Applications of group theory to

1.2 Reducible and irreducible representations of a group 12 1.4 Construction of the force-constant matrix from the 2 Molecular vibrations of isotopically substituted AB2molecules 39 2.1

A P P L I C AT I O N S O F G RO U P T H E O RY TO ATO M S ,

M O L E C U L E S , A N D S O L I D S

The majority of all knowledge concerning atoms, molecules, and solids hasbeen derived from applications of group theory Taking a unique, applications-oriented approach, this book gives readers the tools needed to analyze any atomic,molecular, or crystalline solid system

Using a clearly defined, eight-step program, this book helps readers to stand the power of group theory, what information can be obtained from it, and how

under-to obtain it The book takes in modern under-topics, such as graphene, carbon nanotubes,and isotopic frequencies of molecules, as well as more traditional subjects: thevibrational and electronic states of molecules and solids, crystal-field and ligand-field theory, transition-metal complexes, space groups, time-reversal symmetry,and magnetic groups

With over a hundred end-of-chapter exercises, this book is invaluable for uate students and researchers in physics, chemistry, electrical engineering, andmaterials science

grad-T H O M A S W O L F R A M is a former Chairman and Professor of the Department

of Physics and Astronomy, University of Missouri-Columbia He has founded ascience and technology laboratory for a major company and started a companythat manufactured diode-pumped, fiber-optic transmitters and amplifiers

¸S˙I N A S˙I E L L˙I A L T I O ˘ G L Uis a former Chairman and Professor of Physics at theMiddle East Technical University in Ankara, Turkey He has been a recipient ofHumboldt and Fulbright Fellowships Currently he is a Professor of Physics andDirector of Basic Sciences at TED University in Ankara

A P P L I C AT I O N S O F G RO U P T H E O RY

TO ATO M S , M O L E C U L E S ,

A N D S O L I D S

T H O M A S WO L F R A M

Formerly of the University of Missouri-Columbia

¸S ˙I NA S ˙I E L L ˙I A LT I O ˘G L U

TED University, Ankara Formerly of Middle East Technical University, Ankara

Published in the United States of America by Cambridge University Press, New York Cambridge University Press is part of the University of Cambridge.

It furthers the University’s mission by disseminating knowledge in the pursuit of

education, learning, and research at the highest international levels of excellence.

www.cambridge.org Information on this title: www.cambridge.org/9781107028524

c

 T Wolfram and ¸S Ellialtıo˘glu 2014 This publication is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 2014 Printed in the United Kingdom by CPI Group Ltd, Croydon CR0 4YY

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication data

Wolfram, Thomas, 1936–

Applications of group theory to atoms, molecules, and solids / Thomas Wolfram, ¸Sinasi Ellialtıo˘glu.

pages cm ISBN 978-1-107-02852-4 (hardback)

1 Solids – Mathematical models 2 Molecular structure 3 Atomic structure 4 Group theory.

I Title.

QC176.W65 2013 530.4 1015122–dc232013008434 ISBN 978-1-107-02852-4 Hardback Cambridge University Press has no responsibility for the persistence or

accuracy of URLs for external or third-party internet websites referred to

in this publication, and does not guarantee that any content on such

websites is, or will remain, accurate or appropriate.

1.2 Reducible and irreducible representations of a group 12

1.4 Construction of the force-constant matrix from the

2 Molecular vibrations of isotopically substituted AB2molecules 39

2.1 Step 1: Identify the point group and its symmetry operations 39

2.2 Step 2: Specify the coordinate system and the basis functions 39

2.3 Step 3: Determine the effects of the symmetry operations on

2.4 Step 4: Construct the matrix representations for each element

2.5 Step 5: Determine the number and types of irreducible

2.6 Step 6: Analyze the information contained in the

2.8 Step 8: Diagonalize the matrix eigenvalue equation 50

2.10 Green’s function theory of isotopic molecular vibrations 52

2.11 Results for isotopically substituted forms of H2O 60

References 62

3.4 Decomposition of D (l)in a non-spherical environment 75

3.6 General properties of direct-product groups and representations 79

3.8 General representations of the full rotation group 85

5.3 Irreducible representations of half-integer angular

5.9 The Zeeman effect (weak-magnetic-field case) 153

6.4 An example: The electronic structure of squarene 168

Contents vii

7.1 Bonding and antibonding states: Symmetry functions 193

7.2 The “building-up” of molecular orbitals for diatomic

9.9 Compatibility of the irreducible representations of gk 260

9.10 Energy bands in the plane-wave approximation 265

10.4 Irreducible representations for the perovskite energy bands 284

11 Applications of space-group theory: Lattice vibrations 304

11.1 Eigenvalue equations for lattice vibrations 305

11.4 Lattice vibrations for the perovskite structure 320

12.2 The effect ofT on an electron wavefunction 340

12.6 Co-representations for groups with time-reversal operators 350

12.7 Degeneracies due to time-reversal symmetry 357

13.2 The analogy with the Dirac relativistic theory for massless

Contents ix

Appendix D Tensors, vectors, and equivalent electrons 442

The majority of all knowledge accumulated in physics and chemistry concerningatoms, molecules, and solids has been derived from applications of group theory toquantum systems

