Symmetry and point groups are used to describe the symmetry present in molecules. There are 5 main symmetry elements: rotation axes (Cn), improper rotation axes (Sn), planes of symmetry (s), centers of symmetry (i), and identity (E). Point groups assign the set of symmetry operations for a molecule unambiguously using a flow chart approach. Common point groups include Cnv for molecules with a Cn axis and mirror planes, and Dnh for molecules with a Cn axis and perpendicular C2 axes, as well as sh planes. Symmetry allows predicting properties like bonding, spectra, and chirality.

1. Symmetry and Introduction
to Group Theory
Disclaimer: Some lecture note slides are adopted from CHEM 59-
250 - Originally by Dr. Samuel Johnson
Power point slides from Inorganic Chemistry 4th edition by Gary
L. Miessler and Donald A. Tarr

2. Symmetry and Point Groups
I. Introduction
A. Symmetry is present in nature and in human culture

3. B. Using Symmetry in Chemistry
1. Understand what orbitals are used in bonding
2. Predict IR spectra or Interpret UV-Vis spectra
3. Predict optical activity of a molecule
II. Symmetry Elements and Operations
A. Definitions
1. Symmetry Element = geometrical entity such as a line, a plane, or a point,
with respect to which one or more symmetry operations can be carried out
2. Symmetry Operation = a movement of a body such that the appearance
after the operation is indistinguishable from the original appearance (if you
can tell the difference, it wasn’t a symmetry operation)
B. The Symmetry Operations
1. E (Identity Operation) = no change in the object
a. Needed for mathematical completeness
b. Every molecule has at least this symmetry operation

4. 2. Cn (Rotation Operation) = rotation of the object 360/n degrees about an axis
a. The symmetry element is a line
b. Counterclockwise rotation is taken as positive
c. Principle axis = axis with the largest possible n value
d. Examples:
C23 = two C3’s
C33 = E
C17 axis

5. 3. s (Reflection Operation) = exchange of points through a plane to an
opposite and equidistant point
a. Symmetry element is a plane
b. Human Body has an approximate s operation
c. Linear objects have infinite s‘s
d. s h = plane perpendicular to principle axis
e. s v = plane includes the principle axis
f. s d = plane includes the principle axis, but not the outer atoms
sd
O C O O
H H
sh sv

6. 4. i (Inversion Operation) = each point moves through a common central
point to a position opposite and equidistant
a. Symmetry element is a point
b. Sometimes difficult to see, sometimes not present when you think
you see it
c. Ethane has i, methane does not
d. Tetrahedra, triangles, pentagons do not have i
e. Squares, parallelograms, rectangular solids, octahedra do

7. 5. Sn (Improper Rotation Operation) = rotation about 360/n axis followed by
reflection through a plane perpendicular to axis of rotation
a. Methane has 3 S4 operations (90 degree rotation, then reflection)
b. 2 Sn operations = Cn/2 (S24 = C2)
c. nSn = E, S2 = i, S1 = s
d. Snowflake has S2, S3, S6 axes

8. C2 sd
C. Examples:
1. H2O: E, C2, 2s
O
H H
sv
2. p-dichlorobenzene: E, 3s, 3C2, i
Cl Cl
3. Ethane (staggered): E, 3s, C3, 3C2, i, S6
H H
H C C
H H
H
4. Try Ex. 4-1, 4-2

11. III. Point Groups
A. Definitions:
1. Point Group = the set of symmetry operations for a molecule
2. Group Theory = mathematical treatment of the properties of the group
which can be used to find properties of the molecule
B. Assigning the Point Group of a Molecule
1. Determine if the molecule is of high or low symmetry by inspection
a. Low Symmetry Groups

12. b. High Symmetry Groups

13. 2. If not, find the principle axis
3. If there are C2 axes perpendicular
to Cn the molecule is in D
If not, the molecule will be in C or S
a. If sh perpendicular to Cn then Dnh or Cnh
If not, go to the next step
b. If s contains Cn then Cnv or Dnd
If not, Dn or Cn or S2n
c. If S2n along Cn then S2n
If not Cn

14. C. Examples: Assign point groups of molecules in Fig 4.8

15. Rotation axes of “normal” symmetry molecules

16. Perpendicular C2 axes
Horizontal Mirror Planes

17. Vertical or Dihedral Mirror Planes and S2n Axes
Examples: XeF4, SF4, IOF3, Table 4-4, Exercise 4-3

18. D. Properties of Point Groups
1. Symmetry operation of NH3
a. Ammonia has E, 2C3
(C3 and C23) and 3sv
b. Point group = C3v
2. Properties of C3v (any group)
a. Must contain E
b. Each operation must
have an inverse; doing both
gives E (right to left)
c. Any product equals
another group member
d. Associative property

19. We need to be able to specify the symmetry of molecules clearly.
F H
No symmetry – CHFClBr
Br
Cl
F H
Some symmetry – CHFCl2
H H
Cl
Cl
More symmetry – CH2Cl2
Cl
Cl H Cl
More symmetry ? – CHCl3
Cl
Cl
What about ?
Point groups provide us with a way to
indicate the symmetry unambiguously.

