Decomposing reducible representations

Decomposing Reducible Representation into Irreducible Ones... 

Decomposing reducible representations into irreducible ones... 

Before we deal with drawing out these molecular vibrations in Chapter 6, we will look at a more general way to decompose reducible representations into their irreducible constituents in Chapter 5. 

These six matrices can be verified to multiply just as the symmetry operations do thus they form another three-dimensional representation of the group. We see that in the Ti basis the matrices are block diagonal. This means that the space spanned by the Tj functions, which is the same space as the Sj span, forms a reducible representation that can be decomposed into a one dimensional space and a two dimensional space (via formation of the Ti functions). Note that the characters (traces) of the matrices are not changed by the change in bases. 

The basic idea of symmetry analysis is that any basis of orbitals, displacements, rotations, etc. transforms either as one of the irreducible representations or as a direct sum (reducible) representation. Symmetry tools are used to first determine how the basis transforms under action of the symmetry operations. They are then used to decompose the resultant representations into their irreducible components. 

In the derivation of normal modes of vibration we started with a set of displacements of individual atoms. By determining the reducible representation Ltot and decomposing it, we calculated the number of normal modes of each symmetry species. We could determine what these modes are by solving a secular equation. We could alternatively have used projection operators to determine the symmetry-adapted combinations. 

This reducible representation (the occupancy of two e orbitals in the anion gives rise to more than one state, so the direct product e x e contains more than one symmetry component) can be decomposed into pure symmetry components (labels T are used to denote the irreducible symmetries) by using the decomposition formula given in Appendix E ... 

We can now decompose our previous reducible representation into its irreducible components. For example, tf1=i(7.1 + 3.1+ 7.1+ 3.1) = 5, and similarly for the other three representations, with the result that the reducible representation contains five orbitals of the Ax type (totally symmetric) and two of the B2 type (antisymmetric) ... 

Exercise 11.7 For each nonnegative integer , decompose the representation of S U (F) on 0 mro a Cartesian sum of its irreducible components. Conclude that this representation is reducible. Is there a meaningful physical consequence or interpretation of this reducibility ... 

This reducible representation can be decomposed into irreducible representations as follows ... 

In applying the methods of group theory to problems related to molecular structure or dynamics, the procedure that is followed usually involves deriving a reducible representation for the phenomenon of interest, such as molecular vibration, and then decomposing it into its irreducible components. (A reducible representation will always be a sum of irreducible ones.) Although the decomposition can sometimes be accomplished by inspection, for the more general case, the following reduction... 

It may be asked, of course, whether this is the only way of decomposing the Ti representation. The answer is reassuring The decomposition of any reducible representation is unique. If we find a solution just by inspection of the character table, it will be the only one. Often this is the fastest and simplest way to decompose a reducible representation. 

The reducible representation is then decomposed by using the formula 1 ... 

The potential Vt is invariant under the operations of the group D4h whose characters are listed in Table 6. However, since d-electrons generare representations of even parity only, we may refer to the simpler group D4 whose characters are also contained in Table 6. Moreover, since an irreducible representation in O is generally a reducible representation in D4, it is possible, with the help of the character table andEqs. (42 and 46), to decompose the e- and t2-representations in O into irreducible representations in D4. The results are... 

Suppose that a basis F, of dimension n, can be decomposed into several bases F , whose dimensions are smaller (n,), each of which is globally stable with respect to all the symmetry operations of the group. Suppose also that it is not possible to decompose any of the representations Fi into representations whose dimensions are smaller than n . The reducible representation F is said to have been decomposed into a sum of irreducible representations F,-, which is written ... 

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 ... 

Fig. C.3. Reducible representation, block form, and irreducible representation. In the first row, the matrices F(Ri) are displayed that form a reilucible representation (eadi matrix corresponds to the symmetry operation Rj) the matrix elements are in general nonzero. The central row shows arepresentation F equivalent to the first one i.e., related by a similarity transformation (with matrix P). The new representation exhibits a block form i.e., in this particular case each matrix has two blocks of zeros that are identical in all matrices. The last row shows an equivalent representation F that corresponds to the smallest square blocks (of nonzeros) i.e., the maximum number of the blocks, of the form identical in all the matrices. Not only F, F, and F" are representations of the group, but also any sequence of individual blocks (as that shadowed) is a representation. Thus, F is decomposed into the four irreducible representations.

Any function that is a linear combination of the basis functions of a reducible representation can be decomposed into a linear combination of the basis functions of those irreducible representations that form the reducible representation. 

Stage 2. The reducible representation describes the genuine (internal) vibrations as well as the six apparent vibrations (three translations and three rotations). The apparent vibrations can be easily eliminated by throwing away (from the total reducible representation) those irreducible representations that correspond to x, y, z (translations) and R c, R,., R- (rotations). What the latter ones are, we know from the corresponding table of characters. To summarize, the abovementioned reducible representation has to be decomposed into the irreducible ones. The decomposition yields r = a(F Fi +a(F2)F2 +a(F3)F3... From this decomposition, we have to subtract (in order to eliminate the apparent vibrations) all the irreducihle representations the X, y, z, Ra , Ry R belong to. 

In order to decompose a reducible representation into the iirreducible representations, we do not need the reducible representation be given in details. It is sufficient to know its characters (p. e35). These characters are easy to deduce just by considering what happens to the displacement vectors along x,-, y,-, and Zi (for atom i) under all the symmetry operations. What... 

Thus, the characters of the reducible representation have been found. In order to decompose the representation, we have to know the table of characters for the D3/, symmetry group, shown in Table C.9. 

This means that the reducible representation in question decomposes into... 

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