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Symmetry and Group Theory4

Symmetry and Group Theory4

Symmetry and Group Theory4

Symmetry and Group Theory4

Symmetry Operation and Symmetry Elements

Symmetry Operation: A well-defined, non-translational movement of an

object that produces a new orientation that is indistinguishable from the

original object.

Symmetry Element: A point, line or plane about which the symmetry

operation is performed.

180o rotation: symmetry operation

line: symmetry element

Four kinds of Symmetry Elements and Symmetry

Operations Required in Specifying Molecular Symmetry

Proper (rotation) axis (Cn)

Mirror plane (s)

Center of symmetry or center of inversion (i)

Improper (rotation) axis (Sn)

Four kinds of Symmetry Elements and Symmetry

Operations Required in Specifying Molecular Symmetry (1)

Proper (rotation) axis (Cn):

Rotation about Cn axis by 2p/n

n-fold symmetry axis.

A Cn axis generates n operations.

*1: E : Identity (x1) *2: Right-handed rotation

* Principal rotational axis: highest-

fold rotational axis. If more than

one principal axes exist, any one

can be the principal axis.

Four kinds of Symmetry Elements and Symmetry

Operations Required in Specifying Molecular Symmetry (1)

Proper (rotation) axis (Cn):

Rotation about Cn axis by 2p/n

n-fold symmetry axis.

A Cn axis generates n operations.

Four kinds of Symmetry Elements and Symmetry

Operations Required in Specifying Molecular Symmetry (2)

Mirror plane (s): Reflection about the s plane

How many mirror planes ?

Four kinds of Symmetry Elements and Symmetry

Operations Required in Specifying Molecular Symmetry (2)

*sh : mirror planes perpendicular to the principal axis.

*sv : mirror planes containing the principal axis Unless it is sd.

*sd : mirror planes bisecting x, y, or z axis or bisecting C2 axes

perpendicular to the principal axis.

Mirror plane (s): Reflection about the s plane

How to define molecular axes (x, y, z)?

1. The principal axis is the z axis.

2. If there are more than one possible principal axis, then the

one that connects the most atoms is the z axis.

3. If the molecule is planar, then the z axis is the principal

axis in that plane. The x axis is perpendicular to that plane.

4. If a molecule is planar and the z axis is perpendicular to

that plane, then the x axis is the one that connects the most

number of atoms. (Actually, 3, 4 are arbitrary.)(y in Miessler)

Four kinds of Symmetry Elements and Symmetry

Operations Required in Specifying Molecular Symmetry (3)

Center of symmetry or center of inversion (i) : Inversion of all objects through the center.

i is Pt atom.

i is a point in space.

Four kinds of Symmetry Elements and Symmetry

Operations Required in Specifying Molecular Symmetry (4)

Improper (rotation) axis (Sn) : Rotation about an axis by 2p/n followed by a reflection through a plane perpendicular to that axis or vice versa.

S6

Four kinds of Symmetry Elements and Symmetry

Operations Required in Specifying Molecular Symmetry (4)

Improper (rotation) axis (Sn) : Rotation about an axis by 2p/n followed by a reflection through a plane perpendicular to that axis or vice versa.

Sn generates n operations for even n and 2n operations for odd n.

Point Groups

Point Groups

Point Groups

Point Groups

Point Groups

Point Groups

Point Groups

Point Groups

Properties and Representations of Groups

Properties of a Group

Properties and Representations of Groups

Matrix Representations of Symmetry Operations

100

010

001

:

100

010

001

'

'

'

E

z

y

x

z

y

x

z

y

x

E

z

y

x

z

y

x

E

z

y

x

100

010

001

:

100

010

001

'

'

'

2

2

2

C

z

y

x

z

y

x

z

y

x

C

z

y

x

z

y

x

C

z

y

x

100

010

001

:)(

100

010

001

)(

)(

'

'

'

xz

z

y

x

z

y

x

z

y

x

xz

z

y

x

z

y

x

xz

z

y

x

v

v

v

s

s

s

100

010

001

:)('

100

010

001

)('

)('

'

'

'

yz

z

y

x

z

y

x

z

y

x

yz

z

y

x

z

y

x

yz

z

y

x

v

v

v

s

s

s

Properties and Representations of Groups

Character Tables

h = 4

Schoenfliers Symbol of the Point Group Symmetry Operations (R)

Order of the Group

= S(# of Operations)

Irreducible Representations (G)

Symmetry Types of the

Irreducible

representations

Characters (c)

Classes (# of classes = # of symmetry types)

Character : Numerical representation of the operation

(Extent of originality after the operation)

yzyzyzyzRRRxzzzzC vzzzv ]1[)(' ,]1[)( ,]1[2 ss

222 ]1[)(' ,]1[)( ,]1[)(' ,]1[)( ,]1[

)](1[)( ,]1[ [1]z,zz

2

222222222

2

222

22

zzzvvyzyzyzvxyxyxyvxxx dddyzsssxzdddyzdddxzpppC

zyxzyxzyxCxxxCC

ssss

Properties and Representations of Groups

Character Tables

h = 4

Characters (c)

Character :

Numerical

representation of

the operation

(Extent of

originality after

the operation)

3)()(

100

010

001

:

100

010

001

'

'

'

ETrE

E

z

y

x

z

y

x

z

y

x

E

z

y

x

z

y

x

E

z

y

x

c

1)()(

100

010

001

:

22

2

CTrC

C

c 1)()(

100

010

001

)(

vv

v

Tr

xz

ssc

s

1)'()'(

100

010

001

:)('

vv

v

Tr

yz

ssc

s

3)()(

100

010

001

:

ETrE

E

c

G 3 1 1 1reducible representation (= A1+B1+B2)

Properties and Representations of Groups

Character Tables

* The biggest possible values of c is 3.

