Symmetry is an important concept in science that is commonly observed in nature. It can be described through symmetry operations such as rotations, reflections, and inversions that leave an object in an indistinguishable position. Molecules exhibit different types of symmetry elements including axes of rotation, planes of symmetry, centers of inversion, and improper axes. Understanding molecular symmetry allows classification of molecules and their physical and chemical properties using the mathematical framework of group theory.

2. #Symmetry is common phenomenon in the world
around us
#It is difficult to overestimate the importance of
symmetry in many aspects of science, not only in
chemistry

5. Facial symmetry

6. mirror image relationship
indicate facial symmetry
Mirror image

7. Mirror image

8. (a) Is more symmetrical than (b)

9. Water molecule
revolving anticlock wise
Symmetry operation – a rearrangement
of a body after which it appears
unchanged
A symmetry operation moves molecule
about an axis , a point ,or a plane to a
position indistinguishable

12. Proper axis of rotation Cn
 A rotation about this axis b y an angle Q=360/n leaves
the molecule in an INDISTINGUISHIBLE confrigation
such axis is called n fold proper axis of rotation
 Example water C2 axis

13. Ammonia molecule c3 axis

14. C3 C3
C2
C2

16. Some gumatricle images
C6
C4 C3 C2

17. Plane of symmetry ð
 A molecule posses a plane of symmetry if reflection
through through the plane leaves molecule un
changed
 ðh if plane of symmetry is PERIPENDICULAR to principal
axis
 ðv if plane of symmetry is PARLLEL to principal axis
 Ðd if vertical plane bisect the angle between two C2 axis
 Example in water there is two ðv and no ðh
Red is molecular
plane

18. Plane of symmetry in BF3 molicule
 One ðh i.e molecular plane
 Three ðd

19. Impoper axis of rotation
 A molecule is said to posses an improper axis of
rotation of order n if rotation about the axis by 360/n
followed by REFLECTION in a plane PEPANICULAR
to axis leaves molecule I indistinguisble position
 If molecule has plane of symmetry ðh perpendicular
to proper axis of rotation C2,C4 and C6 the axis
naturally become S2,S4,S6 respectively
 An s2 axis is equivalent to i
 Example staggered ethane

20. C6

21. Inversion i
 If a point exist in center of molecule such that
IDENTICAL atoms are found on either side at equal
distance from it the central point is called center of
inversion
 In H2O and BF3 molecule there is no i
 In C2H4 molecule there is one
ethene

22. i in some geometrical image
i present
i absent

23. Identity E
 If sequence of operation brings molecule back to its
original configuration ,the net operation is called
identity operation E
 Example two C2 rotation in H2O molecule equivalent
to E
1 2
C2

24. Identity E
 If sequence of operation brings molecule back to its
original configuration ,the net operation is called
identity operation E
 Example two C2 rotation in H2O molecule equivalent
to E
1 2 12
C2

25. Identity E
 If sequence of operation brings molecule back to its
original configuration ,the net operation is called
identity operation E
 Example two C2 rotation in H2O molecule equivalent
to E
1 2 112 2
C2 C2

26.  In BF3
 Each and i generate one operation only because two
successive i and bring back the molecule to its
original configuration
EECEC
ECEC
d 

22,,
2
2,
2
2
2
3
3
,,
,




EiE  22
,

27. 
21
EiE  22
,
1
2
i

28.  
121 2
EiE  22
,
1
2
i

29.  
121 1 22
EiE  22
,
1
2
i

30.  
121 1 22
EiE  22
,
1
1
2
2
i i

31.  
121 1 22
EiE  22
,
1
1
12
2
2
i i

32.  But Cn and Sn with n>2 generate number of operation
example C3 generates
 for example in BF3
2
3
1
3 andCC
ECsame
same  EC 3
3
C3
1
2
33

33.  But Cn and Sn with n>2 generate number of operation
example C3 generates
 for example in BF3
2
3
1
3 andCC
ECsame
same  EC 3
3
1
C3
1
22
33
3

34.  But Cn and Sn with n>2 generate number of operation
example C3 generates
 for example in BF3
2
3
1
3 andCC
ECsame
same  EC 3
3
1
C3C3
1
1
2
22
33
3
3

35.  But Cn and Sn with n>2 generate number of operation
example C3 generates
 for example in BF3
2
3
1
3 andCC
ECsame
same  EC 3
3
1
C3C3
C3
1
1
2
22
33
3
3

