Molecular Symmetry and Spectroscopy

430 Pages·1979·22.951 MB·English

by  Philip Bunker (Auth.)

Molecular Symmetry
and Spectroscopy
Philip R. Bunker
Herzberg Institute of Astrophysics
National Research Council
Ottawa, Canada
ACADEMIC PRESS New York San Francisco London 1979
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT © 1979, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
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Library of Congress Cataloging in Publication Data
Bunker, Philip R
Molecular symmetry and spectroscopy.
Bibliography: p.
Includes index.
1. Molecular structure. 2. Molecular spectra.
I. Title.
QD461.B958 54Γ.28 78-51240
ISBN 0-12-141350-0
PRINTED IN THE UNITED STATES OF AMERICA
79 80 81 82 9 8 7 6 5 4 3 2 1
To Eva and Alex
Preface
This is a book about the use of group theory in quantum mechanics with
particular reference to problems in molecular spectroscopy. There are so
many books, so many good books, on this subject that it is hard to believe
that there can be any justification for writing another. The justification lies
in the fact that whereas the existing literature is concerned with the use of
the molecular point group, whose elements consist of rotations and reflec-
tions of vibronic variables, the present volume discusses the use of the
molecular symmetry group, whose elements consist of permutations of
identical nuclei with or without the inversion. The molecular symmetry
group is of more general use than the molecular point group since the effect
of molecular rotation and the effect of tunneling due to nonrigidity (such as
inversion tunneling in the ammonia molecule) are allowed for. Also, because
of the fundamental nature of its elements, the molecular symmetry group
provides a very good pedagogical vehicle for teaching group theory and the
application of group theory to problems in molecular spectroscopy.
The book is aimed at the serious student of molecular spectroscopy, and
although knowledge of the postulatory basis of quantum mechanics is
assumed, group theory is developed from first principles. The idea of the
molecular symmetry group is introduced early in the book (in Chapter 2)
xi
Xll Preface
after the definition of the concept of a group by using permutations. This
is followed by a discussion of point groups and rotation groups. The defini-
tion of the representations of a group and a general account of the use of
representations in classifying (or labeling) molecular states are given in
Chapters 4 and 5. In Chapter 6 the symmetry of the exact molecular
Hamiltonian is considered and the role of identical particle permutation
and overall rotation is stressed. In order to classify molecular states it is
necessary to obtain appropriate approximate molecular wavefunctions and
to understand how the wavefunctions transform under the effects of sym-
metry operations; these topics are discussed in Chapters 7, 8, and 10, and
the coordinates occurring in the approximate molecular wavefunctions (in
particular the Euler angles and vibrational normal coordinates) are dis-
cussed in detail. Chapter 9 is devoted to a detailed discussion of the defini-
tion of the molecular symmetry group and to the application of the definition
in a variety of circumstances. Near symmetry is defined in Chapter 11, and
the use of the energy level classification scheme obtained by using near
symmetry groups (such as the molecular point group), as well as true sym-
metry groups (such as the molecular symmetry group), in understanding
perturbations, optical selection rules, forbidden transitions, the Stark effect,
and the Zeeman effect is described. The special problems present in the
application of these symmetry ideas to linear molecules, to nonrigid mole-
cules, and to molecules in which spin-orbit coupling is strong are also
discussed. The electron spin double groups of molecular symmetry groups
are derived for the first time (in Chapter 10 and in Problem 12-1) and used
for the classification by symmetry of molecular wavefunctions fo rhalf-
integral angular momentum states. In the Appendixes the character tables
and correlation tables of some molecular symmetry groups are given.
A reader already conversant with abstract group theory, the use of point
groups, and the detailed form of molecular wavefunctions can jump straight
from the end of Chapter 2 to Chapter 9 (in which the molecular symmetry
group is defined in detail) and then to Chapters 10-12 in which the ap-
plications of the molecular symmetry group are given. Chapter 11 is the
central chapter of the book and the one in which the relationship of the
molecular symmetry group to the molecular point group is discussed in
detail (see, in particular, Figs. 11-3-11-5). In this chapter the usefulness of
the molecular symmetry group for labeling the states of rigid molecules
(i.e. molecules that do not tunnel between different conformations) is stressed.
I hope that the reader will gain from this book a good understanding of
the place of the molecular symmetry group, in relation to the molecular
point group and the molecular rotation group, in the application of group
theory to problems in molecular spectroscopy. To aid in this understanding
I have included many applications of the ideas as they arise and many
Preface Xlll
figures to show the effect of symmetry operations. I have also included
problems, followed by worked solutions, in the body of the text; as a result
of this the reader can "self pace" his reading by either (a) omitting to read
the problems and solutions, (b) solving the problems as they arise (and then
reading the solution in the text as a check), or (c) reading the problems and
solutions and treating them as a continuing part of the text.
Acknowledgments
I am very grateful to D. M. Bishop, J. M. Brown, R. E. Moss, T. Nakagawa,
and M. Vernon who each read the entire book in manuscript form and gave
much helpful advice in order to improve it. I am also grateful to H. H.
Günthard, J. Hardwick, G. Hills, J. T. Hougen, V. Laurie, I. M. Mills,
A. R. W. McKellar, M. T. Riggin, and S. Novick for their advice on particular
points. M. Herman, J. K. G. Watson, and B. P. Winnewisser each read and
corrected the proofs of the book, and my wife Eva, together with M. B.
Wadsworth and A. M. Lyyra, helped me to compare the proofs with the
typewritten manuscript; I am very grateful for their help.
I would especially like to thank Helen Letaif who typed the entire final
draft of the manuscript and made a special effort to ensure a well produced
and consistent product. Assisting in the typing of the first draft and in
typing the modifications that I made to the second draft were Denise
Charette, Gloria Dumoulin, Marilyn Nadon, and Lorette Ernst; I very much
appreciate all their help.
xv
Introduction
A molecule possesses structural symmetry that is of the same type as that
