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Lecture 35: Symmetry Groups
The material in this lecture covers the following in Atkins.
15 Molecular Symmetry
15.2 Symmetry classification of molecules(d) The groups Sn
(e) The cubic groups
(f) The full rotation group
15.3 Some immediate consequences of symmetry
(a) Polarity
(b)Chirality
Lecture on-line
Symmetry Groups (PowerPoint)
Symmetry Groups (PDF)
Handout for this lecture
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The Symmetry Classification of Molecules Sn
S4
Posesses the inproper rotation axis Sn
S C S
S C
4 2 43
42
2
: , ,S4
=
S2 same as Ci
HO H
COOH
OHHOOC
H
Meso-tartaric
acid
S4 S6S8
C C i S S3 32
65
6, , , ,S C S C
S C S
2 4 83
2
85
43
87
; ; ,
, ,
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The Symmetry Classification of Molecules Cubic groups
These groups have more than one principle axis
C3
C3
C3
C3Td
C2 S4
d
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S4
The Symmetry Classification of Molecules Cubic groups
F
SF F
FF
F
Oh
C4
C3d S6
C2
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The relation of an icosahedron
to a cube.The buckminsterfullerene
molecule (15)
is related to this objectby cutting
off each apex to form a
regular pentagon.
The Symmetry Classification of Molecules Cubic groups
I
C5
C3
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The Symmetry Classification of Molecules Cubic groups
Shapes corresponding to the point
groups (a) T.
The presence of thewindmill-like structures
reduces the
symmetry of the object from Td.
C3
C2
T
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The Symmetry Classification of Molecules Cubic groups
Shapes corresponding to the
point groups (b) O.
The presence of the windmill-like
structures reduces the
symmetry of the object from Oh.
O
C4
C2
C3
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The Symmetry Classification of Molecules Cubic groups
Th
The shape of an object
belonging to the group Th.
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Molecule
Linear ?
i ?ND h C v
N
Twoor morec
nn>2 ?
i ?
Td
N
C5 NIn Oh
N
The SymmetryClassificationof Molecules
C O
N N
C ON N
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n ?
N
N
?
Ni?
s
i 1
Select C withhighest n; than,
are the nC
to C ?
n
2
n
perpendicular
NY
The SymmetryClassification
of MoleculesI
F
ClBr
HO H
COOH
OH
HOOC
H
Meso-tartaricacid
N
Quinoline
N
Quinoline
I
F
ClBr
HO H
COOH
OH
HOOC
H
Meso-tartaricacid
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n ?
N
N
?
Ni?
s
i 1
N
h ?Dnh
Y
Y
N
n d ?YDnd N Dn
The SymmetryClassification
of Molecules
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C v3NH3
n ?
N
Select C withhighest n; than,
are the nC
to C ?
n
2
n
perpendicular
N
The SymmetryClassificationof Molecules
N
h
?Cnh
n v ?CnvS n2 ?
N
S n2
CnB
O
OO
H
H
H
B(OH)3S4
O O
H H
C2
H2O2
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(a) A molecule with a
Cn axis cannot have a dipole
perpendicular to the axis,
The SymmetryClassificationof Molecules
Dipole moments
(but (b) it may have one
parallel to theaxis.
The arrows represent local contributions to the overall
electric dipole, such as may arise from bonds between pairs of
neighbouring atoms with different electronegativities.
r s s s
= ( )r rdr
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The SymmetryClassification
of Molecules
Dipole moments
I
F
ClBr
HO H
COOH
OH
HOOC
H
Meso-tartaricacid
N
Quinoline
O O
H H
C2
H2O2
C
C
Cl H
H Cl
Trans CHCl=CHCl
B
O
OO
H
H
H
B(OH)3
0 = 0
inversion
0
in plane
= 0
hsymmetry
0along C2
0
along C2
0
along C3 = 0inversion
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The SymmetryClassification
of Molecules
Chirality
A chiral molecule is a moleculethat can not be superimposed
on its mirror image
O O
H H
C2
O O
H H
C2
HOOH Mirror image
Canaxis
only contain a Cn
COOH
H2N H
H
Not chiral
COOH
H2N H
CH3
chiral
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A mathematical group, G = {G,}, consists of a set of
elements G = {E, A,B,C,D,....}
A binary relation, called group multiplication is defined such
That:
(a) The product of any two elements A and B in the group
is another element in the group, i.e., we write AB G.