My (T.W.) first encounter with group theory was as an undergraduate in physics,

struggling to understand Wigner’s Group Theory and Its Application to the tum Mechanics of Atomic Spectra (1959) I felt there was something magical about

Quan-the subject It was amazing to me that it was possible to analyze a physical systemknowing only the symmetry and obtain results that were absolute, independent ofany particular model To me it was a miracle that it was possible to find some exacteigenvectors of a Hamiltonian by simply knowing the geometry of the system orthe symmetry of the potential

Many books devote the initial chapters to deriving abstract theorems before cussing any of the applications of group theory We have taken a different approach.The first chapter of this book is devoted to finding the molecular vibration eigen-values, eigenvectors, and force constants of a molecule The theorems required toaccomplish this task are introduced as needed and discussed, but the proofs of thetheorems are given in the appendices (In later chapters the theorems needed for theanalysis are derived within the discussions.) By means of this applications-orientedapproach we are able to immediately give a general picture of how group theory isapplied to physical systems The emphasis is on the process of applying group the-ory The various steps needed to analyze a physical system are clearly delineated

dis-By the end of the first chapter the reader should have an appreciation for the power

of group theory, what information can be obtained, and how to obtain it That is,the “magic” of group theory should already be apparent

In addition to the essential, traditional topics, there are new topics, includingthe electronic and vibrational properties of graphene and nanotubes, the vibra-tions of isotopically substituted molecules, localized vibrations, and discussions ofthe axially symmetric lattice dynamics model The energy bands and vibrational

normal modes of crystals with the perovskite structure are also discussed indetail.

The material in this book was developed in part from group-theory courses andfrom a series of lectures presented in courses on special topics at the University ofMissouri-Columbia It is appropriate for science and engineering graduate studentsand advanced undergraduate seniors The ideal reader will have had a course inquantum mechanics and be familiar with eigenvalue problems and matrix algebra.However, no prerequisite knowledge of group theory is necessary

This book may be employed as a primary text for a first course in group ory or as an auxiliary book for courses in quantum mechanics, solid-state physics,physical chemistry, materials science, or electrical engineering It is intended as aself-teaching tool and therefore the analyses in the early chapters are given in somedetail Each chapter includes a set of exercises designed to reinforce and extend thematerial discussed in the chapter

the-Thomas Wolfram

and

¸Sinasi Ellialtıo˘glu

Introductory example: Squarene

In this chapter we illustrate the solution of a simple physical problem in order tofamiliarize the reader with the procedures used in the group-theoretical analysis.Since this is the initial chapter, we shall give nearly all of the details involved inthe analysis Some readers familiar with group theory may find that the discussionincludes too much detail, but we would rather be clear than brief In later chaptersless detail will be required since the reader will by then be familiar with the analysismethod

Theorems from group theory are stated and discussed when employed in theanalysis, but the proofs of the theorems are not presented in this chapter Read-ers interested in the proofs can find them in Appendix Bor refer to a number ofexcellent standard group-theory texts [1.1]

The procedures employed in this chapter are simple but somewhat tedious andnot the most efficient way to analyze the simple example discussed However, theseprocedures will prove extremely valuable when we are faced with more complexproblems Therefore the reader is encouraged to work through the details of thechapter and the exercises at the end of the chapter

1.1 In-plane molecular vibrations of squarene

As an introductory example we consider a fictitious square molecule we shall call

“squarene” The squarene molecule, shown in Fig.1.1, lies in the plane of the paperwith identical atoms at each corner of a square Our aim in this chapter is to usegroup theory to assist in determining the vibrational frequencies and motions ofthe normal modes (eigenvalues and eigenvectors) of the molecule

In general the molecular vibration problem may be cast into the form of a matrixeigenvalue problem,

F ξ = ω2M ξ,

1 2

3 4

Figure 1.1 The squarene molecule.

whereF is a matrix of “force constants” and ξ is a column vector whose

compo-nents are the vibrational displacements of the atoms The matrixM is a diagonalmass matrix, Mi j = m i δ i j where m i is the mass of the atom with coordinatecomponentξ i andω is the angular frequency of vibration This equation may be

transformed to a standard eigenvalue form,

Fη = ω2η,

withF= M−1/2F M−1/2andη = M1/2 ξ.

In the case of squarene all the masses are the same, soM can be replaced by a

scalar, m, the common mass of the four atoms Therefore the eigenvalue equation

we are concerned with is the 8× 8 matrix equation

One approach to the molecular vibrations of squarene would be to postulate aforce-constant model,F, and solve the resulting eigenvalue problem In its mostgeneral form, F will have 64 non-zero matrix elements If no simplifications aremade, the diagonalization of an 8× 8 matrix could be daunting without the use of

a computer If approximations are employed to simplifyF, it will not be evidentwhether the results are constrained by symmetry or by the assumptions made in thepostulated force-constant model The use of group theory not only greatly simpli-fies the analysis, but also yields the most general form possible forF and providesphysical insight into the nature of the normal modes of vibration