20. Symmetry and Point Groups
Point groups have symmetry about a single point at the center of mass of
the system.
Symmetry elements are geometric entities about which a symmetry
operation can be performed. In a point group, all symmetry elements must
pass through the center of mass (the point). A symmetry operation is the
action that produces an object identical to the initial object.
The symmetry elements and related operations that we will find in
molecules are:
The Identity operation does nothing to the object – it is necessary for
mathematical completeness, as we will see later.
Element Operation
Rotation axis, Cn n-fold rotation
Improper rotation axis, Sn n-fold improper rotation
Plane of symmetry, s Reflection
Center of symmetry, i Inversion
Identity, E

21. n-fold rotation - a rotation of 360°/n about the Cn axis (n = 1 to )
O(1)
180° O(1)
H(2) H(3) H(3) 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.
H(3) H(4)
H(2)
120° 120°
N(1) N(1)
N(1)
H(2) H(4) H(3) H(2)
H(4) H(3)
In ammonia there is a C3 axis so we can perform 3-fold
(120°) rotations to get identical arrangement of atoms.

22. Notes about rotation operations:
- Rotations are considered positive in the counter-clockwise direction.
- Each possible rotation operation is assigned using a superscript integer m
of the form Cnm.
- The rotation Cnn is equivalent to the identity operation (nothing is moved).
H(3)
H(2) H(4)
C31 C32
N(1) N(1)
N(1)
H(2) H(4)
H(4) H(3) H(2)
H(3)
H(2)
C33 = E
N(1)
H(4) H(3)

23. Notes about rotation operations, Cnm:
- If n/m is an integer, then that rotation operation is equivalent to an n/m -
fold rotation.
e.g. C42 = C21, C62 = C31, C63 = C21, etc. (identical to simplifying fractions)
Cl (5) Cl (2) Cl (3)
C41 C42 = C21
Cl (2) Ni (1) Cl (3) Cl (4) Ni (1) Cl (5) Cl (5) Ni (1) Cl (4)
Cl (4) Cl (3) Cl (2)
C43
Cl (4)
Cl (3) Ni (1) Cl (2)
Cl (5)

24. Notes about rotation operations, Cnm:
- Linear molecules have an infinite number of rotation axes C because any
rotation on the molecular axis will give the same arrangement.
C(1) O(2)
O(2)
C(1)
O(3) C(1) O(2)
N(2)
N(1)
N(1) N(2)

25. The Principal axis in an object is the highest order rotation axis. It is
usually easy to identify the principle axis and this is typically assigned to
the z-axis if we are using Cartesian coordinates.
Ethane, C2H6 Benzene, C6H6
The principal axis is the three-fold axis The principal axis is the six-fold axis
containing the C-C bond. through the center of the ring.
The principal axis in a tetrahedron is a
three-fold axis going through one vertex
and the center of the object.

26. Reflection across a plane of symmetry, s (mirror plane)
O(1) sv O(1)
H(2) H(3) H(3) H(2)
These mirror planes are
Handedness is changed by reflection!
called “vertical” mirror
planes, sv, because they
contain the principal axis.
O(1) sv O(1) The reflection illustrated in
the top diagram is through a
H(3) H(3)
mirror plane perpendicular
H(2) H(2)
to the plane of the water
molecule. The plane shown
on the bottom is in the
same plane as the water
molecule.

27. Notes about reflection operations:
- A reflection operation exchanges one half of the object with the reflection of
the other half.
- Reflection planes may be vertical, horizontal or dihedral (more on sd later).
- Two successive reflections are equivalent to the identity operation (nothing is
moved).
A “horizontal” mirror plane, sh, is
sh perpendicular to the principal axis.
This must be the xy-plane if the z-
axis is the principal axis.
In benzene, the sh is in the plane
sd sd of the molecule – it “reflects” each
atom onto itself.
sh
Vertical and dihedral mirror
sv planes of geometric shapes.
sv

28. Inversion and centers of symmetry, i (inversion centers)
In this operation, every part of the object is reflected through the inversion
center, which must be at the center of mass of the object.
F Cl
1 F Cl 2 2 1
Br
Br 2
i 1
1 2 Br 2 1
Br
1 2
Cl F
Cl F
1 2 2 1
i
[x, y, z] [-x, -y, -z]
We will not consider the matrix approach to each of the symmetry operations in this
course but it is particularly helpful for understanding what the inversion operation
does.

