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Symmetry - introMatti Hotokka

Department of Physical ChemistryÅbo Akademi University

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The image looks symmetrical. Why?

Symmetrical or not

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The right hand side is similar to the left handside.

Symmetrical or not

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No! They are mirror images.

Symmetrical or not

If one reflects theleft hand side to theright and vice versa,then one will obtainan image exactlysimilar to theoriginal.

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Such a change of an object that the imagecannot be distinguished from the original.Reflection: The mirror image is similar to theoriginal but we know that changes have beenmade.

Symmetry operation

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IdentityRotationReflectionRotation-reflectionInversion(Translation)

Symmetry operations

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Shönfliess (for chemists) and International (for physicists)

Notations

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n2

Notations

Identity E 1Rotation Cn nReflection �

h, �

v, �

d mRotation-reflection SnInversion i

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The symmetry operations are

� E

� C3� C32

� �

h� S3, S3-1

� C2, C2', C2"� �

v,

�

v’,

�

v”

Equilateral triangle

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Equilateral triangle1

231

231

231

23

1

23

12

3

1

2

3

E

C3

C32

�

h

1

231

231

231

23

S3

C2'

S3-1

C2

1

23

1

2

3

1

231

23

�

V”

C2"

1

3 2

�

V

2

3

1

�

V’

3

21

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( ) ( )

� � � �

A B A B⊗ =∆ ∆

Product of the symmetryoperators A andB, A �B, means that you operate with B onthe original and then with A on theintermediate result.

Multiplication

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EE E

Multiplication table

C3

C3

C3C3C3

2

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EE E

Multiplication table

C3

C3

C3C3C3

2

C32

C32

C3

C32

C32E

E

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Multiplication table

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The set of symmetry operations and themultiplication operator form a group if

� (P1) A �(B �C)=(A �B) �C (Associative law)

� (P2) �E A �E=E �A=A A � (Identity element)

� (P3) A � �A-1 � A �A-1=A-1 �A=E (Inverse element)

Mathematical group

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A huge mathematical literature is basedon the three postulates. A few examples

� (A B)-1 = B-1 A-1

� A B �B A (generally; they may be equal)

� Order of group , # , is number of symmetryoperations in that group– For the equilateral triangle, #D3h= 12

� Isomorphic groups contain different sets ofobjects but the multiplication table is similar

� Generators is a subset of the group that willgenerate all the remaining symmetry operations

Group theory

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D3h has 12 elements:

� E, C3, C32, �

h, S3, S3-1, C2, C2', C2",

�

v,

�

v’,

�

v”The generators are

� C3,

�

h, C2

Another set of generators can be chosen

� C3,

�

h,

�

v

Generators

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Similarity transformation

� C=A-1 �B �AConjugated elements

� B and C are conjugated if there is such an A in thegroup that C=A-1 �B �A (Notation C �B)Class

� All elements for which one can find an A �G such thatthe elements are conjugated form a class– For the equilateral triangle C2, C2' and C2" form a class– Each element can only belong to one class– All elements in a class behave similarly

Group theory

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The classes of the equilateral triangle are

� {E}

� {C3, C32}

� {C2, C2', C2"}� { �

h}� {S3, S3-1}

� { �

v,

�

v’,

�

v”}

Group theory

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C1, Cs, Ci, Cn (n=2,3,4,...)Cnh, Cnv (n=2,3,4,...)Dn, Dnd, Dnh (n=2,3,4,...)S2n (n=2,3,4,...)C �v, D �h

Td, Th, Oh, Ih, K

The point groupsSchönfliess notation

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Schönfliess notation is used by mostchemistsThe international notation lists thegenerators

The point groups

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The point groups

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Classification is done by using a flowdiagram

The point groups

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Special point group?C �v, D �h, Td, Th, Oh, Ih

Finish

Cn (n>1)?

YES

NO �?NO Cs

YES

i?NO

Ci

C1

YES

NO

�?

YES

Collinear S2n?

NO

S2n

YES

� C2 axes?

YES

NO

�h?�

v?

�

h?

�

v?

YES NO

Cnh

Cnv

Cn

Dnd

Dnh

Dn

YES YES

YES YES

NO

NO

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Character table

C2v E C2

�

v (xz) �

v (yx)

A1 +1 +1 +1 +1A2 +1 +1 -1 -1B1 +1 -1 +1 -1B2 +1 -1 -1 +1

x

y

C2

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x x yy x y' cos sin' sin cos= −= +

���

φ φφ φ

xyz

xyz

'''

cos sinsin cos

�

!!"!

#$

%%"% =

−

�

!!"!

#$

%%"%

�

!!"!

#$

%%"%

φ φφ φ

00

0 0 1

A rotation in the xy plane can be given as

Matrix representation

This can be written as a matrix equation

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C3

12

32

32

12

00

0 0 1=

− −−

&'

(((

)*

+++

123

0 1 00 0 11 0 0

123

'''

,-

../.

01

22/2 =

,-

../.

01

22/2

,-

../.

01

22/2

Matrix representationRotation through 120 3 , i.e., C3

Rotations of the corners of an equilateraltriangle 1

23

C3

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C3

12

32

32

12

00

0 0 1=

− −−

45

666

78

999

0 1 00 0 11 0 0

:;

<</<

=>

??/?