(b) If A, B, C are any three elements in the group then(AB)C = A(BC). Therefore, group multiplication
is associative, and frequently, we omit the brackets.
Groups and group multiplications
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(c) There is a unique element E in G such that
EA=AE=A, for every element A in G.
The element E is called the identity element.
(d) For every element A in G, there is a unique element X in G
, such that XA = AX = E.
The element X is referred to as the inverse of A and is denoted A-1.
Groups and group multiplications
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x
y
z
ClCl
C
H
A B
A BH
A B
AB
B
AB
A
H
C
Cl Cl
HH
C2
ClCl
H
C
C
ClClA B
A BHH
C
ClClB
ABHH
A
(yz)
A B
A B
B
B
A
C
ClCl
HH
C
ClCl
HH
(xz)A
Groups and group multiplications
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Groups and group multiplications
A B
A B
B
B
A
AHH
(xz)C2
H
C
H
C
ClCl ClCl
x
y
z
ClCl
C
H
A B
A BH
(yz)
A B
A B
B
B
A
C
ClCl
HH
C
ClCl
HHA
(yz)(xz)
C2
C
ClClA B
A B
ClCl B
B
A
AHH
(yz)C2
H
C
H
(xz)
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C2v E C2 (xz) (yz)
E E C2 (xz) (yz)
C2 C2 E (yz) (xz)
(xz) (xz) (yz) E C2
(yz) (yz) (xz) C2 E
Groups and group multiplications
This table contains all theinformation
about the group and its
structure.
The name of this molecular
point group is C2v.
There are some observations to
makeabout this table.
1. In each row and each column,each operation appears once
and only once
(2) We can identify smaller
groups within the larger one.
For example, {E,C2} is a group.
There are two others;what are they?
(3) In this particular table, we
observe that the group productis commutative.
This is not necessarily true for
other groups.
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Diatomics
The parity of an orbital is even (g) if its wavefunction
is unchanged under inversion in the centre of symmetry
of the molecule, but odd (u) if the wavefunction changes sign.
Heteronuclear diatomic molecules do not have a centre ofinversion, so for them the g,u classification is irrelevant.
Parity of orbitals
The Symmetry
Classification
of orbitals
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Diatomics
The parity of an orbital is even (g) if its wavefunction
is unchanged under inversion in the centre of symmetry
of the molecule, but odd (u) if the wavefunction changes sign.
Heteronuclear diatomic molecules do not have a centre of
inversion, so for them the g,u classification is irrelevant.
Parity of orbitals
The Symmetry
Classification
of orbitals
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Diatomics
The in a term symbol refers
to the symmetry of an orbital
when it is reflected in a plane
containing the two nuclei.
g
= 1
= 1
Reflection index
u
u
g
= 1
u g
= 1
The Symmetry
Classification
of orbitals
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In the language introduced in the next lecture, the
characters of the C2 rotation are +1 and -1 for the and
orbitals, respectively.
A rotation through 180
about the internuclear axis
leaves the sign of a orbital
unchanged
but the sign of a orbital is
changed.
The Symmetry
Classification
of orbitals
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What you should learn from this lecture
1. For a given molecule be able to identify all the symmetry
elements
2. From the list of elements be able to identify the point groupUsing a provided flow chart similar to that given by Atkins
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The SymmetryClassification
of Molecules
Dipole moments
O
H H
Water has local dipoles alongeach bond
+
+
r
OH1
r
OH2
Or
O
H H
r
OHx2
r
OH
y2
They can be decomposedinto x and y components
r
OH
x
1
r
OH
y1
x
y
The
and are
OH
OH
x
x
components
equal in
magnetudebut opposite in sign as
they are related by arotation around C2
r
r
1
2
O
H H
C2
r
OHx1
C2
=r
OHx2
Only dipole alongprinciple axis possible