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Symmetry
Translation
Rotation
Reflection
Slide rotation (Sn)
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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
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Audio-visuals on-line
Symmetry (Great site on symmetry in art and science by MargretJ. Geselbracht, Reed College , Portland Oregon)
The World of Escher:
Wallpaper Groups: The 17 plane symmetry groups
3D Exercises in Point Group Symmetry
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We shall now turn our attention away fromthe 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
sageCharacter Table
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A rotation through 180 about the internuclear axis leaves thesign of a orbital unchanged
Simple caseCharacter Table
but the sign of a orbital is changed.
In the language introduced in this lectture:
The characters of the C2 rotation are +1 and -1 for the and
orbitals, respectively.
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A B
Symmetry label C (i.e. rotation by 18 )
1
- 1
2 0
C2
180
C2
180
Simple caseCharacter Table
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C v3 Character Table Structure of character table
Symmetry groupSymmetry Operations
A
C
B
E A
C
B
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C v3 Character Table Structure of character table
Symmetry groupSymmetry Operations
C C C E3 3 3 =C C C3 33
1= C C E3
13
=
C31 A
C
B
B
A
C
C3 A
C
B
C
B
A
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C v3 Character Table Structure of character table
Symmetry groupSymmetry Operations
A
C
B
A
B
C
v AC
B
C
A
B
v'
A
C
B
B
C
A
v"
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C v3 Character Table Classes of elements
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 conjugatesform a class.
A
B C
v1
A
C B
v1
C3
B
A C
v1
C3v
B
A C
C3
1
B
A C=
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C v3 Character Table Classes of elements
v vC C31
31 =
v vC C' '31
31 =
v vC C'' ''31
31 =
C C C C3 3 3
1
3
=C C C C3
13 3 3
=
EC E C31
3 =
We have in general:
Thus C3 and C3-1 form
a class of dimension 2
The
can
two elements C and C
be related to each other byand
3 3-1
v v' v' , , '
C Cv3 31=
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C v3 Character Table Classes of elements
A
B CC3
1
B
C A
C31v
B
A C
C3
1C3 v
C
B A
=C
B A
v"
E Ev v =1
C Cv v3 31 = "
C Cv v31
3
= ' v v v v
=1
v v v v' ' " =1
v v" =1
In general
Thus
formv
and
a classof dimension
3. The elementsare related by
C and C
v v'
3 3-1
,
"
Elements conjugated to v ?
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C v3 Character Table Structure of character table
Symmetry groupSymmetry Operations
v 'v ' 'v
C3 The symmetry operations are
grouped by classes withthe dimension of each classindicated
Also indicated is the dimensionof the group h
h = total number of symmetry
elements
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2
v2
Name of point group
Symmetry elements
E : identity
C2 : Rotation
(xz) mirror plane
'(yz) mirror plane
Number of symmetry
elements
Character Table Structure of character table
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v2
2 Name of irreduciblerepresentations
A1 A2 B1 B2
Character Table Structure of character table
Characters of irreduciblerepresentations
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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
+-
+-
C2 v
v'
++
-
+
C2 v
v'
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v2
2E =
2=
v xz( ) =
vyz
' ( ) =
Symmetry is a1
pz
pz
pz
pz
pz
pz
pz
pz
Character Table Structure of character table
Irrep is A1
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v2
2E =
2=
v xz( ) =
vyz
' ( ) =
Symmetry is b2
py
py
py
py
py
py
-py
-py
Character Table Structure of character table
Irrep. is B2
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v2
2E =
2=
v xz( ) =
vyz
' ( ) =
Symmetry is b1
px
px
px
px
px
-px
px
-px
Character Table Structure of character table
Irrep. is B1
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v2
2E =
2=
v xz( ) =
vyz
' ( ) =
Symmetry is ?
1s1
1s1
1s2
1s2
1s2
1s2
1s2
1s2
Character Table Structure of character table
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v2
2E =
2 =
v xz( ) =
vyz
' ( ) =
Symmetry is ?
1s1
1s2
1s21s1
1s1
1s1
1s1
1s1
Character Table Structure of character table
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v2
E s s s s( ) ( )1 1 1 11 00 11 2 1 2=
1s1 1s2C s s s s2 1 2 1 21 1 1 1
0 11 0
( ) ( )=
This representation is not reduced
Character Table Structure of character table
v s s s s( ) ( )1 1 1 10 11 01 2 1 2
=
v s s s s'( ) ( )1 1 1 1 0 11 01 2 1 2=
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Character Table Structure of character tableC v2
2E =
2 =
v xz( ) =
vyz
' ( ) =
Symmetry is a1
1s+
1s+
1s+
1s+
1s+
1s+
1s+
1s+
Irrep is A1
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Character Table Structure of character tableC v2
2
1s
1s+
px
py
Only orbitals with samesymmetry label interact
A1 A1 pz A1
B1
B1 B1
B2
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C v2
2
Vibrations andnormal modes
Structure of character tableCharacter Table
H
O
H
O
H H
H
O
H
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Character Table Structure of character tableC v2
2E =
2 =
v xz( ) =
v
yz' ( ) =
Symmetry is a1
H
O
H
H
O
H
H
O
H
H
O
H
H
O
H
H
O
H
H
O
H
H
O
H
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v2
2E =
2 =
v xz( ) =
v
yz' ( ) =
Symmetry is b1
O
H H
O
H H
O
H H
O
H H
O
H H
O
H H
O
H H
O
H H
Character Table Structure of character table
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A1
A1
B1
v2
Vibrations andnormal modes
C
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v3
We have three classes ofsymmetry elements :
E the identity
Two three fold rotations
C and C3 3-1
Three mirror planes
v v v , ' , ' '
Character Table
Ch T bl
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v3
Molecular orbitals of NH3
a1 ex ey
Normal modes of NH3
Character Table
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What you must learn from this lecture
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
1. You are not expected to derive any of the theorem of group
theory. However, you are expected to use it as a tool
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.
Ch t T bl
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Appendix C von 3
Symmetry operations in
the same class are related to one
another by the symmetry operationsof 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 v.
Character Table
Character Table
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The dimension is 6 since wehave 6 elements.
We have three different symmetryrepresentations as we have three
different classes of symmetry elements
Character TableAppendix C von 3
Character Table
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The p orbital
does not change
with E, C C
z
3 3-1
,, ' , " v v v
The symmetry
rep. is A1
px pydoes not change
with E, C C3 3
-1,
, ' , " v v v
X
Y
X
Y
Character TableAppendix C von 3
Character Table Appendix Con
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X
Y
X
Y
X
Y
px p'x p' 'x
Ep = p ; C p p' ; C p = p"x x 3 x x 3-1
x x=
X
Y
X
Y Y
Character Table Appendix C von 3
Character Table
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p p p px y x y( ) = ( )
D C( )3
1
2
3
23
2
1
2
p p p px y x y( ) = ( )
D C( )3
1
1
2
3
23
2
1
2
The trace is - 1 forboth matrices
p p p px y x y( ) = ( )
D E( )1 0
0 1The trace is 2
which is also thedimension ofthe representation
Character TableAppendix C von 3
Character TableC
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p p p px y x y( ) = ( )
D v( )1 0
0 1
p p p px y x y( ) = ( )
D v( )'
12
32
3
2
1
2
p p p px y x y( ) = ( )
D v( )"
1
2
3
23
2
1
2
The trace is -1 for
both matrices
Character TableAppendix C von 3
Character TableA di C
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Typical symmetry
-adapted
linear combinationsof
orbitals in a
C 3v molecule.
Character TableAppendix C von 3