Lecture 34: Symmetry Elements The material in this lecture ... - Cobalt

Lecture 34: Symmetry Elements
The material in this lecture covers the following in
Atkins.
15 Molecular Symmetry
The symmetry elements of objects
15.1 Operations and symmetry elements
15.2 Symmetry classification of molecules
(a) The groups C 1 ,C i , and C s
(b) The groups C n ,C nv , and C nh
(c) The groups D n ,D nh , and D nd
Lecture on-line
Symmetry Elements (PowerPoint)
Symmetry Elements (PDF)
Handout for this lecture

Symmetry elements
The systematic discussion of
symmetry is called
Some objects are more
symmetrical than others
Any rotation of sphere
around axis through
center brings sphere
over into itself
Some of the symmetry elements of a cube.
The twofold, threefold, and fourfold axes
are labelled with the conventional symbols.
Only some rotations of sphere
brings cube into itself

Symmetry elements
(a) An NH 3 molecule has a
threefold (C 3 ) axis
(b) an H 2 O
molecule has a twofold (C 2 ) axis.
Both have other symmetry
elements too.
NH3 has more rotation
symmetry than H 2 O

Symmetry elements
An action that leaves an object the same after
it has been carried out is called a :
Symmetry operations are :
For each symmetry operation there is a :
symmetry element
Which is
with respect to which the operation is performed

Operators
and Symmetry elements
The classification of objects according to
symmetry elements corrsponding to
operations that leave at least one common
point unchanged gives rise to the :
There are five kinds of symmetry
operations and corresponding
symmetry elements
1. The identity E Consists of doing nothing
Cl
I
F
Br
Only element for CFClBrI

Operators
and Symmetry elements
C 3 and C
3 -1
C 6
C ; C ; C
and C ; C
(b) an H2O
molecule has a twofold (C2) axis.
6 3 2
6 -1 3 -1 The principle rotation axis
is the axis of the higest fold
C 5

Operators
and Symmetry elements
The H 2O
molecule
has two mirror
planes. They are
both vertical
(that is contain
the vertical axis)
and as so are
denoted
σ and σ '
v
v

C 6
Dihedral mirror planes ( bisect
Operators
and Symmetry elements
C 2
C 2
C 2
Benzene has one
mirror plane
perpendicular
to the principle
C6 - axis ( σ h)
σ d)
the C 2 axis perpendicular to the
principle axis

Operators
and Symmetry elements
A regular octahedron has a centre of inversion (i).

Operators
and Symmetry elements
(a). A CH 4 molecule
has a fourfold improper
rotation axis (S 4 ):
the molecule is indistinguishable
after a 90° rotation followed by
a reflection across the horizontal
plane, but neither operation
alone is a symmetry operation.
C 4
σ

Operators
and Symmetry elements
(b) The staggered form of
ethane has an S 6 axis
composed of a 60°
rotation followed by
a reflection.

The Symmetry Classification of Molecules
The Schoenflies system
Hermann − Mauguin

The Symmetry Classification of Molecules
C , C , C 1 i s
The groups C , C , C 1 i s
F
A molecule belongs to C if it
has only the identitity E 1
Cl
I
Br
COOH
HO
H
A molecule belongs to C i if it
has only the identitity E and i
H
HOOC
OH
Meso-tartaric
acid
A molecule belongs to Cs if it
has only the identity and a mirror
plane
N
Quinoline

The Symmetry Classification of Molecules C n
The groups C n,
C nv , and C nh
A molecule belongs to C n if it
has a C n axis and the identitity E only
C C C
4 5 6
C 2 C 3
C 2
H
O
O
H
H 2 O 2

The Symmetry Classification of Molecules
The groups C n,
C nv , and C nh
A molecule belongs to C nv if it
has a C n axis and n vertical
mirror planes
C nv
cone
C
C C 4v
C 5v
C6v
C ∞v
2v
3v
NH 3
C 3v
H 2 O C2v
CO
Heteronuclear
diatomics are
C ∞v

The Symmetry Classification of Molecules
The groups C n,
C nv , and C nh
A molecule belongs to C nh if it
has a C n axis and a horizontal
mirror plane
C nh
C 2h
C 3h
C 4h
C C 5h
6h
Cl
C
H
O
H
C
H Cl
Trans CHCl=CHCl
H
O
B
O
H
B(OH) 3

The Symmetry Classification of Molecules
The groups C n,
C nv , and C nh
C nh
C 2
σ
i
The presence of a twofold axis and a horizontal mirror plane
jointly imply the presence of a centre of inversion in the
molecule.

The Symmetry Classification of Molecules
D n
The groups D n,
D nv , and D nh
A molecule belongs to D n if it
has a C n axis and n two fold axes
(C ) perpendicular to C .
2 n
N
Pt
N
N
N
N
D 3
N
A molecule with n twofold rotation axes perpendicular to an
n-fold rotation axis belongs to the group Dn.

The Symmetry Classification of Molecules
The groups D n,
D nv , and D nh
D 2 D3
D 4 D5 D 6
D n

The Symmetry Classification of Molecules
The groups D , D , and D
A molecule belongs to D nh if it
has a C n axis and n two fold axes
(C 2) perpendicular to C n as well as
a horizontal mirror plane.
n nv nh
D nh
A molecule
with a
Mirror plane
perpendicular
to a C n axis,
and with n
twofold axes in
the plane, belongs
to the group
D nh .

The Symmetry Classification of Molecules
The groups D , D , and D
n nv nh
A molecule belongs to D nh if it
has a C n axis and n two fold axes
(C 2) perpendicular to C n as well as
a horizontal mirror plane.
D 2h
D 3h
D 4h
D 5h
D h
D nh
C 2
H H
C
σ h
6
C 2
C
C 4
H H D 2h
C 4
C 2
Cl
Cl
σ h Au
C'
Cl
Cl
2
C 2
D 4h C 2

The Symmetry Classification of Molecules
The groups D , D , and D
A molecule belongs to D nh if it
has a C n axis and n two fold axes
(C 2) perpendicular to C n as well as
a horizontal mirror plane.
n nv nh
F
B
D nh
F
F
E , C 3, 3C 2 , σh
E , C 6, 3C 2 ,3C' 2, σh

The Symmetry Classification of Molecules
The groups D n,
D nv , and D nh
D nh
F
σ h
C 3
C 2
C 2
F
C 2
F
P
F
F
D 3h

The Symmetry Classification of Molecules
The groups D , D , and D
A molecule belongs to D nd if
in addition to the elements of
D n it posesses n dihedral mirror
planes
n nv nh
D nd
D 2d
D 3d
D 4d
D 5d
D 6d
C 2
C 2
C C C C 2
D 2d

What you must learn from this lecture
1. Given a molecule, you should be able to identify the different
symmetry elements (C n ,σ v ,σ h ,σ d ,i,S n , etc. )
2. Having identified the symmetry elements of a molecule,
you should be able to establish the point group to which
the molecule belong