36.  But Cn and Sn with n>2 generate number of operation
example C3 generates
 for example in BF3
2
3
1
3 andCC
ECsame
same  EC 3
3
1
C3C3
C3
1
1
2
22
33
3
3
E

37. Total symmetry of some molecule
 The symmetry of molecule can be determined by
listing all the element present on it
 H2O E,C2, v (yz), v(xz)

38. BF3
 E, C3, S3, 3C2, h, 3 d 

39. Ethene
 E, 3C2, h, v, v ,
 ,,

C CC
H H
H
H

40. benzene
 E, C6, C3 ,C2 , i , S6, S3, h, ,,3 2

C ,3 2

C 
C6
,2

C
,2

C
d3 v3

41. WARNING
 Differentiating between a v and d can be little
tricky at first, but with practice it should become clear
 Both v and d are collinear with the principle
rotation axis
 In a molecule that has both v and d mirror plane,
the d planes bisect as many atom possible.
 
 
 


42. Exercise
 NH3 choose write option 1.E,C2,3 v, h
 Di chloroethylene trans cis



2.E,C2,3 v
3.E,C3,3 v
4.E,C4,3 h
1.E,C2, d, v 1.E,C2, h, v
2.E,C2, h,i
3.E,C3, d, 
d
4.E,C2, v, ,i 
v
2.E,C2, v,

 
 
v
3.E,C1, v, i
4.E,C1, d, i
Cl
Cl
Cl
Cl

43. Exercise
 NH3 choose write option 1.E,C2,3 v, h
 Di chloroethylene trans cis



2.E,C2,3 v
3.E,C3,3 v
4.E,C4,3 h
1.E,C2, d, v 1.E,C2, h, v
2.E,C2, h,i
3.E,C3, d, 
d
4.E,C2, v, ,i 
v
2.E,C2, v,

 
 
v
3.E,C1, v, i
4.E,C1, d, i
Cl
Cl
Cl
Cl

44. Exercise
 NH3 choose write option 1.E,C2,3 v, h
 Di chloroethylene trans cis



2.E,C2,3 v
3.E,C3,3 v
4.E,C4,3 h
1.E,C2, d, v 1.E,C2, h, v
2.E,C2, h,i
3.E,C3, d, 
d
4.E,C2, v, ,i 
v
2.E,C2, v,

 
 
v
3.E,C1, v, i
4.E,C1, d, i
Cl
Cl
Cl
Cl

45. Exercise
 NH3 choose write option 1.E,C2,3 v, h
 Di chloroethylene trans cis



2.E,C2,3 v
3.E,C3,3 v
4.E,C4,3 h
1.E,C2, d, v 1.E,C2, h, v
2.E,C2, h,i
3.E,C3, d, 
d
4.E,C2, v, ,i 
v
2.E,C2, v,

 
 
v
3.E,C1, v, i
4.E,C1, d, i
Cl
Cl
Cl
Cl

46. Combination of operations
 Two operation over the molecule combined to get a third
operation
 Example in water molecule C2 is followed by operation
the result is same as single operation
v
v
1
1 2
2
C2

47. Combination of operations
 Two operation over the molecule combined to get a third
operation
 Example in water molecule C2 is followed by operation
the result is same as single operation
v
v
1
1
12
2
2
C2

48. Combination of operations
 Two operation over the molecule combined to get a third
operation
 Example in water molecule C2 is followed by operation
the result is same as single operation
v

v

v
v
1
1
12
2 12
2
C2 v

49. Combination of operations
 Two operation over the molecule combined to get a third
operation
 Example in water molecule C2 is followed by operation
the result is same as single operation
v

v

v
v
1
1
12
2 12
2
C2 v

v

50. Group theory
 The relationships among the symmetry elements can
be treated very elegantly in terms of group theory
 A group in mathematics consists of a set of members
which obey the following four rules
 Rule 1 The combination of any two members A
and B result in C ,which also belong to same group .
 AB = C
 Example water [ E, C2 , , ]

vv

 vvC  2

51. Group theory
 The relationships among the symmetry elements can
be treated very elegantly in terms of group theory
 A group in mathematics consists of a set of members
which obey the following four rules
 Rule 1 The combination of any two members A
and B result in C ,which also belong to same group .
 AB = C
 Example water [ E, C2 , , ]

