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Symmetry
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Reflection
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Slide rotation (Sn)
Lecture 36: Character Tables
The material in this lecture covers the following in Atkins.
15 Molecular Symmetry
Character tables
15.4 Character tables and symmetry labels
(a) The structure of character tables
(b) Character tables and orbital degeneracy
(c) Characters and operators
Lecture on-line
Character Tables (PowerPoint)
Character tables (PDF)
Handout for this lecture
Audio-visuals on-line
Symmetry (Great site on symmetry in art and science by Margret
J. Geselbracht, Reed College , Portland Oregon)
The World of Escher:
Wallpaper Groups: The 17 plane symmetry groups
3D Exercises in Point Group Symmetry
Character Table
Usage
We shall now turn our attention away from
the symmetries of molecules themselves
and direct it towards the symmetry characteristics of :
1. Molecular orbitals
2. Normal modes of vibrations
This discussion will enable us to:
I. Symmetry label molecular orbitals
II. Discuss selection rules in spectroscopy
Character Table
Simple case
A rotation through 180° about the internuclear axis leaves the
sign of a s orbital unchanged
but the sign of a p orbital is changed.
In the language introduced in this lectture:
The characters of the C2 rotation are +1 and -1 for the s and p
orbitals, respectively.
Character Table
A
Simple case
B
180°
C2
Symmetry label
s
C 2
s
1
p
-1
180°
C2
(i.e. rotation by 180)
p
Character Table
Symmetry group
Structure of character table
C3v
Symmetry Operations
C
C
E
A
A
B
B
Character Table
Structure of character table
Symmetry group
Symmetry Operations
A
C
C31
B
C3C3  C31
B
C
B
A
C3v
C3
C
C3C3C3  E
C
A
B
C31C3  E
A
Structure of character table
Character Table
Symmetry Operations
Symmetry group
sv
B
C
sv
A
A
C
C
C
B
sv"
B
'
A
C
C
A
B
A
C3v
A
B
B
Character Table
Classes of elements
C3v
In a group G={E,A,B,C,...},
we say that two elements B and C
are conjugate to each other if :
ABA-1 = C,
for some element A in G.
An element and all its conjugates
form a class.
A
A
B
C C

B A
s 1
v
B
B
B
C
C 3s 1
v
A
C
s v C 3s 1
v
A
C
C 31
Character Table
Classes of elements
C3v
We have in general:
EC3E1  C3
C3C3C1
3  C3
C31C3C3  C3
1
s vC3s 1

C
v
3
1
s v''C3sv''
 C31
1
s v'C3sv'
 C31
Thus C3 and C3-1 form
a class of dimension 2
The two elements C 3 and C-1
3
can be related to each other by
s v ,s v' , and s v' '
C3  sv C1
3
Character Table
Classes of elements
Elements conjugated to sv ?
A
B
C
C
In general
EsvE1  s v
C3s vC31  s v"
C31s vC3  s v'
s vsv s1
v  sv
s v'svs 1
v'  sv"
1
s v"s vsv"
 s v'
C
B
B
C31
A
A
C3v

C B
A
s v C 1 C 3 s v C 1
3
3
Thus s v , sv'
and s v"
form a class
of dimension
3. The elements
are related by
C 3 and C-1
3
C
B
s v"
A
Character Table
Structure of character table
C3v
Symmetry Operations
Symmetry group
The symmetry operations are
grouped by classes with
the dimension of each class
indicated
C3
Also indicated is the dimension
of the group h
sv
s' v
s'' v
h  total number of symmetry
elements
Character Table
C2
Structure of character table
C2v
Name of point group
Number of symmetry
elements
Symmetry elements
E : identity
C2 : Rotation
s(xz) mirror plane
s' (yz) mirror plane
Character Table
C2
Structure of character table
C2v
Name of irreducible
representations
A1 A2 B1 B2
Characters of irreducible
representations
of character table
Character Table C Structure
2 sv
sv'
++
C2
-
sv
sv'
+
+
-
The px,py, and pz orbitals
on the central atom of
a C2v molecule and the
symmetry elements of the
group.
Character Table
Structure of character table
C2v
Irrep is A 1
Symmetry is a1
C2

E
pz
pz

C2
pz
pz

sv (xz)
pz
pz
sv' (yz )

pz
pz
Character Table
Structure of character table
C2v
Irrep. is B2
Symmetry is b2
C2
E
C2
sv (xz)
sv' (yz )
py
py
py
py


py
-py

-py

py
Character Table
Structure of character table
C2v
Irrep. is B1
Symmetry is b1
C2
E
C2
sv (xz)
sv' (yz )
px


px
px
px


px
-px
px
-px
Character Table
Structure of character table
C2v
Symmetry is ?
C2
E
C2
1s2
1s2
sv (xz) 1s
2
sv' (yz )



