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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.