vv

 vvC  2

 vvC  2 vvC  
2

52. Group theory
 The relationships among the symmetry elements can
be treated very elegantly in terms of group theory
 A group in mathematics consists of a set of members
which obey the following four rules
 Rule 1 The combination of any two members A
and B result in C ,which also belong to same group .
 AB = C
 Example water [ E, C2 , , ]

vv

 vvC  2

 vvC  2 vvC  
2

53. Commutative
 There is two possibilities AB=BA AB BA

54. Commutative
 There is two possibilities AB=BA AB BA
Commutative Noncommutative


55. Commutative
 There is two possibilities AB=BA AB BA
Commutative Noncommutative
Abelian group Nonabelian
group


56. Commutative
 There is two possibilities AB=BA AB BA
Commutative Noncommutative
Abelian group Nonabelian
group
EXAMPLE H2O NH3


57. H2O
 Example C2 = C2
v
v v
1 2
1 2
abelian

58. H2O
 Example C2 = C2
v
v v
1 12
1
2
2
abelian

59. H2O
 Example C2 = C2
v
v v
v
1 1 1
2
1
2 2
2
abelian

60. H2O
 Example C2 = C2
v
v v
v
1 1 1
2
1 1
2 2
2 2
abelian

61. H2O
 Example C2 = C2
v
v v
v
1 1 1
1
2
1 1
2 2
2
2 2
same
abelian

62. Ammonia
 . Pass thru H1 and thru H2vvvv  

H1
v 
v
H2H3
Nonabelian

63. Ammonia
 . Pass thru H1 and thru H2vvvv  

H1
v 
v
H2H3
v
Nonabelian

64. Ammonia
 . Pass thru H1 and thru H2vvvv  

H1 H1
v 
v
H2 H2H3 H3
v
Nonabelian

65. Ammonia
 . Pass thru H1 and thru H2vvvv  

H1 H1
v 
v
H2 H2H3 H3
v 
v
Nonabelian

66. Ammonia
 . Pass thru H1 and thru H2vvvv  

H1 H1
H1
v 
v
H2 H2 H2H3 H3
H3
v 
v
Nonabelian

67. Ammonia
 . Pass thru H1 and thru H2vvvv  

H1 H1
H1
H1
v 
v
H2 H2 H2
H2
H3
H3
H3
H3
v 
v
Nonabelian

68. Ammonia
 . Pass thru H1 and thru H2vvvv  

H1 H1
H1
H1
v 
v
H2 H2 H2
H2
H3
H3
H3
H3
v 
v

v
Nonabelian

69. Ammonia
 . Pass thru H1 and thru H2vvvv  

H1 H1
H1
H1
H2
v 
v
H2 H2 H2
H2 H1
H3
H3
H3
H3
H3
v 
v

v
Nonabelian

70. Ammonia
 . Pass thru H1 and thru H2vvvv  

H1 H1
H1
H1
H2 H3
v 
v
H2 H2 H2
H2 H1
H2
H3
H3
H3
H1
H3
H3
v 
v

v
v
Nonabelian

71. Ammonia
 . Pass thru H1 and thru H2vvvv  

H1 H1
H1
H1
H2 H3
v 
v
H2 H2 H2
H2 H1
H2
H3
H3
H3
H1
H3
H3
v 
v

v
v
Not
same
Nonabelian

72. Rule 2
 There must be a member E in group such that
 Example in H2O EC2=C2E=C2
Rule 3 When more then two member
combine they do in associative manner
AE=EA=A ,BE=EB =B
Identity
(AB)C=A(BC)

73. Rule 4
 Every member A must have inverse which is also
member of group
Example in H2O
is anti clock wise rotation and C2 is clock wise
rotation and cancel each other
EAAAA   11
1
A
ECC 
2
1
2
1
2

C

74. Symmetry Points group
 The group of symmetry operation of molecule is called
point group
 several molecule may have the same set of opeation
 For example following molecule have same set of
operation i.e E,C2, , hence belong to same
point group
v 
v
H
O
O
H
C
HH
O
Cl Cl

75.  The symbols for most of the point group can be based of
principle axis of rotation as Cnv,Cnh,Dnh,Dnd
 If the Cn principal axis is accompanied by nC2 right angle to
it the latter D is used
 If a plane is present the symbol h is used (Cnh,Dnh)
If n plane are present with out the symbol used is Cnv
If n are present than Dnd when is absent
If no symmetry plane is present than drop small latter
Dihedral group
d
hv
h
h

