Symmetry Translation Rotation Reflection Slide rotation (S n )
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Transcript of Symmetry Translation Rotation Reflection Slide rotation (S n )
Symmetry
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Translation
Rotation
Reflection
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 operatorsLecture 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 MargretJ. Geselbracht, Reed College , Portland Oregon) The World of Escher: Wallpaper Groups: The 17 plane symmetry groups 3D Exercises in Point Group Symmetry
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
UsageCharacter Table
A rotation through 180° about the internuclear axis leaves the sign of a orbital unchanged
Simple case Character 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.
A B
Symmetry label C 2 (i.e. rotation by 180°)
σ 1
π -1
C2
180°
σ
C2
180° π
Simple case Character Table
C3v Character Table Structure of character table
Symmetry group Symmetry Operations
A
C
B
E A
C
B
C3v Character Table Structure of character table
Symmetry group Symmetry Operations
C3C3C3 =EC3C3 =C3−1 C3
−1C3 =E
C3−1 A
C
B
B
A
C
C3 A
C
B
C
B
A
C3v Character Table Structure of character table
Symmetry group Symmetry Operations
A
C
B
A
B
C
v A
C
B
C
A
B
v'
A
C
B
B
C
A
v"
C3v 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 conjugates form a class.
A
B C
v−1
A
C B
v−1C3
B
A C
v−1C3σv
B
A C
C3−1
B
A C=
C3v Character Table Classes of elements
σvC3σv−1=C3
−1
σv'C3σv'−1=C3
−1
σv''C3σv''−1=C3
−1
C3C3C3−1=C3
C3−1C3C3 =C3
EC3E−1=C3
We have in general:
Thus C3 and C3-1 form
a class of dimension 2
The two elements C3 and C3-1
can be related to each other byσv,σv', and σv''
C3 =σvC3−1
C3v Character Table Classes of elements
A
B CC3
−1
B
C A
C3−1v
B
A C
C3−1C3 v
C
B A
=C
B A
σv"
EσvE−1=σv
C3σvC3−1=σv"
C3−1σvC3 =σv'
σvσvσv−1=σv
σv'σvσv'−1=σv"
σv"σvσv"−1=σv'
In general Thus σv,σv' and σv"form a class of dimension 3. The elements are related by
C3 and C3-1
Elements conjugated to v ?
C3v Character Table Structure of character table
Symmetry group Symmetry Operations
σv σ'vσ''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 symmetryelements
C2
C2v
Name of point group
Symmetry elements
E : identity
C2 : Rotation
σ(xz) mirror plane
σ'(yz) mirror plane
Number of symmetry elements
Character Table Structure of character table
C2v
C2 Name of irreducible representations
A1 A2 B1 B2
Character Table Structure of character table
Characters of irreducible representations
The px,py, and pz orbitals on the central atom ofa C2v molecule and thesymmetry elements of the group.
Character Table Structure of character table
+ -+ -
C2 σv
σv'
++
-
+
C2 σv
σv'
C2v
C2 E =
C2=
σv(xz) =
σv' (yz) =
Symmetry is a1
pz
pz
pz
pz
pz
pz
pz
pz
Character Table Structure of character table
Irrep is A1
C2v
C2 E =
C2=
σv(xz) =
σv' (yz) =
Symmetry is b2
py
py
py
py
py
py
-py
-py
Character Table Structure of character table
Irrep. is B2
C2v
C2 E =
C2=
σv(xz) =
σv' (yz) =
Symmetry is b1
px
px
px
px
px
-px
px
-px
Character Table Structure of character table
Irrep. is B1
C2v
C2 E =
C2=
σv(xz) =
σv' (yz) =
Symmetry is ?
1s1
1s1
1s2
1s2
1s2
1s2
1s2
1s2
Character Table Structure of character table
C2v
C2 E =
C2=
σv(xz) =
σv' (yz) =
Symmetry is ?