of a macroscopic body, and it can be described in terms of rotation axes
and reflection planes. For example, a methane molecule and a macroscopic
tetrahedron both have the same structural symmetry. This symmetry is
specified by saying that the molecule belongs to a certain point group, and
the group consists of a definite set of rotation and reflection operations (or
elements); for the methane molecule the point group is called T . In molecu-
d
lar physics we make great use of symmetry in order to label (or classify)
molecular energy levels. Point group symmetry is not the only symmetry
present in a molecule, and in this volume the various types of symmetry are
discussed at length. The use of these symmetries in labeling molecular states
and the application of these labels in understanding molecular processes
are also described.
The treatment given in this volume concerning the symmetry classification
of the states of polyatomic molecules can be understood by reference to
the symmetry classification scheme used for the states of diatomic molecules.
The reader unfamiliar with this scheme will have to take the next two
sentences on trust, but the scheme will be explained fully in Chapters 7 and
12. For a homonuclear diatomic molecule such as H the vibronic (vibration-
2
electronic) states are classified or labeled using the particular molecular
1
2 Introduction
point group D^. On the other hand the rovibronic (rotation-vibration-
electronic) energy levels (sometimes called rotational levels) are labeled
simply as + s, + a, — s, or — a depending on whether the rovibronic wave-
function is invariant ( + ) or changed in sign ( —) by the inversion of the mo-
lecular coordinates in the center of mass, and on whether it is invariant (s) or
changed in sign (a) by the interchange or permutation of the two identical
nuclei. Two such classification schemes are also possible for the states
of polyatomic molecules, but although the use of the molecular point group
to classify vibronic states is the subject of a vast textbook literature, the use
of inversion and nuclear permutation symmetry to classify rovibronic
states is but little described. Both classification schemes are useful in under-
standing and categorizing molecular interactions, and in this book, for a
change, emphasis is given to the use of inversion and nuclear permutation
symmetry; these symmetry elements constitute the molecular symmetry
group.
The molecular point group, whose elements are rotations and reflections
of vibronic variables, is used when studying the vibronic levels of a molecule
in a given electronic state that has a unique equilibrium configuration with
no observable tunneling between configurations (i.e., a rigid molecule). This
group is useful in understanding, for example, the infrared and Raman
activity of vibrational fundamentals, the terms that can occur in the molecular
potential function, and the atomic orbital functions that can be included
in a particular molecular orbital. Although the molecular point group is
introduced and defined in Chapter 3 here, and its use discussed in Chapter 11,
the reader is referred to the texts listed at the end of the chapter for a more
exhaustive discussion.
The molecular symmetry group, whose elements consist of the permuta-
tions of identical nuclei with or without inversion, is used when studying
rovibronic levels of a molecule and the presence of a unique equilibrium
configuration is immaterial. The molecular symmetry group has its origins
in the work of Christopher Longuet-Higgins and Jon Hougen. The impor-
tance of the molecular symmetry group lies not only in its use for studying
nonrigid molecules such as ammonia that have large amplitude vibrations,
or for studying electronic transitions in which there are changes in the
nuclear geometry, but also in its application to rovibronic as well as vibronic
states.
Apart from the limitations on using the molecular point group imposed
by the need for there to be a unique equilibrium configuration, it is very
important to realize that the molecular point group cannot be used if we
wish to allow for molecular rotation. The operations of the molecular point
group are rotations of the vibronic variables about molecule fixed axes
and reflections of the vibronic variables in planes defined by the molecule
Bibliographical Notes 3
fixed axes. For a rotating molecule the molecule fixed axes are not inertial
axes, and in this molecule fixed frame there will be centrifugal and Coriolis
forces. As a result the operations of the molecular point group are only
true symmetry operations (in the sense of not affecting the molecular energy)
for the nonrotating molecule. If we wish to allow for the effects of molecular
rotation (i.e., for the effects of centrifugal distortions and Coriolis forces)
we cannot use the molecular point group but must use the molecular sym-
metry group instead.
A simple example of the effect of molecular rotation is provided by the
methane molecule. The methane molecule has a tetrahedral equilibrium
geometry in its electronic ground state and we use the T point group to
d
classify the vibrational states. On the basis of point group symmetry con-
siderations, we would say that this molecule has no electric dipole moment
and no allowed electric dipole rotational spectrum. However centrifugal
distortion in the rotating molecule can give rise to a nonvanishing electric
dipole moment so that the molecule does have a rotational spectrum. The
molecular symmetry group of the methane molecule helps us to understand
which rovibration states can interact as a result of centrifugal distortion
and to determine which rotational transitions can occur in the spectrum.
The most important general idea that this book is designed to convey
is that in molecular physics we use two types of symmetry: true symmetry
and near symmetry. The molecular symmetry group is a group of true
symmetry operations for an isolated molecule, whereas the molecular point
group is a group of near symmetry operations. True symmetry is symmetry
that remains when every detail in the molecule is considered, and near
symmetry is the symmetry when certain details are neglected; for the molec-
ular point group the small details that are neglected are the effects of molec-
ular rotation. A true symmetry group is not "better" than a near symmetry
group in molecular applications, and it should be emphasized that the two
types of groups complement each other. However, the molecular symmetry
group provides a more fundamental and simpler vehicle for teaching group
theory and its use in molecular spectroscopy than does the molecular point
group.
BIBLIOGRAPHICAL NOTES
Diatomic Molecules
Herzberg (1950a).1 The symmetry of a diatomic molecule is discussed on pages 128-131
and 212-218.
1 References are given in full at the end of the book.