This document discusses group theory and symmetry elements in chemistry. It defines symmetry elements as geometrical entities like points, lines, or planes that objects can be rotated or reflected around. Common symmetry operations are rotation, reflection, and inversion. Examples of symmetry elements include axes of rotation, planes of reflection, and centers of inversion. Several molecules like water, carbon dioxide, ethene, benzene, and ruthenium complexes are analyzed to identify their specific symmetry elements and point groups.

1. GROUP THEORY
TONY FRANCIS
DEPARTMENT OF CHEMISTRY
St. MARY'S COLLEGE,
MANARKADU

2. Mathematical study of symmetry is called
Group Theory
• Symmetry Element– A symmetry element is a
geometrical entity such as a point, a line or a
plane about which a symmetry operation is
performed.
• Symmetry operation– A symmetry operation
is a movement such as inversion about a
point, rotation about a line or a reflection
about a plane in order to get an equivalent
orientation.

3. An equivalent orientation is an orientation similar to
the original orientation but not the identity.
Equivalent
orientation Identity

4. Symmetry Elements
Element Symmetry Operation Symbol
Identity E
Proper axis Rotation by 2π/n Cn
Plane of symmetry Reflection σ
Center of symmetry Inversion i
Improper axis of Rotation by 2π/n Sn
symmetry followed by reflection
perpendicular to the
axis of rotation

5. Identity, E
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.

6. Centre of symmetry (i)
• It is a point within the molecule from which
lines drawn to opposite direction meet similar
points at exactly the same distance and
direction.

7. Proper axis of symmetry
• It is an axis passing through the molecule
about which the molecule is rotated through
360◦, if we get n times equivalent orientations
the molecule has an n-fold axis of symmetry.

9. Principal axis
• If there are more than one axis of symmetry in many
cases one of the axis is identified as principal axis. The
selection will be on the following basis:-
1. Highest order axis
2. Unique axis
3. The axis passing through maximum no of molecule.
4. The axis perpendicular to the plane of the molecule
The other axis are known as subsidiary axis

10. Plane of symmetry
• Plane of symmetry is a plane which divide the
molecule into two equal halves such that one
half is the mirror image of the other half.
• On the basis of the principal axis they are of two
types vertical and horizontal plane.
• HP:-plane perpendicular to the principal axis(σh )
• VP:-plane which is along the principal axis or
involving the principal axis (σv )

12. Rotational axes and mirror planes of the water
molecule:
C2
principal axis
C2
C2σv
mirror plane
σv
mirror plane
The water molecule has only one rotational axis, its C2 axis,
which is also its principal axis. It has two mirror planes that
contain the principal axis, which are therefore σv planes. It
has no σh mirror plane, and no center of symmetry.

13. A rotation-reflection operation
(Sn) required
rotation of 360° /n, followed by
reflection through
a plane perpendicular to the axis of
rotation.

14. Equivalent and non-equivalent operations
14
O
H H
C2
sv
sv’
H
N
H
H
C3
sv
sv
sv
• sv : No atom moves
sv ‘: H atoms interchange
• sv : Two atoms move
Other two don’t
• sv and sv ‘ do not interchange by
C2
• The three sv planes interchange by
C3
Non-equivalent
planes
Equivalent planes
(Same class)

16. Boron trifluoride
C3
principal axis
C3
principal axis
σh
σh
σv σv
C2
C2 C2
boron trifluoride has a C3 principal
axis and three C2 axes, a σh mirror plane
three σv mirror planes, but no center of inversion
E,2C3,3C2,3σv,σh,2S3—D3h

20. Carbon dioxide-Dαh

22. E,C2,2C2,2σv,σh,i--- D2h
Ethene

26. C6
principal axis
C2
C2
C2
C6
C2
σv
σv
Rotational axes and mirror planes of benzene
σh
C6
principal axis
C6
principal axis

28. Ruthenium triethylenediamine
- Ru(en)3 - D3

37. Distinct operations
• D.O are operations that cannot be
represented by any other axis of lower
symmetry.

38. • Order- Total number of symmetry
operation
• Classes- It is the number of distinct
symmetry operations