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Symmetry planes and reflections Center of inversion Proper axes and proper rotations Improper axes and improper rotations
Products and relations of symmetry operations The symmetry point groups
III. III. Molecular symmetry andMolecular symmetry andthe symmetry groupsthe symmetry groups
IIIIII--5. Products and relations 5. Products and relations of symmetry operations of symmetry operations
iyzxyxziyzxzxy
111111
111111
111111
zyxzyxyzzyxzyxxzzyxzyxxy
Reflection on the plane
Inversion through the center
111111 zyxzyxi x
z
y
1 1 1( , , )x y z
Rotation about the proper axis
1
2
2
1
1
1
zyx
zyx
An
1000cossin0sincos
Cn
z
contour clockwise
1000cossin0sincos
Cn
z
clockwise
1000cossin0sincos
Sn
z
1000cossin0sincos
Sn
z
Rotation about the proper axis
zCxyzS nn
n2 x
z
y
1 1 1( , , )x y z
x
z
y
1 1 1( , , )x y z
Example:
100010001
1000cossin0sincos
C2
z
1111112 zyxzyxzC The same way we can derive
1111112 zyxzyxxC
1111112 zyxzyxyC 1112111111211122
1112111111211122
zyxzCzyxzyxxCzyxyCxCzyxzCzyxzyxyCzyxxCyC
Example: 11111111121112
1111111111112
zyxizyxzyxzCzyxxyzCzyxizyxzyxxyzyxzCxy
2S
x
y(x1, y1)
d d
45 45
(-y1, -x1)
(y1, x1)
111111
111111
zxyzyxzxyzyx
d
d
x
y
(x1, y1)
d d
4545
(y1, x1)
(-y1, -x1)
111111
111111
zxyzyxzxyzyx
d
d
Example: clockwise d
d
zCxzxzzC
4
4
1 2
34
x
y
xz
zC 4 4 1
23
x
y
xz
xz 1 4
32
x
y
xz
d
1 2
34
x
y
xzd
GTAchem4-1
Calculate
1 2
3
4
x
y
v 5
6
zC 6
v
?
???
6
6
6
6
zCzC
zCzC
v
v
v
v
and then specify where they are in Figure
Basically we need study Cn and Sn only
As Cn is subgroup of Sm , so the most important is nS
*The symmetry element can be thought of as the whole of space+Note the equivalences and
perpendicular to rotation axis
Rotation by followed by reflection
n-Fold axis ofimproper rotation+
iInversionCenter of inversionReflectionMirror planeRotation by n-Fold symmetry axis
EIdentity*SymbolSymmetry operationSymmetry element
Summary Important Symmetry operations and symmetry elements
2 / n nC
2 / nnS
1 =S 2 =S
Enantiomer
Not superimposed (by proper rotation)
Chiral (different molecule)
Achiral (same molecule)
superimposed
3H COH
H3CHHO
H
3H CH
3CHHH
H
Molecules that are not superimposable on their mirror image
These molecules are called dissymmetric
Theorem: A molecule that has no improper rotation axis must be dissymmetric
If molecule has a Sn axis, then it includes the following collinear axes
oddnfor ,,,evennfor ,,,
22
2
ESSSSESSS
nn
nnnn
nnnn
So if n = odd, there is hnnS
If n = even, iCS h 22
Mirror plane
A BCD
AB
C D
A B
CD
AB
C D
A B
CD
ABC D
i Rotate by
A BCD
AB
C D
superimposed Proper rotationis OK
Example
34
1
23
4
1
23
4
12
C
4
Mirror plane
34
1
2 34
1
2C 4
4S
superimposed
Example
See textbook page 37
1,3,5,7-tetramethylcyclooctatetraene has neither plane symmetry nor center symmetry
Has plane symmetry
Example
4S
4S3CH
3CH
3CH
3H C
In general if molecule has S2m symmetry
Mirror plane
mS 2
C2m
