Symmetry, symmetry elements, Symmetry operations, point groups.

1. Dr. BASAVARAJAIAH S. M.
Assistant Professor and Coordinator
P. G. Department of Chemistry
Vijaya College.
drsmbasu@gmail.com
9620012975

7. Understanding of symmetry is essential in discussions of
molecular spectroscopy and calculations of molecular
properties.
consider the structures of BF3, and BF2H, both of which are
planar
BF bond distances
are all identical (131
pm) trigonal planar
the BH bond is shorter (119 pm)
than the BF bonds (131 pm).
pseudo-trigonal planar

8. The molecular symmetry properties are not the same
In this chapter,
Symmetry Element,
Symmetry Operation,
Point Group
Group theory is the mathematical treatment of
symmetry.

11. • Identity (E)
• Proper Axis of Rotation (Cn)
• Mirror Planes (σ)
• Center of Symmetry (i)
• Improper Axis of Rotation (Sn)

12. All molecules have Identity. This
operation leaves the entire molecule
unchanged. A highly asymmetric
molecule such as a tetrahedral carbon
with 4 different groups attached has
only identity, and no other symmetry
elements.

13. Proper Axis of Rotation (Cn)
The symmetry operation of rotation about an n-fold axis (the
symmetry element) is denoted by the symbol Cn, in which the angle
of rotation is:
– where n = 2, 180o
rotation
– n = 3, 120o rotation
– n = 4, 90o rotation
– n = 6, 60o rotation
– n = , (1/ )o
rotation
• principal axis of rotation,

14. H(2)
O(1)
H(3) H(3)
O(1)
H(2)
In water there is a C2 axis so we can perform a 2-fold (180°) rotation
to get the identical arrangement of atoms.
180°
For H2O

15. For NH3

16. Applying this notation to the BF3 molecule
BF3 molecule contains a C3 rotation axis

17. If a molecule possesses more than one type of n-axis, the axis
of highest value of n is called the principal axis; it is the axis of
highest molecular symmetry. For example, in BF3, the C3 axis
is the principal axis.

18. Ethane, C2H6 Benzene, C6H6
The principal axis is the three-fold axis
containing the C-C bond.
The principal axis is the six-fold axis
through the center of the ring.

19. Mirror planes (σ)
sh => mirror plane perpendicular to a
principal axis of rotation
sv => mirror plane containing principal
axis of rotation
sd => mirror plane bisects dihedral angle made
by the principal axis of rotation and two
adjacent C2 axes perpendicular to principal
rotation axis
The symmetry operation is one of reflection and the
symmetry element is the mirror plane (denoted by s ). If reflection
of all parts of a molecule through a plane produces an
indistinguishable configuration, the plane is a plane of symmetry.

20. The reflection of
the water molecule in
either of its two mirror
planes results in a
molecule that looks
unchanged.
The subscript “v” in
σv, indicates a vertical
plane of symmetry. This
indicates that the mirror
plane includes the
principal axis of rotation
(C2).

22. Reflection through a plane of symmetry
(mirror plane)

23. Center of Symmetry (i)
If reflection of all parts of a molecule through the
centre of the molecule produces an indistinguishable
configuration, the centre is a centre of symmetry,
also called a centre of inversion; it is designated by
the symbol i.
CO2 SF6Benzene
[x, y, z]
i
[-x, -y, -z]

24. Improper Axis of Rotation (Sn)
If rotation through about an axis, followed by
reflection through a plane perpendicular to that
axis, yields an indistinguishable configuration, the
axis is an n-fold rotation–reflection axis, also
called an n-fold improper rotation axis. It is
denoted by the symbol Sn.

27. For example, in planar BCl3, the S3 improper
axis of rotation corresponds to rotation about
the C3 axis followed by reflection through the
sh plane.

28. Summary Table of Symmetry Elements and Operations

30. Group theory is the mathematical treatment of
symmetry.

31. Group
A group is a set, G, together with an operation • (called the group
law of G) that combines any two elements a and b to form another
element, denoted a • b or ab. To qualify as a group, the set and
operation, (G, •), must satisfy four requirements known as the group
axioms:
Closure
For all a, b in G, the result of the operation, a • b, is also
in G.
Associativity
For all a, b and c in G, (a • b) • c = a • (b • c).
Identity element
There exists an element e in G such that, for every
element a in G, the equation e • a = a • e = a holds. Such
an element is unique , and thus one speaks of the identity
element.

32. Inverse element
For each a in G, there exists an element b in G, commonly
denoted a−1 (or −a, if the operation is denoted "+"), such
that a • b = b • a = e, where e is the identity element.
The result of an operation may depend on the order of the
operands. In other words, the result of combining element a with
element b need not yield the same result as combining
element b with element a; the equation
a • b = b • a

34. Abelian and Non-abelian Group
An abelian group is a set, A, together with an operation • that
combines any two elements a and b to form another element
denoted a • b. The symbol • is a general place holder for a concretely
given operation. To qualify as an abelian group, the set and
operation, (A, •), must satisfy five requirements known as
the abelian group axioms:
Closure
Associativity
Identity element
Inverse
Commutativity
For all a, b in A, a • b = b • a
A group in which the group operation is not commutative is called a
"non-abelian group" or "non-commutative group".