1.1 In-plane molecular vibrations of squarene 3

Often, the normal-mode frequencies can be measured by spectroscopic means;infrared or Raman spectroscopy, for example However, the force constants (thematrix elements of F) can not be measured directly They are usually inferred

by using a specific model and choosing the constants so that the eigenfrequenciesmatch the observed frequencies This method does not always produce a unique set

of force constants, and other spectroscopic data must be employed; for example,the frequencies of isotopically substituted molecules and/or elastic constants

In Chapter 2 we shall investigate how group theory can be used to predictisotopic frequency shifts In fact, in some cases the frequencies of an isotopic mole-cule can be expressed in terms of frequencies of the non-isotopic molecule [1.2]without ever having to specify the force-constant matrix.1

There is a general order to the way in which group theory is employed to lyze an eigenvalue problem We shall label the various steps as “Step 1” through

ana-“Step 8”

1.1.1 Step 1: Identify the point group and its symmetry operations

At rest, the squarene molecule has the symmetry of a square For simplicity weconfine our discussion to the two dimensions of the plane of the molecule (There

is no loss of generality in this assumption because the in-plane vibrational modes

do not mix with the motions perpendicular to the plane of the molecule (see cise 1.16).) The operations that rotate or reflect an object into a configuration

Exer-indistinguishable from its original configuration are called the covering tions and constitute a group For squarene these operations are rotations about an axis perpendicular to the plane of the molecule and reflections through lines in the

opera-plane of the molecule For example we can rotate the square by 90 degrees or anymultiple of 90 degrees about an axis perpendicular to the plane passing throughthe center of the square So there are four rotations: 90-degree, 180-degree, 270-degree, and 360-degree rotations Rotation by 360 degrees is taken to be the same

as “doing nothing” The “doing-nothing” operation is usually called the “identity”

operation and denoted by the symbol E There are seven other symmetry

opera-tions If we draw a diagonal line across the square and reflect the corners throughthis line, the square remains unchanged There are two different diagonal lines thatpass through the center and two corners of the square Another pair of reflectionlines passes through the middle of the square and bisects two opposite sides Thesesymmetry operations are shown schematically in Fig 1.2 The point group cor-

responding to these operations is “C4v ” (In three dimensions the group is D 4h

1 Assuming harmonic motions and neglecting very small changes in the force constants due to isotope substitution.

3 4

Figure 1.2 The squarene molecule and its symmetry elements.

Methods for identifying the group of a molecule are discussed in AppendixG.) For

C4vthere are in total eight symmetry elements.2

(1) E= a 0-degree or 360-degree rotation

(2) C4= a 90-degree rotation

(3) C2= a 180-degree rotation

(4) C43= a 270-degree rotation

(5) σ v1= reflection in a diagonal line

(6) σ v2= reflection in a diagonal line

(7) σ d1 = reflection in a line cutting the sides of the square

(8) σ d2 = reflection in a line cutting the sides of the square

For the remainder of this chapter we shall regard the symbols E, C4, σ v1 etc as

symmetry operators that act on the molecule, not the coordinate system.3Thus, for

example C4 rotates the molecule and its nuclear displacement by 90 degrees in aclockwise manner

The formal definition of a group is given in Theorem1.1

2 Inversion through the center of the square (i.e., r → −r) is a symmetry operation However, the effect of

inversion is the same as the effect of a C2 rotation in this case and so may be omitted.

3 There is a distinction between the symmetry elements and the symmetry operators, but the group of the operators is isomorphic to the group of the elements, which means that the two groups are equivalent group theoretically.

1.1 In-plane molecular vibrations of squarene 5

Theorem 1.1 (Definition of a group)

1 A group is a set of elements that are closed under group multiplication That is, the product of any two elements in the group produces another element of the group If A and B are in the group then so is A B.

2 The associative law holds: A(BC) = (AB)C.

3 There is a unit or identity element, E, such that E A = AE = A.

4 Every element in the group has an inverse element that is also in the group.

If A is an element then A−1 is also an element in the group, and A A−1 =

1.1.2 Step 2: Specify the coordinate system and basis functions

The coordinate system to be employed in the analysis is often a set of Cartesiancoordinates,4but there are many other choices; Euler angles, internal coordinates,cylindrical coordinates, and spherical coordinates, to name a few In general thechoice of basis functions depends on the physical phenomena to be analyzed Inthe case of molecular vibrations we are concerned with the motions of each of theatoms comprising the molecule Therefore an appropriate choice of basis functions

is the atomic displacement vectors To measure the displacements of each atomfrom its equilibrium position we label the corner positions of the square (not the

atoms of the molecule) and construct X –Y Cartesian coordinate systems anchored

to these corner positions as shown in Fig.1.3(a)

1.1.3 Step 3: Determine the effects of the symmetry operations

on the basis functions

We consider how a vector representing the displacement of one of the atoms istransformed when the molecule is subjected to one of the symmetry operations

Note that the X –Y coordinate systems as well as the position numbers of the

cor-ners are fixed Imagine the molecule placed on top of the square in Fig 1.3(a)and rotated or reflected under the action of a symmetry operation If we rotate the

molecule clockwise by 90 degrees (the C4 operation) the atoms of the moleculemove to different corner positions as indicated by Table1.1

4 In chemistry texts internal coordinates are often used See reference [ 1.3 ].

34

X3 X4

Y3 Y4

Figure 1.3 (a) The coordinate system Here 1, 2, 3, and 4 label the fixed corners

of the square, and X i and Y i label the fixed coordinate axes (b) The effect of C4

rotation on the displacement vector, r, on an atom initially at corner 1.