superimposed
IIIIII--6. The symmetry point groups6. The symmetry point groups
For a given molecule, we collect a complete list of symmetry operations
h321 A A A A
Those operations form a Group
This group is called a point group
3C
v
2C
2C 2C h
vv
E
For example: planar AB3 molecule
3C 3S23C23
53 CS h
Totally 12 symmetry operations
53
53
33
hh
vv
vv
vv
22
22
22
23
23
33
S
S S C CC CC CC C
C CE E
S
533hvvv222
233
533hvvv222
233
S S C C C C C E
S S C C C C C E
533hvvv2222333 S S C C C C C E C
2C2C
2C v
v
v
B1
B2
B3
B1
B2
B3
2CB1
B2
B3
3CB1
B2 B3
B1
B2
B3
2C 223 CCC
count clockwise
23C
clockwise
It looks like we can use vCC 23 , but this is not true
If we use coordinate system, we can know that
Rotation * rotation = roatation
321 axisCaxisCaxisC kmn General law
axiscrossCplaneplane kvv 21
533hvvv2222333 S S C C C C C E C
2C
2C
2C v
v
v
B1
B2
B3
v
B1
B2
B3
3CB1
B2 B3
B1
B2
B3
v vvC 3
axiscrossCplaneplane kvv 21
axiscrossCplaneplane kvv 12
Should not be 2C
count clockwise
23C
clockwise
533hvvv2222333 S S C C C C C E C
2C
2C
2C v
v
v
B1
B2
B3
v
B1
B2
B3
v B1
B2B3
B1
B2
B3
23C 2
3Cvv
axiscrossCplaneplane kvv 21
Should not be 2C
count clockwise
3C clockwise
223 CCC 232 CCC
533hvvv2222333 S S C C C C C E C
C C C C C C 2222223 C C C C C C C 2223222 C
C C C C C C 22223222 C 23222222 C C C C C C C
223 CCC 223 CCC
vvC 3 vv C 3
vvvvv3 vC v3vvv vvC
23Cvv 3Cvv
2 ' '' ' '' 53 3 2 2 2 3 3
2 ' '' ' '' 53 3 2 2 2 3 32 '' ' '' ' 5
3 3 3 2 2 2 3 32 2 ' '' ' '' 53 3 3 2 2 2 3 3
' '' 2 5 ' ''2 2 2 2 3 3 3 3' ' '' 2 5 ' ''2 2 2 2 3 3 3 3'2
v v v h
v v v h
v v v h
v v v h
h v v v
h v v v
E C C C C C S SE E C C C C C S SC C C E C C C S SC C E C C C C S SC C C C E C C S SC C C C C E C S SC
' '' ' 2 5 '' '
2 2 2 3 3 3 3' '' 5 2 ' ''
3 3 3 3 2 2 2' ' '' 5 2 ' ''
3 3 3 3 2 2 2'' '' ' 5 2 '' '
3 3 3 3 2 2 25 ' '' ' '' 2
3 3 2 2 2 3 35 '' ' '' ' 2
3 3 3 2 2 2 3 3
h v v v
v v v v h
v v v v h
v v v v h
h h v v v
h v v v
C C C C C E S SS S E C C C C C
S S C E C C C CS S C C E C C C
S S C C C E C CS S S C C C C C E
5 5 ' '' ' '' 23 3 3 2 2 2 3 3h v v vS S S C C C C E C
2 ' ''3 3
2 ' ''3 32 ' ''
3 3 32 2 '' '3 3 3
'' ' 23 3
' ' '' 23 3
'' '' ' 23 3
v v v
v v v
v v v
v v v
v v v v
v v v v
v v v v
E C CE E C CC C C EC C E C
E C CC E CC C E
C3v
subgroup
v
v v
Molecular has no symmetry operations other than E 1CE C1 (1)
Elements Group name (order)
Molecular has only one reflection plane
E2 Cs (2)
Molecular has only one inversion center
Eii 2 Ci (2)
Nomenclature of point group
Molecular has only one proper axis Cn
E CC C CC nnnnnnn 132 Cn (n)
Elements Group name (order)
Molecular has only one improper axis Sn (n=even)
hnnS
Eii 2 Ci (2)
E SS S SS nnnnnnn 132
Sn (n)
Molecular has only one improper axis Sn (n=odd)
ES SS SS nnnn
nnnn
212
Cnh (2n) E CC C CC
E CC C CCnn
nnnnnh
nn
nnnnn
132
132
as
Where S2
Find all symmetry operations of AB3 non-planar moleculeand make multiplication table (give detailed procedures)
GTAchem4-2
