In this chapter we illustrate the solution of a simple physical problem in order to familiarize the reader with the procedures used in the group-theoretical analysis. Since this is the initial chapter, we shall give nearly all of the details involved in the analysis. Some readers familiar with group theory may find that the discussion includes too much detail, but we would rather be clear than brief. In later chapters less detail will be required since the reader will by then be familiar with the analysis method.

Theorems from group theory are stated and discussed when employed in the analysis, but the proofs of the theorems are not presented in this chapter. Readers interested in the proofs can find them in Appendix B or refer to a number of excellent standard group-theory texts [1.1].

The procedures employed in this chapter are simple but somewhat tedious and not the most efficient way to analyze the simple example discussed. However, these procedures will prove extremely valuable when we are faced with more complex problems. Therefore the reader is encouraged to work through the details of the chapter and the exercises at the end of the chapter.

In-plane molecular vibrations of squarene

As an introductory example we consider a fictitious square molecule we shall call “squarene”. The squarene molecule, shown in Fig. 1.1, lies in the plane of the paper with identical atoms at each corner of a square.