Helpful Mathematical Tools

This introductory chapter gives the reader the tools necessary to understand the next three chapters in which molecular vibrations, electronic structure, and chemical reactions are discussed. Further reading is recommended for broader knowledge of the subject 

A mathematical group is a very general idea. It is a collection (set) of symbols or objects together with a rule telling us how to combine them. A simple example is a set of two numbers and addition for the rule. The theory of groups has a wide range of applications far beyond pure mathematics especially in physics and chemistry. Symmetry and group theory are inherently related to each other. When the symmetries of molecules are characterized by Schoenflies 

Hargittai, I. Hargittai, Symmetry through the Eyes of a Chemist, 3rd ed., 

A mathematical group is a set of elements related by certain rules. They will be illustrated on the symmetry operations. 

One element in the group must commute with all other elements in the group and leave them unchanged. This is the identity element, E. Thus, 

A mathematical group is a very general idea. It is a special case when the elements of the group are symmetry operations. When the symmetries of molecules are characterized by Schoenflies symbols, for example, C, C3, or 

The products of the elements in a group are always associative. That means that if there is a consecutive application of several symmetry operations, their application may be grouped in any way without changing the final result as long as the order of application remains the same. Thus, for example, 

For each element in a group, there is an inverse or reciprocal operation which is also an element of the group and satisfies the following condition  

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