Why do we need to recognize symmetry elements

In Fig. 3.4 we showed that a Ci axis is coincident with the C4 axis in Xep4. The Ci operation is equivalent to C4. The character table summarizes this information by stating 2C4 C2 , referring to C4 and C, and C4 = C2. The operation C4 is 

The left-hand column in a character table gives a list of symmetry labels. These are used in conjunction with the numbers, or characters, from the main part of the table to label the synunetry properties of, for example, molecular orbitals or modes of molecular vibrations. As we shall see in Chapter 5, although the symmetiy labels in the character tables are upper case (e.g. A, E, Tig), the corresponding symmetry labels for orbitals are lower case (e.g. a, e, hg)- Symmetry labels give us information about degeneracies as follows  

In Chapter 5, we use character tables to label the symmetries of orbitals, and to understand what orbital symmetries are allowed for a molecule possessing a particular symmetiy. 

Appendix 3 gives character tables for the most commonly encountered point groups, and each table has the same format as those in Tables 3.2 and 3.3. 

So far in this chapter, we have described the possible symmetry elements that a molecule might possess and, on the basis of these symmetry properties, we have illustrated how a molecular species can be assigned to a particular point group. Now we address some of the reasons why the recognition of symmetry elements in a molecule is important to the inorganic chemist. 

Most of the applications of symmetry fall into one of the following categories  

The next two sections deal briefly with the consequences of symmetry on observed bands in infrared spectra and with the relationship between molecular symmetry and chirality. In Chapter 21, we consider the electronic spectra of octahedral and tetrahedral J-block metal complexes and discuss the effects that molecular symmetry has on electronic spectroscopic properties. 

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