What’s common The common feature between the matrices is trace

Sum = 0 Sum = 0

Trace is called character ( @) in symmetrytheory

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− −−

AB

CCC

DE

FFF

− −−

AB

CCC

DE

FFF =

−− −

AB

CCC

DE

FFF

12

32

32

12

12

32

32

12

12

32

32

12

00

0 0 1

00

0 0 1

00

0 0 1

C32

12

32

32

12

00

0 0 1=

−− −

GH

III

JK

LLL

Point groups: C3

MC3=C32

Matrix groups: Rotation through 240 N

Isomorphy

Indeed,

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Two set of matrices both represent thepoint groupTherefore the matrices must be relatedYes, they are, through a similaritytransformation

Similarity transformation

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S =−

−

OP

QQRQ

ST

UURU

2 0 11 3 11 3 1

S − =−

−V

WXXRX

YZ

[[R[1

13

16

16

36

36

13

13

13

0

S C S− ⋅ ⋅ =−

−

\]

^^_^

`a

bb_b

\]

^^c^

`a

bbcb

−

−

\]

^^c^

`a

bbcb =

− −−

\]

^^^

`a

bbb1

3

13

16

16

36

36

13

13

13

12

32

32

120

0 1 00 0 11 0 0

2 0 11 3 11 3 1

00

0 0 1

Example

Choose therefore

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The simplest possible matrices: diagonal

Goal

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Like isomorphy but one matrix for severalsymmetry operationsStill a similar multiplication tableMakes smaller matrices possibleGoal: still the simplest possible matriceswhich are 1x1, in some cases 2x2 or 3x3Still, several representations can beconstructed of a set of matrices

Homomorphy

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Example

C2v E C2

d

v (xz) d

v (yx)

A1 +1 +1 +1 +1A2 +1 +1 -1 -1B1 +1 -1 +1 -1B2 +1 -1 -1 +1

The simplest matrices are (+1) and (-1).Still, four different combinations can be formedso that the multiplication tables are similar. Thus there are four representations.

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The simplest possible matrix representation

e Cannot be reduced to a simpler form: irreducibleAny matrix representation

e Can be formed out of these simplest buildingblocks (and a similarity transformation)

e Conversely, any matrix can be reduced to itsbuilding blocks. This is called “reducing arepresentation”.

Irreducible representation

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C2

1 0 00 0 10 1 0

=−

−−

fg

hh/h

ij

kk/kE =

lm

nn/n

op

qq/q

1 0 00 1 00 0 1

σ v =

rs

tt/t

uv

ww/w

1 0 00 0 10 1 0

σ 'v =−

−−

xy

zz/z

{|

}}/}

1 0 00 1 00 0 1

ExampleWater, C2v

C2v E C2

~

v~’vE E C2

~v

~’vC2 C2 E ~’v ~

v~

v

~

v~’v E C2~’v ~’v ~

v C2 E

Please verify the multiplication table!

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S S= −

��

����

��

���� � = −

��

��"�

��

��"�−

1 0 00 1 10 1 1

1 0 000

1 12

12

12

12

S C S− = −

��

��"�

��

��"�

−−

−�

�����

��

���� −

��

����

��

���� =

−

−

��

����

��

����1

212

12

12

12

1 0 000

1 0 00 0 10 1 0

1 0 00 1 10 1 1

1 0 00 1 00 0 1

ExmpleWater

Let

Then

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C2

1 0 00 1 00 0 1

=−

−

��

����

��

����E =

��

��/�

��

��/�

1 0 00 1 00 0 1

σ v = −�

���/�

��

��/�

1 0 00 1 00 0 1

σ 'v =−

−−

� 

¡¡/¡

¢£

¤¤/¤

1 0 00 1 00 0 1

Γ = +A B2 12

ExampleWater

Summarising

The matrices consist of B1 A2 B1

C2v E C2

¥

v (xz) ¥

v (yx)

A1 +1 +1 +1 +1A2 +1 +1 -1 -1B1 +1 -1 +1 -1B2 +1 -1 -1 +1

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Γ = aµµ

µ ag

gi i ii

µµχ χ= 1

¦ Notations

§ The general (reducible) representation is ¨§ The irreducible representations of the point group are ©

– In C2v, A1, A2, B1, B2§ Number of symmetry operations in the point group is g = # ª

§ The classes of symmetry operations in the point group are i§ The number of symmetry operations in class i is g i§ The character of class i in irreducible representation © is «i

¬

§ The character of class i in ¨ is «i§ The integer weight of irreducible representation © in ¨ is a ¬

¦ Thus

Reducing a representation

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[ ]aA1

14 1 1 3 1 1 1 1 1 1 1 1 3 0= ⋅ ⋅ + ⋅ ⋅ − + ⋅ ⋅ + ⋅ ⋅ − =( ) ( )

[ ]aB1

14 1 1 3 1 1 1 1 1 1 1 1 3 2= ⋅ ⋅ + ⋅ − ⋅ − + ⋅ ⋅ + ⋅ − ⋅ − =( ) ( ) ( ) ( )

[ ]aA2

14 1 1 3 1 1 1 1 1 1 1 1 3 1= ⋅ ⋅ + ⋅ ⋅ − + ⋅ − ⋅ + ⋅ − ⋅ − =( ) ( ) ( ) ( )

[ ]aB2

14 1 1 3 1 1 1 1 1 1 1 1 3 1= ⋅ ⋅ + ⋅ ⋅ − + ⋅ − ⋅ + ⋅ ⋅ − =( ) ( ) ( )

ag

gi i ii

µµχ χ= 1

Example

C2v (1)E (1)C2 (1) ­

v (xz) (1) ­

v (yx)

A1 +1 +1 +1 +1A2 +1 +1 -1 -1B1 +1 -1 +1 -1B2 +1 -1 -1 +1

® +3 -1 +1 -3