Table 1.1 The effect of C4on squarene

Atom’s initial corner Atom’s final corner

Consider a vector displacement r of an atom initially located at corner 1 The

com-ponents of this vector in the X1–Y1coordinate system are r x and r y, respectively,

as shown in Fig.1.3(b) After rotation the displacement vector r (anchored to the

atom) is now centered on the X4–Y4 coordinate system associated with corner 4

In the X4–Y4coordinate system, the X4component is r y and the Y4 component is

−r x In symbolic notation,

C4r x (1) → −r y (4),

C4r y (1) → r x (4), meaning that the C4 operation transforms the component of r along the X1 axisinto a component along the−Y4axis and transforms the component of r along the

Y1axis into a component along the X4axis The displacement vector of any of theatoms of squarene will transform in the same way, so, in general,

C4r x (i) → −r y ( f ),

C4r y (i) → r x ( f ),

1.1 In-plane molecular vibrations of squarene 7

Table 1.2 The effect of σ v1 on squarene

Atom’s initial corner Atom’s final corner

Next consider the action of the reflection operation, σ v1 For this symmetry

operator we have the results shown in Table 1.2 For σ v1 the displacements are

transformed so that r x (i) → r y ( f ) and r y (i) → r x ( f ) Symbolically,

σ v1 r x (i) → r y ( f ),

σ v1 r y (i) → r x ( f ),

where the initial and final corner numbers are given in Table1.2

A convenient device for determining the effects of symmetry operations is the

action table shown in Table1.3 The columns are labeled by the group operators

The rows are labeled by r x and r y and the corner numbers The intersection of acolumn and a row is the action of the column operator on the displacements (firsttwo rows) and the corner labels (last four rows) The table makes it easy to generatethe effect of any operator For example, considerσ d1 acting on the displacementvector initially at corner 2 From the table (column 7) we see that under action of

σ d1corner 2→ corner 1, r x → −r x , and r y → r y Therefore, r x (2) → −r x (1) and

r y (2) → r y (1) Using the action table the effect of any operator of the group on the

basis functions can be determined

Another useful device is the group multiplication table shown in Table 1.4 In

this table the rows and columns are labeled by the operators of the group, O i

(i = 1, 2, , 8) The intersection of the ith row with the jth column is the ator obtained by the sequential application of the operators in the order O i O j

oper-The order is important because O i O j is often different from O j O i The group

Table 1.4 The group multiplication table for C4v

multiplication table is easily generated from the action table For example, C4

transforms r x , r y , 1, 2, 3, 4 into −r y , r x , 4, 1, 2, 3, respectively If we now apply

σ d1 to this result then −r y , r x , 4, 1, 2, 3 is transformed into −r y , −r x , 3, 2, 1, 4.

Therefore,σ d1 C4 transforms r x , r y , 1, 2, 3, 4 into −r y , −r x , 3, 2, 1, 4 On looking

at the action table, we see that this is the same action asσ v2 Thusσ d1 C4 = σ v2.

This result may be seen as the operation at the intersection of the row labeledσ d1

and the column labeled C4 in the group multiplication table Similarly, one may

see from the same table that C4σ d1 = σ v1, which shows that with a change of order

in multiple operations one does not necessarily obtain the same result

The action table can also be applied to products such as r x (1) r y (2), [r x (1)]2,

or products of more than two displacements For example, C4r x (1) r y (2) →

−r y (4) r x (1) and C4[r x (1)]2→ [r y (4)]2

1.1.4 Step 4: Construct matrix representations for each element

of the group using the basis functions

It will prove useful to employ an eight-component column vector so that we canshow the effect of a symmetry operation on all the atomic displacement vectors at

the same time To do this, we use, as the components of the 8-vector, the r x and r y

displacement components of all four atoms Using the action table we have, in oursymbolic notation,

1.1 In-plane molecular vibrations of squarene 9

The rows of the column vectors in (1.2) are labeled from top to bottom by ex (1),

ey (1), e x (2), e y (2), e x (3), e y (3), e x (4), and e y (4), unit vectors along the fixed

Cartesian coordinates in Fig.1.3(a) It is important to understand the meaning of(1.2) The rows of the column vectors are labeled by the unit vectors of the fixed

coordinate system Therefore a column vector such as

c

000

is a displacement of length d for the x-component on corner 1 What

equa-tion (1.2) means is that a clockwise rotation, C4, rotates the molecule so that the

x-component of the molecular displacement on corner 1 after rotation appears as

the−y-component on corner 4 In other words, we assign to the new vector

(right-hand side) the value−r x (1) for its y-component on corner 4 This is indicated by

the arrow in (1.2) Similarly, we assign to the new vector the value r y (4) for its x-component on corner 3 In more compact notation,

where(C4) is an 8 × 8 matrix.5 The eight-dimensional column vector, Vi, has

as its components the entries of the column on the left-hand side of (1.2) The

components of the column vector, Vf, are the entries of the right-hand side of (1.2)

