation r can be made using the data for the set of irreducible representations available in the standard character tables. 

Section 4.11 used the matrix representation to deal with a set of three basis vectors jc, y, z on the central atom of a square planar D41, complex. It was shown that this basis can be reduced to + A2U by inspection of the matrices for the operations in the Ah group. The characters for the reducible and irreducible representations are shown in Table 5.2. 

The breakdown of the 3 x 3 matrices into 2x2 and 1 x 1 does not affect any of the diagonal elements, which are the characters of our irreducible representations. This means that the irreducible characters in each class must add up to the character in the reducible representation they were derived from. 

By inspection of Table 5.2, it can be seen that this is indeed the case. For any class of operations in the group, the characters of the irreducible representations sum to give 

As we have already seen ( 6.2.5), the charaaer table gives us information on orbital symmetry properties. If the molecule contains a central atom, the symmetries of the orbitals of this atom are indicated in the last two columns of the table. However, the orbitals on non-central atoms, for example the Ish orbitals in H2O or NH3, are not individually bases for an irreducible representation (Tables 6.1 and 6.3). These AO form a basis for a reducible representation that can be decomposed into a sum of irreducible representations of the point group. Although the character table does not give the result immediately, it does enable us to find it by using the reduction formula. 

If the characters /r associated with a reducible representation F are known, it can be decomposed into a sum of irreducible representations (F = fl, Fj) of the point group by using the reduction formula  

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

We need intially to establish the characters of a reducible representation r. For a given symmetry operation R, only the diagonal terms of the matrix associated with this operation contribute to the character xr ( 6.2.4.2). If the symmetry operation transforms the orbital under consideration into itself, the contribution to the character is +1. If however, it transforms the orbital into its opposite, the contribution is —1. If the orbital is transformed into another one, the contribution is zero. We have already given an example of the calculation of these characters for the Ish orbitals on the hydrogen atoms in NH3 (Table 6.4). We shall now consider tow other examples, for the molecules H2O and C2H4. 

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. 

---

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... 

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)... 

In this expression, there are just sufficient equations to determine the complex system of partial fractions, by equating the coefficients of like powers of x. The integration of many of the resulting expressions usually requires the aid of one of the reduction formulae ( 76). 

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

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). 

The order of the group (h, the number of symmetry operations) is 16 (first line of Table 6.18). The reduction formula (6.5) enables us to decompose the four-dimensional representation F into a sum of irreducible representations, The only non-zero values of at are ... 

---