Group Theory for Chemists: Fundamental Theory and Applications

229 Pages·2011·11.69 MB·English

by  Kieran C Molloy

Group Theory for Chemists
'Talking of education, people have now a-days" (said he) "got a strange opinion that every
thing should be taught by lectures. Now, I cannot see that lectures can do so much good as
reading the books from which the lectures are taken. I know nothing that can be best taught
by lectures, except where experiments are to be shewn. You may teach chymestry by
lectures — You might teach making of shoes by lectures!"
From James Boswell's Life of Samuel Johnson, 1766
About the Author
Kieran MoIIoy was born in Smethwick, England and educated at Halesowen
Grammar School after which he studied at the University of Nottingham where he
obtained his BSc, which was followed by a PhD degree in chemistry, specialising
in main group organometallic chemistry. He then accepted a postdoctoral position
at the University of Oklahoma where he worked in collaboration with the late
Professor Jerry Zuckerman on aspects of structural organotin chemistry of
relevance to the US Navy.
His first academic appointment was at the newly established National Institute for
Higher Education in Dublin (now Dublin City University), where he lectured from
1980 to 1984. In 1984, Kieran Molloy took up a lectureship at the University of
Bath, where he has now become Professor of Inorganic Chemistry. His many
research interests span the fields of synthetic and structural inorganic chemistry
with an emphasis on precursors for novel inorganic materials.
In 2003 he was joint recipient of the Mary Tasker prize for excellence in teaching,
an award given annually by the University of Bath based on nominations by
undergraduate students. This book Group Theory for Chemists is largely based
on that award-winning lecture course.
G r o u p T h e o r y f o r C h e m i s t s
Fundamental Theory and Applications
Second Edition
Kieran C. Molloy
W P
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First edition 2004, Horwood Publishing Limited
Second edition 2011, Woodhead Publishing Limited
©K. C. Molloy, 2011
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ISBN 978-0-85709-241-0 (online)
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Table of Contents
Preface viii
Parti
Symmetry and Groups
1. Symmetry
1.1 Symmetry 2
1.2 Point Groups 7
1.3 Chirality and Polarity 13
1.4 Summary 14
Problems 15
2. Groups and Representations
2.1 Groups 16
2.2 Transformation Matrices 18
2.3 Representations of Groups 19
2.4 Character Tables 24
2.5 Symmetry Labels 26
2.6 Summary 27
Problems 28
Part Π
Application of Group Theory to Vibrational Spectroscopy
3. Reducible Representations
3.1 Reducible Representations 30
3.2 The Reduction Formula 34
3.3 The Vibrational Spectrum of S0 35
2
3.4 Chi Per Unshifted Atom 38
3.5 Summary 41
Problems 41
4. Techniques of Vibrational Spectroscopy
4.1 General Considerations 43
4.2 Infrared Spectroscopy 45
4.3 Raman Spectroscopy 46
4.4 Rule of Mutual Exclusion 47
4.5 Summary 50
Problems 50
5. The Vibrational Spectrum of Xe(0)F
4
5.1 Stretching and Bending Modes 52
5.2 The Vibrational Spectrum of Xe(0)F 57
4
5.3 Group Frequencies 60
Problems 62
Part ΙΠ
Application of Group Theory to Structure and Bonding
6. Fundamentals of Molecular Orbital Theory
6.1 Bonding in H 66
2
6.2 Bonding in Linear H 67
3
6.3 Limitations in a Qualitative Approach 70
6.4 Summary 72
Problems 72
7. H0 - Linear or Angular ?
2
7.1 Symmetry-Adapted Linear Combinations 74
7.2 Central Atom Orbital Symmetries 75
7.3 A Molecular Orbital Diagram for H0 76
2
7.4 A C / Dooh Correlation Diagram 77
2v
7.5 Summary 80
Problems 80
8. NH - Planar or Pyramidal ?
3
8.1 Linear or Triangular H ? 81
3
8.2 A Molecular Orbital Diagram for BH 84
3
8.3 Other Cyclic Arrays 86
8.4 Summary 90
Problems 90
9. Octahedral Complexes
9.1 SALCs for Octahedral Complexes 93
9.2 (/-Orbital Symmetry Labels 95
9.3 Octahedral P-Block Complexes 96
9.4 Octahedral Transition Metal Complexes 97
9.5 -πBonding and the Spectrochemical Series 98
9.6 Summary 100
Problems 101
10. Ferrocene
10.1 Central Atom Orbital Symmetries 105
10.2 SALCs for Cyclopentadienyl Anion 105
10.3 Molecular Orbitals for Ferrocene 108
Problems 111
Part IV
Application of Group Theory to Electronic Spectroscopy
11. Symmetry and Selection Rules
11.1 Symmetry of Electronic States 115
11.2 Selection Rules 117
11.3 The Importance of Spin 119
vii
11.4 Degenerate Systems 120
11.5 Epilogue - Selection Rules for Vibrational Spectroscopy 124
11.6 Summary 125
Problems 125
12. Terms and Configurations
12.1 Term Symbols 128
12.2 The Effect of a Ligand Field - Orbitals 131
12.3 Symmetry Labels for cf Configurations - An Opening 133
12.4 Weak Ligand Fields, Terms and Correlation Diagrams 136
12.5 Symmetry Labels for cf Configurations - Conclusion 142
12.6 Summary 143
Problems 145
13. d-d Spectra
13.1 The Beer-Lambert Law 146
13.2 Selection Rules and Vibronic Coupling 147
13.3 The Spin Selection Rule 150
13.4 d-d Spectra - High-Spin Octahedral Complexes 151
13.5 d-d Spectra - Tetrahedral Complexes 154
13.6 d-d Spectra - Low-Spin Complexes 156
13.7 ascending Symmetry 158
13.8 Summary 163
