Molecular Symmetry and Group Theory

A Programmed Introduction to Chemical Applications

Alan Vincent

• Publisher: Wiley
• Year: 2013
• ISBN:
  9781118723388
• Edition: 2
• Topic: Physical Sciences
• Subtopic: Inorganic Chemistry

This substantially revised and expanded new edition of the bestselling textbook, addresses the difficulties that can arise with the mathematics that underpins the study of symmetry, and acknowledges that group theory can be a complex concept for students to grasp.

Written in a clear, concise manner, the author introduces a series of programmes that help students learn at their own pace and enable to them understand the subject fully. Readers are taken through a series of carefully constructed exercises, designed to simplify the mathematics and give them a full understanding of how this relates to the chemistry.

This second edition contains a new chapter on the projection operator method. This is used to calculate the form of the normal modes of vibration of a molecule and the normalised wave functions of hybrid orbitals or molecular orbitals.

The features of this book include:

• A concise, gentle introduction to symmetry and group theory
• Takes a programmed learning approach
• New material on projection operators, and the calcultaion of normal modes of vibration and normalised wave functions of orbitals

This book is suitable for all students of chemistry taking a first course in symmetry and group theory.

Programme 1

Symmetry Elements and Operations

Objectives

After completing this programme, you should be able to:

1. Recognise symmetry elements in a molecule.

2. List the symmetry operations generated by each element.

3. Combine together two operations to find the equivalent single operation.

All three objectives are tested at the end of the programme.

Assumed Knowledge

Some knowledge of the shapes of simple molecules is assumed.

Symmetry Elements and Operations

1.1 The idea of symmetry is a familiar one, we speak of a shape as being “symmetrical”, “unsymmetrical” or even “more symmetrical than some other shape”. For scientific purposes, however, we need to specify ideas of symmetry in a more quantitative way.

Which of the following shapes would you call the more symmetrical?

1.2 If you said A, it shows that our minds are at least working along similar lines!

We can put the idea of symmetry on a more quantitative basis. If we rotate a piece of cardboard shaped like A by one third of a turn, the result looks the same as the starting point:

Since A and A′ are indistinguishable (not identical) we say that the rotation is a symmetry operation of the shape.

Can you think of another operation you could perform on a triangle of cardboard which is also a symmetry operation? (Not the anticlockwise rotation!)

1.3 Rotate by half a turn about an axis through a vertex i.e. turn it over

How many operations of this type are possible?

1.4 Three, one through each vertex.

We have now specified the first of our symmetry operations, called a PROPER ROTATION, and given the symbol C. The symbol is given a subscript to indicate the ORDER of the rotation. One third of a turn is called C₃, one half a turn C₂, etc.

What is the symbol for the operation:

1.5 C₄. It is rotation by

of a turn.

A symmetry operation is the operation of actually doing something to a shape so that the result is indistinguishable from the initial state. Even if we do not do anything, however, the shape still possesses an abstract geometrical property which we term a symmetry element. The element is a geometrical property which is said to generate the operation. The element has the same symbol as the operation.

What obvious symmetry element is possessed by a regular six-sided shape:

1.6 C₆, a six-fold rotation axis, because we can rotate it by

of a turn

One element of symmetry may generate more than one operation e.g. a C₃ axis generates two operations called C₃ and

:

What operations are generated by a C₅ axis?

1.7

What happens if we go one stage further i.e.

?

1.8 We get back to where we started i.e.

The shape is now more than indistinguishable, it is IDENTICAL with the starting point. We say that

, or indeed any

, where E is the IDENTITY OPERATION, or the operation of doing nothing. Clearly this operation can be performed on anything because everything looks the same after doing nothing to it! If this sounds a bit trivial I apologise, but it is necessary to include the identity in the description of a molecule’...