(C4) can easily be constructed For example, (1.2) shows that r x (1) is

trans-formed to−r y (4), therefore the first column of (C4) has only one entry, namely

5 An equally valid method of describing the group and its representations is to fix the molecule and rotate

the coordinate framework Clearly, rotating the coordinate axes counter-clockwise with the molecule fixed is

equivalent to fixing the coordinate axes and rotating the molecule clockwise However, the usual convention in most texts is rotation of the molecule or function rather than the coordinate axes.

a “−1” in the eighth row Similarly, r y (3) is transformed to r x (2), so the sixth

col-umn of(C4) has a single entry, a “+1” in the third row Proceeding in this way

produces the 8× 8 matrix

−1

Matrix multiplication of Vi by (C4) produces the vector V f Using the action

table we can readily construct matrices for each of the eight operations of the C4v

group These matrices are shown in Table1.5

The eight matrices in Table1.5form a matrix representation of the C4vgroup.

In the parlance of group theory we say that the displacements r x ( j) and r y ( j) ( j = 1, 2, 3, and 4) are basis functions for an eight-dimensional representation of

the group C4v The symbols r x and r y without regard to the corner numbers form

a two-dimensional representation of C4v Finally, we note that the corner numbers

(the last four rows in Table1.3) form the basis for a four-dimensional representation

of C4v.

From the group multiplication table we can find the inverse of any representationmatrix by finding the intersection that produces the identity element For example,

the inverse of C4is C3

4 and the reflection operators are their own inverses

Inspection of the representation matrices in Table1.5reveals a number of erties (1) All of the matrices are unitary, meaning that the inverse,(O i )−1, is the

prop-transpose, complex conjugate of(O i ), where O i denotes the i th operator of the

group Since, in this case, all the matrix elements are real, the inverse is simplythe transpose (2) The rows (or columns) of(O i ) are orthogonal to each other;

where h, the order of the group, is the number of elements in the group and (O i ) r s

is the r s matrix element of (O i ) (3) The determinant of each matrix is +1 The

results (2) and (3) are consequences of (1), namely of the matrices being unitary

1.1 In-plane molecular vibrations of squarene 11

Table 1.5 Matrix representations of the group C4v : (a) through (h),

representation matrices based on the displacement coordinates of squarene

−1

( f ) (σ v1 ) =

1 1

1 1 1 1 1 1

−1 1

−1 1

−1 1

(d) (C3

4) =

−1 1

(h) (σ d2 ) =

1

−1 1

−1 1

−1 1

−1

To verify that the matrices in Table1.5form a proper representation of the group,one can simply multiply them together to show that they obey the properties of agroup representation given in Theorem1.2.

Theorem 1.2 (Properties of a matrix representation)

1 For any operators A, B, and C in the group there are associated matrices (A),

(B), and (C) such that if A B = C then (A) (B) = (A B) = (C) (This requires that (A)−1= (A−1).)

2 For every point group there exists a representation consisting of unitary ces Unitary matrices are square matrices for which (A)−1 is the complex

matri-conjugate, transpose of (A).

It follows from Theorem 1.2 that matrices given in Table 1.5 obey the samegroup multiplication table (Table1.4) as do the operators with(O i ) instead of O i

labeling the rows and columns

1.2 Reducible and irreducible representations of a group

A group can have many matrix representations If we perform the same similaritytransformation,T−1(O i ) T, on all of the matrices of Table 1.5we will obtain adifferent set of 8× 8 matrices,  (a) (O i ), that also represents the group (This could

be written asT−1 T =  (a), where it is understood that the similarity

transforma-tion,T, is applied to all of the representation matrices.) Two representations related

by a similarity transformation are said to be equivalent If we use different basis

functions we may obtain a representation whose dimensionality is different fromthat of For example, we could base a representation on how the atoms of the

molecule move without regard to atomic displacements That would generate (b),

a 4× 4 matrix representation of C4v Finally we could combine all these

repre-sentations into a 20× 20 representation by placing (O i ),  (a) (O i ), and  (b) (O i )

as blocks along the diagonal to form the matrix (c) (O i ) This process is shown

schematically in Fig 1.4(a) ,  (a),  (b), and  (c) are all valid representations

of the group  (c) is composed of three uncoupled representations (unconnected

blocks of matrices) If an arbitrary similarity transformation were performed on

 (c), the block-diagonal form would usually be lost even though the transformed

representation would be equivalent to (c).

We see from the previous paragraph how larger representation matrices can becomposed from smaller ones The reverse process can also be accomplished Given

an N × N matrix representation,  N, it may be possible to transform all its matricesinto block-diagonalized forms by a similarity transformation = S−1 NS That

1.2 Reducible and irreducible representations of a group 13

Figure 1.4 (a) The(20×20) representation  (c) = + (a) + (b) , (b) the N × N

representation N, and (c) decomposition of N into smaller representations by

a similarity transformation.

is, by a similarity transformation we can decompose the  N representation intosmaller representations Symbolically, we have

 N ≡  (1) ⊕  (2) ⊕ · · · ⊕  (m) (1.4)Equation (1.4) is an equivalence statement meaning that there exists a similaritytransformation that will decompose N into the series of uncoupled blocks shown

on the right-hand side In more detail,

with the understanding that simple matrix addition is not implied We will use this

notation when there is no chance for confusion

Since each  (i) appearing in (1.5) is itself a representation, it may be

possi-ble to decompose it into smaller blocks Eventually, minimum-sized blocks must

be achieved These minimal blocks are said to be irreducible representations.