29. n-fold improper rotation, Snm (associated with an improper rotation axis or
a rotation-reflection axis) This operation involves a rotation of 360°/n
followed by a reflection perpendicular to the axis. It is a single operation and
is labeled in the same manner as “proper” rotations.
F1 F2 H1
F4 S4 1 F1 H4 C H2
H3
F2 F3 S41
F2 H2
F3 F4
F1 H1 C H3
90° sh
H4
C21
F3
S42
F4 H3
H2 C H4
H1
Note that: S1 = s, S2 = i, and sometimes S2n = Cn (e.g. in box) this makes
more sense if you examine the final result of each of the operations.

30. Identifying point groups
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.

31. Identifying point groups

32. Identifying point groups
Special cases:
Perfect tetrahedral (Td) e.g. P4, CH4
Perfect octahedral (Oh) e.g. SF6, [B6H6]-2
Perfect icosahedral (Ih) e.g. [B12H12]-2, C60

33. Identifying point groups
Low symmetry groups:
Only* an improper axis (Sn)
e.g. 1,3,5,7-tetrafluoroCOT, S4
F1
F4
F2
F3
Only a mirror plane (Cs)
e.g. CHFCl2
F H
Cl
Cl

34. Identifying point groups
Low symmetry groups:
Only an inversion center (Ci)
e.g. (conformation is important !)
F Cl
Br
Br
Cl F
No symmetry (C1)
e.g. CHFClBr F H
Br
Cl

35. Identifying point groups
Cn type groups:
A Cn axis and a sh (Cnh)
e.g. B(OH)3 (C3h, conformation is important !)
H
O
H
B
O O
H
O O
H
H
e.g. H2O2 (C2h, conformation is important !)
Note: molecule does not have to be planar
e.g. B(NH2)3 (C3h, conformation is important !)

36. Identifying point groups
Cn type groups:
Only a Cn axis (Cn)
e.g. B(NH2)3 (C3, conformation is important !)
H H
N
B
H
N N
H H
H
e.g. H2O2 (C2, conformation is important !)
H
O O
H

37. Identifying point groups
Cn type groups:
A Cn axis and a sv (Cnv)
e.g. NH3 (C3v) H
N
H H
e.g. H2O2 (C2v, conformation is important !)
O O
H H

38. Identifying point groups
Cn type groups:
A Cn axis and a sv (Cnv)
e.g. NH3 (C3v, conformation is important !)
H
H
e.g. carbon monoxide, CO (Cv)
H
There are an infinite number of possible
H
N
H
Cn axes and sv mirror planes.
H H
H H C O
e.g. trans-[SbF4ClBr]- (C4v) Cl
F
F F
Sb O
C
F Sb
Br
Cl F F F
Br
F

39. Identifying point groups
Dn type groups:
A Cn axis, n perpendicular C2 axes
and a sh (Dnh)
e.g. BH3 (D3h)
H
H B
H H
B
H H
e.g. NiCl4 (D4h)
Cl (2)
lC lC
iN
lC lC
Cl (4) Ni (1) Cl (5)
lC
lC iN lC
lC
Cl (3)

40. Identifying point groups
Dn type groups:
e.g. pentagonal prism (D5h)
A Cn axis, n perpendicular C2 axes
and a sh (Dnh)
e.g. Mg(5-Cp)2 (D5h in the eclipsed conformation)
Mg
Mg
View down the C5 axis
e.g. square prism (D4h)
e.g. carbon dioxide, CO2 or N2 (Dh)
There are an infinite number of possible Cn
axes and sv mirror planes in addition to the
sh.
O O O
C

41. Identifying point groups
Dn type groups:
A Cn axis, n perpendicular C2 axes
and no mirror planes (Dn)
-propellor shapes
e.g. Ni(CH2)4 (D4) H H H H
Ni
H H H H H
H
H H
Ni
H H
H H
H H H H
Ni
H H H H

42. e.g. (SCH2CH2)3 (D3 conformation is important!)
e.g. propellor (D3)
e.g. Ni(en)3 (D3 conformation is important!) en = H2NCH2CH2NH2

43. Identifying point groups
Dn type groups:
A Cn axis, n perpendicular C2 axes
and a sd (Dnd)
e.g. ethane, H3C-CH3
(D3d in the staggered conformation)
H
H H
H H
H
H
H
H
H
H
H
dihedral means between sides or planes –
this is where you find the C2 axes

44. e.g. Mg(5-Cp)2 and other metallocenes in the staggered conformation (D5d)
Fe Mg Al M
View down the C5 axis
These are pentagonal
antiprisms
e.g. triagular e.g. square
e.g. allene or a tennis ball (D2d)
antiprism (D3d) antiprism (D4d)

45. Summary of point group identification
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.