Problems 164
Appendix 1 Projection Operators 166
Appendix 2 Microstates and Term Symbols 175
Appendix 3 Answers to SAQs 178
Appendix 4 Answers to Problems 196
Appendix 5 Selected Character Tables 211
Index 216
viii
PREFACE
The book I have written is based on a course of approximately 12 lectures and 6
hours of tutorials and workshops given at the University of Bath. The course deals
with the basics of group theory and its application to the analysis of vibrational
spectra and molecular orbital theory. As far as possible I have tried to further refer
group theory to other themes within inorganic chemistry, such as the links between
VSEPR and MO theory, crystal field theory (CFT) and electron deficient molecules.
The book is aimed exclusively at an undergraduate group with a highly focussed
content and thus topics such as applications to crystallography, electronic spectra etc
have been omitted. The book is organised to parallel the sequence in which I present
the material in my lectures and is essentially a text book which can be used by
students as consolidation. However, group theory can only be mastered and
appreciated by problem solving, and I stress the importance of the associated
problem classes to my students. Thus, I have interspersed self-assessment questions
to reinforce material at key stages in the book and have added additional exercises at
the end of most chapters. In this sense, my offering is something of a hybrid of the
books by Davidson, Walton and Vincent.
I have made two pragmatic decisions in preparing this book. Firstly, there is no
point in writing a textbook that nobody uses and the current vogue among
undergraduates is for shorter, more focussed texts that relate to a specific lecture
course; longer, more exhaustive texts are likely to remain in the bookshop, ignored
by price-conscious purchasers who want the essentials (is it on the exam paper ?) and
little more. Secondly, the aim of a textbook is to inform and there seems to me little
point in giving a heavily mathematical treatment to a generation of students for
whom numbers are an instant turn-off. I have thus adopted a qualitative, more
pictorial approach to the topic than many of my fellow academics might think
reasonable. The book is thus open to the inevitable criticism of being less than
rigorous, but, as long as I have not distorted scientific fact to the point of falsehood, I
am happy to live with this.
Note for Students
Group theory is a subject that can only be mastered by practising its application. It
is not a topic which lends itself to rote-learning, and requires an understanding of the
methodology, not just a knowledge of facts. The self-assessment questions (SAQs)
which can be found throughout the book are there to test your understanding of the
information which immediately precedes them. You are strongly advised to tackle
each of these SAQs as they occur and to check your progress by reference to the
answers given in Appendix 3. Longer, more complex problems, some with answers,
can be found at the end of each chapter and should be used for further consolidation
of the techniques.
Note for Lecturers
In addition to the SAQs and problems for which answers have been provided there
are a number of questions at the end of most chapters for which solutions have not
been given and which may be useful for additional tutorial or assessment work.
ix
Acknowledgements
In producing the lecture course on which this book is based I relied heavily on the
textbook Group Theory for Chemists by Davidson, perhaps naturally as that author
had taught me the subject in my undergraduate years. Sadly, that book is no longer
in print - if it were I would probably not have been tempted to write a book of my
own. I would, however, like to acknowledge the influence that book primus inter
pares* has had on my approach to teaching the subject.
I would also like to offer sincere thanks to my colleagues at the University of Bath
- Mary Mahon, Andy Burrows, Mike Whittlesey, Steve Parker, Paul Raithby - as
well as David Cardin from the University of Reading, for their comments, criticisms
and general improvement of my original texts. In particular, though, I would like to
thank David Liptrot, a Bath undergraduate, for giving me a critical student view of
the way the topics have been presented. Any errors and shortcomings that remain
are, of course, entirely my responsibility.
Kieran Molloy
University of Bath, August 2010
* Other texts on chemical group theory, with an emphasis on more recent works,
include:
J S Ogden, Introduction to Molecular Symmetry (Oxford Chemistry Primers 97),
OUP.2001.
A Vincent, Molecular Symmetry and Group Theory: A Programmed Introduction
to Chemical Applications, 2nd Edition, John Wiley and Sons, 2000.
Ρ Η Walton, Beginning Group Theory for Chemistry, OUP, 1998.
Μ Ladd, Symmetry and Group Theory in Chemistry, Horwood Chemical Science
Series, 1998.
R. L. Carter, Molecular Symmetry and Group Theory, Wiley and Sons, 1998.
G Davidson, Group Theory for Chemists, Macmillan Physical Science Series,
1991.
F A Cotton, Chemical Applications of Group Theory, 3rd Edition, John Wiley and
Sons, 1990.