(We shall use the abbreviation “IR” for “irreducible representation” throughoutthis book.)

Theorem 1.3 (Definitions of reducible and irreducible representations)

• A representation that can be reduced or decomposed by a similarity formation into smaller, uncoupled, submatrix representations is a reducible representation.

• Conversely, a representation that can not be reduced by any similarity formation into smaller, uncoupled, submatrix representations is an irreducible representation.

trans-For most science problems the basis functions normally employed to describethe system are bases for a reducible representation,  The role of group theory

is to provide a method of finding the matrix S such that the similarity formationS−1 S = S produces a representation that has only IRs along the

trans-diagonal of eachS(O i ) The notation S−1 S = Sis symbolic It means that

the same similarity transformation is applied to every representation matrix of , that isS−1(O i ) S = S(O i ) (i = 1, 2, , h), and every S(O i ) has the same block-diagonal form consisting of submatrices that are irreducible representation matrices.

The columns ofS or the rows of S−1are the symmetry functions for the system.

These symmetry functions are linear combinations of the original basis functions.When a physical problem is expressed in terms of its symmetry functions theeigenvalue equation is block-diagonal and a great simplification of the mathemat-ics usually results A method for deriving the symmetry functions is discussed in

“Step 7” below in Subsection1.2.6

1.2 Reducible and irreducible representations of a group 15

Symmetry functions are usually not eigenfunctions If the decomposition of the

representation based on the physical coordinates contains the same IR r times, then there will be r sets of symmetry functions for that IR and the eigenfunctions

will be linear combinations of these symmetry functions On the other hand, if thedecomposition contains a particular IR only once, then the basis functions for that

IR are also eigenfunctions This is a rather amazing result It means that some ofthe eigenfunctions of a physical system may be entirely determined by symmetry,independently of the details of the system’s Hamiltonian or force-constant matrix.6

We shall see examples of this as we proceed with the analysis of the vibrations ofsquarene

1.2.1 Step 5 Determine the number and types of irreducible representations

The matrix representation,, we have constructed for squarene is reducible into

unconnected IR blocks by the similarity transformation S−1 S, where S is the

matrix of symmetry functions The reduction of  to diagonal blocks need not

result in distinct IRs, and in many cases the same (or equivalent) IR will occurmore than once among the diagonal blocks The number and types of IRs thatoccur among the diagonal blocks depend on the basis functions used to generate What is the connection between the IRs of the C4v group and the physical

problem of the molecular vibrations? The connection is this: The similarity formation S which block-diagonalizes the reducible representation  will also

trans-block-diagonalize the matrixF of (1.1)

To determine the number and type of IRs contained in we need to explore a

few other properties of a group

1.2.2 Classes of a group

An important feature of a group is its classes The definition of a class is given in

Theorem1.4

Theorem 1.4 (Definition of a class)

• Two operators, A and B, are conjugate if A = C BC−1, where C is any member

of the group A class is a set of all of the distinct operators of a group that are conjugate to each other.

• From the definition of a representation, Theorem 1.2 , it follows that two resentation matrices (A) and (B) belong to the same class if (A) =

rep-(C) (B) rep-(C)−1, where (C) is a matrix representation of any operator of the group.

6 A valid Hamiltonian is not arbitrary It must commute with the operations of the group Similarly, F must commute with the operations of the group.

To find the classes of C4v we can make use of the group multiplication table

(Table1.4) The inverse of any operation can be found from the table by observing

which product produces the identity, E:

E−1= E,

C4−1= C3

4,

C2−1= C2, (C3

{E}; {C4, C3

4}; {C2}; {σ d1 , σ d2 }; and {σ v1 , σ v2} Similarly, for the representationmatrices the classes are{(E)}; {(C4), (C3

4)}; {(C2)}; {(σ d1 ), (σ d2 )}; and {(σ v1 ), (σ v2 )}.

1.2.3 The character of a representation matrix

Another important property of a representation matrix is its trace The trace of a

matrix is the sum of the diagonal elements of the matrix The trace of a

represen-tation matrix is called its character The usual symbol for a character isχ and the

character of α (O i ) is denoted by χ α (O i ) or χ α

i

If A and B are in the same class then A = C BC−1 for some element C in

the group This requires that(A) = (C) (B) (C)−1and therefore(A) and

(B) are related by a similarity transformation The trace of a matrix is unchanged

by a similarity transformation and therefore the character of(A) must be equal to

the character of(B) From this result it follows that all representation matrices belonging to the same class have the same character.