35. Determining the point group of a molecule or molecular ion

36. We can use a flow chart such as this one to determine the point
group of any object. The steps in this process are:
1. Determine the symmetry is special (e.g. octahedral).
2. Determine if there is a principal rotation axis.
3. Determine if there are rotation axes perpendicular to the principal
axis.
4. Determine if there are mirror planes.
5. Assign point group.
IDENTIFYING POINT GROUPS

39. Point group
Symmetry
operations
Simple description
of typical geometry
Example 1 Example 2
C1 E
no
symmetr
y, chiral Bromofluorochloro
methane
C2H2F2Cl2
Dichlorodifluoro
ethane
Cs E, σh
mirror
plane, no
other
symmetr
y
SOCl2
Thionyl
dichloride
Chloroiodometh
ane
Ci E, i
inversion
center
(S,R) 1,2-
dibromo-1,2-
COMMON POINT GROUPS

40. Point group
Symmetry
operations
Simple
description of
typical geometry
Example 1 Example 2
C2 E, C2
"open
book
geometry
," chiral
Hydrogen
peroxide
C3 E, C3
Propeller,
chiral
PPh3
Triphenylphophi
ne
C∞v
E, 2C∞,
∞σv
Linear
HCl, HCN, HI, CO, NC, NCS, HCN,
HCCH

41. Point group Symmetry operations
Simple
description of
typical geometry
Example 1 Example 2
C2v
E,
C2, σv(xz),
σv'(yz)
Angular
H2O
Sulfur dioxide
(SO2),
Dichlorometha
ne
CH2Cl2
C3v E, 2C3, 3σv
Trigonal
pyramidal
or
Tetrahedr
al Ammonia
(NH3)
Phosphane
(PH3)
Chloroform
(CHCl3)
C4v E, 2C4 , C2 ,
2σv , 2σd
Square
pyramidal
Xenon
oxytetrafluoride
(XeOF4)
Pentaborane
(B5H9)

42. Point group Symmetry operations
Simple description
of typical
geometry
Example 1 Example 2
C2h E C2 i σh
Planar with
inversion
center trans-1, 2-
Dichloroethylen
e
B(OH)3
C3h
E,
C3,C3
2,σh,
S3, S3
5
Propelle
r
Boric acid Phloroglucinol
D2
E, C2(x),
C2(y), C2(z)
twist,
chiral
Biphenyl
Cyclohexane
(twist)
D3 E, C3(z), 3C2,
triple
helix,
chiral
Tris
(ethylenediamin
e)
cobalt(III) cation

43. Point group Symmetry operations
Simple
description of
typical geometry
Example 1 Example 2
D2h
E, C2(z)
,C2(y),
C2(x), i, σ(xy
), σ(xz),
σ(yz)
Planar
with
inversion
center Ethylene (C2H4) Diborane (B2H6)
D3h
E, 2C3, 3C2,
σh, 2S3,3σv
Trigonal
planar or
trigonal
bipyramid
al
Boron trifluoride
(BF3) (PCl5)
D4h
E, 2C4,
C2 ,2C2'
2C2 i 2S4 σh
2σv 2σd
Square
planar
Xenon
tetrafluoride

44. Point group Symmetry operations
Simple
description of
typical geometry
Example 1 Example 2
D6h
E
2C6 2C3 C2 3C
2'
3C2‘’ i 2S32S6
σh 3σd 3σv
Hexagona
l
Benzene (C6H6)
Coronene
(C24H12)
D2d
E, 2S4,C2, 2C2'
, 2σd
90° twist
Allene
D3d
E, 2C3 , 3C2 ,
i ,2S6 3, σd
60° twist
Ethane
(Staggered)
D∞h
E, C∞,
∞σv, ∞C2, i Linear
N2, O2, F2, H2, Cl2, CO2, BeH2, N3

45. Point group Symmetry operations
Simple description
of typical geometry
Example 1 Example 2
S2 E, 2S2 , C2 -
Tetraphenylmetha
ne
Td
E, 8C3 , 3C2 ,
6S4 , 6σd
Tetrahedral
Methane
Phosphorus
pentoxide
Oh
E,
8C3 ,6C2 ,6C4
, 3C2 , i ,
6S4 ,8S6
3σh ,6σd
Octahedral
or cubic
Sulfur hexafluoride
Ih
E
12C5 12C5
2 20
C3 15C2 i12S10
12S10
3 20S6 1
Icosahedral
or
Dodecahedr
al
Buckminsterfullerene