Group Theory for Chemists

Fundamental Theory and Applications

Kieran C Molloy

• Publisher: Woodhead Publishing
• Year: 2010
• ISBN:
  9780857092410
• Edition: 2
• Topic: Mathematics
• Subtopic: Group Theory
• Index: Mathematics

The basics of group theory and its applications to themes such as the analysis of vibrational spectra and molecular orbital theory are essential knowledge for the undergraduate student of inorganic chemistry. The second edition of Group Theory for Chemists uses diagrams and problem-solving to help students test and improve their understanding, including a new section on the application of group theory to electronic spectroscopy.Part one covers the essentials of symmetry and group theory, including symmetry, point groups and representations. Part two deals with the application of group theory to vibrational spectroscopy, with chapters covering topics such as reducible representations and techniques of vibrational spectroscopy. In part three, group theory as applied to structure and bonding is considered, with chapters on the fundamentals of molecular orbital theory, octahedral complexes and ferrocene among other topics. Additionally in the second edition, part four focuses on the application of group theory to electronic spectroscopy, covering symmetry and selection rules, terms and configurations and d-d spectra.Drawing on the author's extensive experience teaching group theory to undergraduates, Group Theory for Chemists provides a focused and comprehensive study of group theory and its applications which is invaluable to the student of chemistry as well as those in related fields seeking an introduction to the topic.

• Provides a focused and comprehensive study of group theory and its applications, an invaluable resource to students of chemistry as well as those in related fields seeking an introduction to the topic
• Presents diagrams and problem-solving exercises to help students improve their understanding, including a new section on the application of group theory to electronic spectroscopy
• Reviews the essentials of symmetry and group theory, including symmetry, point groups and representations and the application of group theory to vibrational spectroscopy

Part I

Symmetry and Groups

1

Symmetry

While everyone can appreciate the appearance of symmetry in an object, it is not so obvious how to classify it. The amide (1) is less symmetric than either ammonia or borane, but which of ammonia or borane – both clearly “symmetric” molecules – is the more symmetric ? In (1) the single N-H bond is clearly unique, but how do the three N-H bonds in ammonia behave ? Individually or as a group ? If as a group, how ? Does the different symmetry of borane mean that the three B-H bonds will behave differently from the three N-H bonds in ammonia ? Intuitively we would say “yes”, but can these differences be predicted ?

This opening chapter will describe ways in which the symmetry of a molecule can be classified (symmetry elements and symmetry operations) and also to introduce a shorthand notation which embraces all the symmetry inherent in a molecule (a point group symbol).

1.1 SYMMETRY

Imagine rotating an equilateral triangle about an axis running through its midpoint, by 120° (overleaf). The triangle that we now see is different from the original, but unless we label the corners of the triangle so we can follow their movement, it is indistinguishable from the original.

The symmetry inherent in an object allows it to be moved and still leave it looking unchanged. We define such movements as symmetry operations, e.g. a rotation, and each symmetry operation must be performed with respect to a symmetry element, which in this case is the rotation axis through the mid-point of the triangle.

It is these symmetry elements and symmetry operations which we will use to classify the symmetry of a molecule and there are four symmetry element ⁄ operation pairs that need to be recognised.

1.1.1 Rotations and Rotation Axes

In order to bring these ideas of symmetry into the molecular realm, we can replace the triangle by the molecule BF₃, which valence-shell electron-pair repulsion theory (VSEPR) correctly predicts has a trigonal planar shape; the fluorine atoms are labelled only so we can track their movement. If we rotate the molecule through 120° about an axis perpendicular to the plane of the molecule and passing through the boron, then, although the fluorine atoms have moved, the resulting molecule is indistinguishable from the original. We could equally rotate through 240°, while a rotation through 360° brings the molecule back to its starting position. Each of these rotations is a symmetry operation and the symmetry element is the rotation axis passing through boron.

Fig. 1.1 Rotation as a symmetry operation.

Remember, all symmetry operations must be carried out with respect to a symmetry element. The symmetry element, in this case the rotation axis, is called a three-fold axis and is given the symbol C₃. The three operations, rotating about 120°, 240° or 360°, are given the symbols C₃¹, C₃² and C₃³, respectively. The operations C₃¹ and C₃² leave the molecule indistinguishable from the original, while only C₃³ leaves it identical. These two scenarios are, however, treated equally for identifying symmetry.

In general, an n-fold C_n axis generates n symmetry operations corresponding to rotations through multiples...