1s2 
1s2
1s1
1s1
1s2
Character Table
Structure of character table
C2v
Symmetry is ?
C2
E
C2
1s1


1s1
1s1
1s2
sv (xz)
1s1 
1s2
sv' (yz )
1s1 
1s1
Character Table
Structure of character table
 1 0
E(1s1 1s2 )  (1s1 1s2 )

 0 1
 0 1
C2 (1s1 1s2 )  (1s1 1s2 )

1
0


1s1
C2v
1s2
 0 1
s v (1s1 1s2 )  (1s1 1s2 )

 1 0
 0 1
s v' (1s1 1s2 )  (1s1 1s2 )

1
0


This representation is not reduced
Character Table
Structure of character table
C2v
Irrep is A1
Symmetry is a1
C2
E
C2
1s+
1s+
sv (xz) 1s+
sv' (yz )



1s+ 
1s+
1s+
1s+
1s+
Character Table
Structure of character table
C2v
Irrep. is B1
Symmetry is b1
C2
E
C2
1s1s-


sv (xz) 1s
-

sv' (yz )

1s-
1s-
-1s-1s1s-
Character Table
C2
Structure of character table
C2v
1s+
A1
A1
pz
py
1sB1
B2
B1
px
Only orbitals with same
symmetry label interact
A1
B1
Character Table
Structure of character table
C2v
Vibrations and
normal modes
O
C2
H
H
O
H
H
O
H
H
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Character Table
Structure of character table
C2v
Symmetry
is a1
O
O
C2
E
H

H
H
H
O
O
C2

H
H
H
H
O
O
sv (xz)H
H

H
O
sv' (yz )
H
H
O
H

H
H
Character Table
Structure of character table
C2v
Symmetry is b1
O
O
C2
E
H
H

H
H
O
O
C2
H
H

H
H
O
O
H
sv (xz)
H

O
sv' (yz )
H
H
H
O
H
H
H
C2v
Vibrations and
normal modes
A1
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A1
B1
Character Table
C3v
We have three classes of
symmetry elements :
E
the identity
Two three fold rotations
C3 and C-1
3
Three mirror planes
sv , s' v , s' ' v
C3v
Character Table
Molecular orbitals of NH 3
a1
ex
Normal modes of NH3
ey
What you must learn from this lecture
1. You are not expected to derive any of the theorem of group
theory. However, you are expected to use it as a tool
2.. You must understand the different parts of a character
table for a symmetry group: (a) Name of symmetry group;
(b)Classes of symmetry operators; (c) Names of irreducible
symmetry representations. (d) The irreducible characters
3. For simple cases you must be able to deduce what irreducible
representation a function or a normal mode belongs to by the
help of a character table.
Character Table
Appendix on C3v
Symmetry operations in
the same class are related to one
another by the symmetry operations
of the group. Thus, the
three mirror planes shown here
are related by threefold
rotations, and the two rotations
shown here are related by
reflection in sv.
Character Table
Appendix on C3v
The dimension is 6 since we
have 6 elements.
We have three different symmetry
representations as we have three
different classes of symmetry elements
Character Table
Appendix on C3v
py
px
does not change
The pz orbital
does not change
with E, C3, C-1
3
sv , s'v , s"v
The symmetry
rep. is A1
with E, C3 , C -1
3
sv , s'v , s"v
Y
Y
X
X
Character Table
Appendix on C3v
Y
Y
Y
X
X
X
px
p'x
p' ' x
Epx = px ; C3px  p'x ; C-1
3 px = p"x
Y
Y
Y
X
X
Character Table
Appendix on C3v
 1 0
px py D(E)  px py 

 0 1
The trace is 2
which is also the
 1
3
 
 
dimension of
1
2
2

px py D(C3 )  p x p y 
the representation
1
 3

 
 2
2



px







py D(C3 )  px
 1
3


py  2 2 
1
 3

 
 2
2

The trace is - 1 for
both matrices
Character Table
px
Appendix on C3v
 1 0
py D(s v )  px p y 

 0 1



 1
3



2 
px py D(s'v )  px p y  2
3
1


 
 2
2 
 1
3


2 
px py D(s"v )  px p y  2
1
 3

 
 2
2








The trace is - 1 for
both matrices
Character Table
Appendix on C3v
Typical symmetry
-adapted
linear combinations
of
orbitals in a
C 3v molecule.