Molecular has one proper axis Cn and one C2 axis ( Cn C2 )
E CC C CC nnnnnnn 132
Dn (2n)
Elements Group name (order)
All n twofold axes in the same plane and perpendicular to the prAll n twofold axes in the same plane and perpendicular to the principal axis incipal axis CCnn
Cnv (2n) vn
vv vWhen n = odd , n the same type of planes
12222 n CC C C
Molecular has one proper axis Cn and one reflection plane that contain this axis
E CC C CC nnnnnnn 132
d
ndd
vn
vv
2/
d
2/v
When n = even , n/2 the same another n/2 same
All n vertical reflection planes intersect at one line which is Cn axis
All n twofold axes in the same plane and perpendicular to the prAll n twofold axes in the same plane and perpendicular to the principal axis incipal axis CCnn
Cnv (2n)
v
nvv vn = odd , n the same planes
E CC C CC nnnnnnn 132
d
ndd
vn
vv
2/
d
2/v
n = even , n/2another n/2
h = Dnh (4n)
Horizontal reflection plane
n improper axes n improper axes E CC C CC nnnnnnnh 132 n proper axes n proper axes
2v nCvnvvh 22/d
22/
v
)2/(
)2/(
Cn
Cnn
dh
nvh
n vertical planes n vertical planes
n Cn C22 axes axes
nhh D
zC 2
111111 zyxzyxxyxyh
yCxC
hvvh
hvvh
2
2
Blackboard
1111112
1111112
zyxzyxyCzyxzyxxC
111111
111111
zyxzyxyzyz
zyxzyxxzxz
v
v
x
y
h
yCyC
xCxC
hhv
hhv
22
22
Multiplied by h
vvvv zC if 2
xCv
2
yCv
2
3C
v
2C
2C2C
v v
In general two planes intersectin a line that is Cn axis
If two planes has dihedral angle is
22
nC nvv
vv
33223/ Cn
Example
All n twofold axes in the same plane and perpendicular to the prAll n twofold axes in the same plane and perpendicular to the principal axis incipal axis CCnn
Dn (2n) ndh Dd = Dnd (4n)
Vertical reflection plane
n improper axes n improper axes collinear with collinear with CCnn
n proper axes n proper axes
n vertical planes n vertical planes
n Cn C22 axes axes
E CC C CC nnnnnnn 132
22222 C CC C C
n
122523222222 n-nnnnnd S S SSC CC C
dndnnnnnnnd E CC C CC d132
22
2C
If principal axis is C2(z) , then there are two C2 in xy-plane
xC 2
yC 2
2
So two C2 in xy-plane has angle andthen dihedral angle (, C2(x)) is /2
d
32
3C
If principal axis is C3(z) , then there are three C2 in xy-plane
xC 2
2C
2 So two C2 in xy-plane has angle and
then dihedral angle (, C2(x)) is /2d
2C
11 yx
2211 yx yx d
dx 22 yx
1
n2 with nC
2cos2sin2sin2sin2cos2cos
1112
1112
yxryyxrx
10002cos2sin02sin2cos
d
1
1
1
2
2
2
zyx
zyx
d
xC2
2C
This Cn(z) with n C2
100010001
2 xC
2
10002cos2sin02sin2cos
S
10002cos2sin02sin2cos
d
4
nn 222222
nd SxC 22
Dnh (4n) Dnd (4n)
nhh D ndh DBig difference
Example: ethane
4
5
6
3
1
2
C3(z) S6(z) symmetry2C
3
5
26
1
42C
3
5
2
6
1
4
There areThe other two C2
There is no h symmetry D3d (4*3)
d
For example: planar AB3 molecule
There is h symmetry D3h (4*3)
3C
v
2C
2C 2C h
vv
E
3C 3S23C23
53 CS h
EXERCISES Set A. pages 61-62
A3.1, A3.2, A3.3
EXERCISES Set C. page 64
(1), (3), (6), (9)
GTAchem4-3
GTAchem4-4
A3.2(e) excluded
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