8 - Group Theory

Abstract

This chapter explores the application of group theory of interest to physicists and chemists. The most obvious application of group theory is its use in specifying the geometry of a molecule— such as, planar and tetrahedral. The chapter shows that this theory is equally capable of describing the symmetry of mathematical functions and discusses the conditions necessary for defining a group in the mathematical sense with examples. The chapter presents permutation operations that are carried out on symbols rather than physical objects used to identify infants. To identify the symmetry group, it is necessary to visualize all of the operations, which characterize the structure of the molecule. The symmetry operations and their descriptions as well as the notation established by Schönflies are given in the chapter. The chapter explains the application of character in group theory. The character of a matrix transformation is invariant under a similarity transformation, but also that all of the representations of the same class have the same character. The basic theorem of representation theory is used for irreducible representations. The chapter illustrates the “magical formula” used to reduce a representation. The general principle of direct product representation is applicable to representations where the term character is used to describe the trace. The chapter explains the concepts of hybridization of atomic orbital's and crystal symmetry in group theory.