1s1
1s2
1s21s1
1s1
1s1
1s1
1s1
Character Table Structure of character table
C2vE(1s1 1s2)=(1s1 1s2)
1 00 1
⎛ ⎝ ⎜
⎞ ⎠ ⎟
1s1 1s2C2(1s1 1s2)=(1s1 1s2)
0 11 0
⎛ ⎝ ⎜
⎞ ⎠ ⎟
This representation is not reduced
Character Table Structure of character table
σv(1s1 1s2)=(1s1 1s2)0 11 0
⎛ ⎝ ⎜
⎞ ⎠ ⎟
σv'(1s1 1s2)=(1s1 1s2)0 11 0
⎛ ⎝ ⎜
⎞ ⎠ ⎟
Character Table Structure of character table C2v
C2 E =
C2=
σv(xz) =
σv' (yz) =
Symmetry is a1
1s+
1s+
1s+
1s+
1s+
1s+
1s+
1s+
Irrep is A1
Character Table Structure of character table C2v
C2 E =
C2=
σv(xz) =
σv' (yz) =
Symmetry is b1
1s-
1s-
1s-
1s-
1s-
1s-
-1s-
-1s-
Irrep. is B1
Character Table Structure of character table C2v
C2
1s-
1s+
px
py
Only orbitals with same symmetry label interact
A1 A1 pzA1
B1B1 B1
B2
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C2v
C2
Vibrations and normal modes
Structure of character table Character Table
H
O
H
O
H H
H
O
H
Character Table Structure of character table C2v
C2 E =
C2=
σv(xz) =
σv' (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
C2v
C2 E =
C2=
σv(xz) =
σv' (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
C2v
Vibrations andnormal modes
C3v
We have three classes ofsymmetry elements:
E the identity
Two three fold rotations
C3 and C3-1
Three mirror planesσv,σ'v ,σ''v
Character Table
C3v
Molecular orbitals of NH3
a1 ex eyNormal modes of NH3
Character Table
What you must learn from this lecture
2.. You must understand the different parts of a charactertable 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 grouptheory. However, you are expected to use it as a tool
3. For simple cases you must be able to deduce what irreduciblerepresentation a function or a normal mode belongs to by the help of a character table.
Appendix on C3v
Symmetry operations in the same class are related to oneanother by the symmetry operations of the group. Thus, thethree mirror planes shown here are related by threefold rotations, and the two rotations shown here are related byreflection in v.
Character Table
The dimension is 6 since wehave 6 elements.
We have three different symmetryrepresentations as we have threedifferent classes of symmetry elements
Character Table Appendix on C3v
The pz orbitaldoes not change
with E, C3, C3-1
σv, σ'v ,σ"vThe symmetryrep. is A1
px pydoes not change
with E, C3, C3-1
σv, σ'v ,σ"v
X
Y
X
Y
Character Table Appendix on C3v
X
Y
X
Y
X
Y
px p'x p''x
Epx =px ; C3px =p'x; C3-1 px =p"x
X
Y
X
Y Y
Character Table Appendix on C3v
px py( )D(C3)= px py( )−
12
32
3
2−
1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
px py( )D(C3−1 )= px py( )
−12
−32
−3
2−
1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
The trace is -1 forboth matrices
px py( )D(E)= px py( )1 0
0 1
⎛
⎝ ⎜
⎞
⎠ ⎟
The trace is 2which is also thedimension ofthe representation
Character Table Appendix on C3v
px py( )D(σv)= px py( )−1 0
0 1
⎛
⎝ ⎜
⎞
⎠ ⎟
px py( )D(σv' )= px py( )
12
−32
−3
2−
1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
px py( )D(σv" )= px py( )
12
32
3
2−
1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
The trace is -1 forboth matrices
Character Table Appendix on C3v
Typical symmetry-adapted linear combinations of orbitals in aC 3v molecule.
Character Table Appendix on C3v