The character table for the C4v group is given in Table1.6 The first column

of the table lists the names of the IRs of the C4v group The next five columns

give the characters (the χs) of the classes of C4v for the five different IRs The

last row lists the results for, our 8 × 8 representation for squarene obtained from

1.2 Reducible and irreducible representations of a group 17

Table 1.6 The character table for C4v

to which C4 belongs Since all the members of a class have the same character,

it is not necessary to specify each element The last column of Table1.6 lists afew functions that can serve as basis functions for the corresponding IR Charactertables for the different point groups are given in AppendixC

The character table for C4v gives all the information we need to determine the

number and types of IRs of our 8×8 representation,  However, to find the

decom-position of and to construct the S matrix that block-diagonalizes the eigenvalue

equation we need a few more results from group theory

1.2.4 Orthogonality of the characters of an irreducible representation

C4v has five classes If the characters of the classes of a given IR of C4v are

con-sidered as components of a five-dimensional unit vector, five normalized vectors

can be constructed, one for each of the IRs of C4v We define the kth component of

k is the character of the kth class of the αth IR, and h is the number of

elements in the group The vectors are

√2

√2

−√√22

−200

k=1N k (χ α

k )∗χ k β = δ αβ The orthogonality of the characters of the IRs is ageneral feature of a group A statement of this property is given in Theorem1.5

Theorem 1.5 (Orthogonality of the characters of irreducible representations) The

characters of the representation matrices of the IRs of a group are orthogonal to one another:

Theorem1.5 is the general statement of the result we found for the characters

in (1.7) To proceed with our analysis we also need the results from group theorygiven in Theorem1.6

Theorem 1.6 (Properties of irreducible representations)

1 The number of different irreducible representations of a group is equal to Nc, the number of classes.

1.2 Reducible and irreducible representations of a group 19

2 The sum of the squares of the dimensions of all the distinct IRs of a group is equal to the number of elements in the group:

We see from the character table for C4vthat there are five IRs Since the character

E is equal to the dimensionality of the IR, it is clear the first four IRs are

one-dimensional and the last is two-one-dimensional The first result quoted in Theorem1.6

tells us that there are five IRs since there are five classes for C4v The second result

of Theorem1.6tells us that the sum of the squares of the dimensions of the five IRsmust be 8 There can’t be a three-dimensional IR (since 32 = 9 is already greater

than 8) and the smallest dimension possible has l i = 1 So we need five integers,all of which are either 1 or 2 and whose squares add up to 8 The only solution is

12+ 12+ 12+ 12+ 22= 8

Our next step is to determine the number and types of IRs contained in our

8× 8 representation,  In general, if  is any representation of the group we know

that there exists a similarity transformation,S−1(O i ) S = (O i ), where (O i )

is block-diagonal with various IR matrices along the diagonal The character of

(O i ) is equal to the sum of the traces of the IR blocks along the diagonal Since

the trace of(O i ) is unchanged by the similarity transformation, its trace is equal

to the trace of(O i ) Therefore, it follows that

in the same way

If we multiply both sides of (1.8) by the factor(1/h)(χ i β )∗and sum over all of

the operations of the group, we obtain

To obtain the final result in (1.9) we used the equation (GT1.5a) from Theorem1.5.Equation (1.9) can be rewritten in terms of the characters of the classes:

where the sum of k is over the classes, N k is the number of elements in the kth

class,Ncis the number of classes,χ 

k is the character of the kth class of , and χ k β

is the character of the kth class of the βth IR.

To see how (1.10) works, we apply it to our representation for squarene Withthe help of the character table (Table1.6), we find that

 =  A1 +  A2+  B1+  B2 + 2 E (1.12)Since the E IR is two-dimensional, it requires two basis functions That is, S−1

will have two rows of symmetry functions for each E IR To distinguish the

two functions they are called the “row-1” and “row-2” basis functions ofE (An

N -dimensional IR will have N rows of symmetry functions.) For squarene E occurs

twice in the decomposition, so there will be two different row-1 and two differentrow-2 symmetry functions inS−1.

1.2.5 Step 6: Analyze the information contained in the decomposition

One of the most important results of group theory is the result stated below inTheorem1.7

1.2 Reducible and irreducible representations of a group 21

Theorem 1.7 (Matrix elements of a commuting operator) If H is a Hermitian operator that commutes with the operations of the group, the matrix elements of

H vanish between basis functions for different IRs or between basis functions for different rows ofS−1(or columns of S) of the same IR.

Let V α j be a basis vector for the j th row of the αth IR and W β k be a basis vector for the kth row of the βth IR, then

α

j|H|Wβ k = MV W δ j k δ αβ , whereMV W is a constant that depends on the two functions.

Note that V and W can both be basis functions for the same row of the same IR

even if they are different functions.