(= A1+B1+B2)

h = 4

Character Tables

Properties and Representations of Groups

Great Orthogonality Theorem

Co

H3N

H3N NH3

NH3

Br

Cl

Ez =z

C4z=z

C2z=z

svz=z

sdz=z

A1 1 1 1 1 1

Ex =x

C4x=y

C2x=-x

svx=x

sdx=y

z

x y

Ey =y

C4y=-x

C2y=-y

svy=-y

sdy=x

x, y are interconvertable.

degenerate

0)( 01

10

'

'

0)( 10

01

'

'

-2)( 10

01

'

'

0)( 01

10

'

'

2(E) 10

01

'

'

22

44

dd

vv

y

x

x

y

y

x

y

x

y

x

y

x

y

x

y

x

Cy

x

y

x

y

xC

y

x

Cy

x

x

y

y

xC

y

x

y

x

y

x

y

xE

y

x

scs

scs

c

c

c

Properties and Representations of Groups

Character Tables

s(yz)

100

010

001

:

100

010

001

'

'

'

E

z

y

x

z

y

x

z

y

x

E

z

y

x

z

y

x

E

z

y

x

100

02

1

2

3

02

3

2

1

:

100

02

1

2

3

02

3

2

1

100

03

2cos

3

2sin

03

2sin

3

2cos

3

2cos

3

2sin

3

2sin

3

2cos

cossin

sincos

'

'

'

3

3

3

C

z

y

x

z

y

x

z

y

x

C

z

yx

yx

z

yx

yx

z

y

x

C

z

y

x

pp

pp

pp

pp

100

010

001

:)(

100

010

001

)(

)(

'

'

'

xz

z

y

x

z

y

x

z

y

x

xz

z

y

x

z

y

x

xz

z

y

x

v

v

v

s

s

s

Properties and Representations of Groups

Character TablesC3v

h = 6

100

010

001

:E

100

02

1

2

3

02

3

2

1

:3C

block diagonalized

x, y are degenerated.

Properties and Representations of Groups

Character TablesC3v

100

010

001

:)(xzvs

Symmetry Labels

1. Degeneracies Symmetry Labels

1 A : symmetric about Cn (c(Cn) = 1)

B : antisymmetric about Cn (c(Cn) = -1)

2 E

3 T

2. Subscript labels Meanings

1 symmetric about C2(⊥Cn), (c(C2) = 1)

2 antisymmetric about C2(⊥Cn), (c(C2) = -1)

3. Subscript labels (when no C2(⊥Cn)Meanings

1 symmetric about sv, (c(sv) = 1)

2 antisymmetric about sv, (c(sv) = -1)

4. Subscript labels Meanings

g symmetric about i, (c(i) = 1)

u antisymmetric about i, (c(i) = -1)

5. Superscript labels Meanings

' symmetric about sh, (c(sh) = 1)

" antisymmetric about sh, (c(sh) = -1)

Properties and Representations of Groups

Character Tables

Polar molecule: a molecule with a permanent electric dipole moment.

A molecule with a center of inversion (i) cannot have a permanent dipole.

A molecule cannot have a permanent dipole perpendicular to any mirror plane.

A molecule cannot have a permanent dipole perpendicular to any axis of

symmetry.

Therefore, molecules having both a Cn axis and a perpendicular C2 axis or σh

cannot have a dipole in any direction. ⇒Molecules belonging to any Cnh, D,

T, O or I groups cannot have permanent dipole moment.

Applications of Symmetry Polarity

Applications of Symmetry Chirality

A chiral molecule is a molecule that is distinguished from its mirror image in

the same way that left and right hands are distinguishable.

A molecule that has no axis of improper rotation (Sn) is chiral.

Sn: including S1 = σ and S2 = i

Applications of Symmetry Molecular Spectroscopies

UV/VIS

Fluorescence

IR Raman

Stokes anti-

Stokes

Phosphorescence

S0

S1

v0

v1

v2

v3

v0

v1

v2

v3

T1

Intersystem Crossing

Virtual state

Applications of Symmetry Molecular Vibrations

IR Raman

Stokes anti-

Stokes

S0

S1

v0

v1

v2

v3

v0

v1

v2

v3

Virtual state

IR-active: when there is a change in dipole

moment in a molecule as it vibrates

Raman-active: when there is a change in

polarizability a molecule as it vibrates

Applications of Symmetry Molecular Vibrations

H2O

z1

x1

y1

z

x

y

z2

x2

y2

G 9 1 3 1

(Little) Orthogonality Theorem

GR

ii RRRgh

n )()()(1

cc

2]1113)1(1)1()1(1911[4

1

3]1)1(1311)1()1(1911[4

1

1]1)1(13)1(1)1(11911[4

1

3]111311)1(11911[4

1

2

1

2

1

B

B

A

A

n

n

n

n

2121 233 BBAA G

212

211

BBA

BBA

rot

trans

G

G

112 BAvib G

Water

Applications of Symmetry Molecular Vibrations

H2O

z1

x1

y1

z

x

y

z2

x2

y2

G 9 1 3 1

112 BAvib G

Water

IR active Raman active

Both IR and Raman active

Applications of Symmetry Molecular Vibrations

Selected Vibrational Modes

Gn(CO) 2 0 2 0

Gn(CO) = A1 + B1

A1 : B1 :

IR, Raman active

Gn(CO) 2 0 0 2 0 2 2 0

Gn(CO) = Ag + B3u

Ag : B3u :

Raman active IR active

Applications of Symmetry Molecular Vibrations

Selected Vibrational Modes

Exclusion Rule : In a molecule with i symmetry element,

IR-active and Raman -active vibrational modes are not

coincident.