For our vibration problem Theorem1.7tells us that there are no non-zero matrixelements of F between symmetry functions that belong to different IRs or todifferent rows ofS−1(or columns ofS) of the same IR

A great deal of information can be gleaned from the decomposition of 

(Eq (1.12)) First, we note that  A1,  A2,  B1, and  B2 are one-dimensionalIRs and  E is a two-dimensional IR The number of degrees of freedom is

(4 × 1 + 2 × 2) = 8 We know that the similarity transformation S which

block-diagonalizes must also block-diagonalize the force-constant matrix, F, of (1.1).ThusS−1F S will have four 1 × 1 blocks Since, according to Theorem1.7, therecan be no matrix elements between basis functions for different IRs, the columns of

S corresponding to these one-dimensional blocks must be eigenvectors For thesefour normal modes the eigenvectors are the same as the symmetry functions andare completely determined, independently of the values of the force constants Theeigenvalues (mass times frequencies squared), of course, depend on the force con-stants, since they are diagonal matrix elements ofS−1F S corresponding to the A1,

A2, B1, and B2vibrations

We also know thatS−1F S has two (2 × 2) E-representations Since E appears

twice in the decomposition of, there will be two different symmetry functions

that are the bases for row 1 and two that are the bases for row 2 ofE There can be

matrix elements between two functions that are bases for the same row of the same

IR ThereforeS−1F S could have what appears to be a 4 × 4 block

1.2.6 Step 7: Generate the symmetry functions

In order to determine theS matrix we need to find the symmetry functions theoretical results can be used to generate the symmetry functions starting from anarbitrary function of the displacement coordinates The theorem that allows us to

Group-do this is given below in Theorem1.8

Theorem 1.8 (The symmetry-function-generating machine: The projection

opera-tor) Let V be an arbitrary column vector and generate the vector V α m according to the rule

the function V by the matrix(O k ) If we choose V to be any linear function of the

displacement coordinates for squarene, Vα m(when normalized) will be a symmetry

function that is a basis for row m of the αth IR Therefore Theorem 1.8 provides

a direct method for obtaining the symmetry functions that are needed in order to block-diagonalize the eigenvalue matrix equation of ( 1.1 ).

To simplify the notation we shall use Vxk (Vyk) to indicate a column vector

having a single entry of “1” in the row corresponding to x (y) displacement on the kth corner.

As an example we generate the symmetry coordinate that transforms according

to the B1irreducible representation. B1is one-dimensional, so the matrix elementsare the same as the characters listed in Table1.6 We select as our arbitrary column



Vx3

(1.13)

1.2 Reducible and irreducible representations of a group 23 (a)A1 (b)A2

Figure 1.5 The squarene molecule and its symmetry functions.

Expressing VB1 as a normalized column vector, we have

We could have chosen different displacements for V and generated the same result.

Since  B1 appears only once in the decomposition, VB1 must be an eigenvector

of the molecular vibration problem of (1.1) Figure1.5(c) shows schematically therelative motions of the atoms for this normal mode

Now let us find the symmetry functions that transform according toE The

rep-resentation is two-dimensional and, according to (1.12), it is contained twice inthe decomposition of This means that there must be four symmetry functions

(2 IRs× 2 rows) To use the symmetry-function-generating machine we need thediagonal matrix elements of the IR forE According to the character table (the last

column of Table1.6) X and Y (or the displacements r x and r y) can be used as basisfunctions for a two-dimensionalE representation.

We have already done the work for this representation The first two rows ofTable1.3show how r x and r y transform under the group operations We have, forexample, that

C4r x → −r y ,

C4r y → r x ,

Table 1.7 IR matrices for the E representation

All of the 2× 2 matrices for the E representation are shown in Table1.7

To generate a row-1 symmetry function we proceed as we did in finding VB1,except now we use the diagonal elements, E (O i )11, instead of the characters togenerate the function The matrix elements are given in Table1.7 Since there are

several coordinates, we shall label these vectors according to the IR and the row

of the matrix element used to generate the function Using Vx1 as our arbitraryfunction and the first-row diagonal matrix elements, we obtain

1.2 Reducible and irreducible representations of a group 25

The physical motion represented by this vector is translation in the x-direction as

shown in Fig.1.5(e) If we use Vy1to generate a symmetry function, we find

VE

1b = 12

−1010

1b are the two row-1 symmetry functions The vectors are orthogonal

to each other The remaining two E-functions transform according to the row-2

symmetry functions of theE irreducible representation For the row-2 vectors we

use the (2,2) diagonal elements with Vx1and Vy1 as the generating functions Weobtain

VE

2a = 12

−1010

−10

2b represents a translation in the y-direction.

Using Vx1 as the arbitrary function, we find the A2symmetry function to be

−11

The motion associated with VA2, shown in Fig.1.5(b), is a rotation about a z-axis

through the center of the square.7This result could have been anticipated from the

7 The VA2 mode represents a rotation in the limit of small displacements.

Table 1.8 The S matrix The top row labels the symmetry functions according to the IR to which they belong The left-most column lists the displacement

coordinates and the bottom row lists the normalization factors The center 8× 8

The symmetry functions can be used to construct theS and S−1matrices needed

for the similarity transformation that block-diagonalizes the eigenvalue matrixequation (1.1) The 8× 8 matrix S is given in Table1.8 The inverse,S−1, is simply

the transpose ofS

1.2.7 Step 8 Block-diagonalize the eigenvalue matrix equation

A vibrational normal mode must leave the center of mass stationary This can beexpressed by the conditions

Center-of-mass considerations

The similarity transformation carries theF matrix of (1.1) into the block-diagonalform We shall denote this block-diagonal matrix by the symbol, F, where