76.  If there is S2n collinear with Cn and there is no
the symbol user is S2n
 Using Principal axis
d
No symmetry
element except
E i
C1 Cs Ci

77.  Linear molicule

vC hD
But no present
v h h
i apsent i present
C OS CO O

84. Multiplication Table
 The elements of group can be listed in a table known
as multiplication table
 This reveals the different relation between the
element of a group
 Table for water


v
v
1
2
v
1
22 CE



C
E
C vv

85. 
 Any thing multiply by E is equal to itself

 E(anything)=anything



vv
vv
1
2
1
2
vv
1
2
v
1
22
C
CE
CE




C
E
C vv

86.  We know that C2 C2 = E
Evv 
Evv 

E
E
EC
CE
CE
vv
vv
1
2
1
2
vv
1
2
v
1
22







C
E
C vv

87.  We know that C2 C2 = E
Evv 
Evv 

E
E
EC
CE
CE
vv
vv
1
2
1
2
vv
1
2
v
1
22







C
E
C vv

88.  We know that C2 =

 vv C  2
E
E
EC
CE
CE
vvv
vv
v
1
2
1
2
vv
1
2
v
1
22










v
v
vv
C
E
C
v 
v

89.  We know that = C2
EC
CE
EC
CE
CE
1
2vvv
1
2vv
v
1
2
1
2
vv
1
2
v
1
22










v
v
vv
C
E
C
v
v

90. For ammonia
ECC
CEC
CCE
CEC
ECCC
CCE
CCE
2
33vvv
3
2
3vvvv
2
3
1
3vvv
vv
1
3
3
3
2
3
vvv
2
3
1
3
1
3
vvv
2
3
1
3
v
2
3
1
33














v
v
v
vvv
C
E
C

91. Similarity of symmetry
operations
 It can be said on the basis of common sense that in the
point group the C3v operation C3 and similar
,thought not identical ,as are
 Similar member of group obey following rule
 Where R is any member of group including A and B
 This rule is called rule of similarity transformation
2
3C

vvv  ,,
BRRA -1


92.  For example in C3v
 Theoretical language they belong to same class
likewise belong to same class
2
3
1
3v
v
-1
v
1
3
A
RthenR,CB
CC vvv 




similarareC2
3
1
3andC

vvv  ,,

93. Classes of some point group


vvvh
vvv
h
SSCCCC
C
i



,,,,,,,,,CE,:D
,,CE,:
,,CE,:C
,
5
33222
2
333h
22
22h

94. Classes of some point group


vvvh
vvv
h
SSCCCC
C
i



,,,,,,,,,CE,:D
,,CE,:
,,CE,:C
,
5
33222
2
333h
22
22h
4 Classes

95. Classes of some point group


vvvh
vvv
h
SSCCCC
C
i



,,,,,,,,,CE,:D
,,CE,:
,,CE,:C
,
5
33222
2
333h
22
22h 4 Classes
4 Classes

96. Classes of some point group


vvvh
vvv
h
SSCCCC
C
i



,,,,,,,,,CE,:D
,,CE,:
,,CE,:C
,
5
33222
2
333h
22
22h 4 Classes
4 Classes
6 Classes

97. Representation of groups
 Each symmetry operation in the point group can be
represented by number or more generally , by matrix
of numbers

98. Representation of groups
 Each symmetry operation in the point group can be
represented by number or more generally , by matrix
of numbers
 A set of numbers will be true representation of group
all the operation over symmetry element are satisfy by
these number
 For example C2h group


99. Representation of groups
 Each symmetry operation in the point group can be
represented by number or more generally , by matrix
of numbers
 A set of numbers will be true representation of group
all the operation over symmetry element are satisfy by
these number
 For example C2h group

iCE h2 

100. Representation of groups
 Each symmetry operation in the point group can be
represented by number or more generally , by matrix
of numbers
 A set of numbers will be true representation of group
all the operation over symmetry element are satisfy by
these number
 For example C2h group

iCE h2 
i2C
This operation should
be satisfy in C2h

101. Trivial representation
 Assigning a number 1 or unit matrix of any dimantion
to each operation

102. Trivial representation
 Assigning a number 1 or unit matrix of any dimantoin
to each operation
 For example C2h point group
:
:
iCE:C
1
1
h22h





103. Trivial representation
 Assigning a number 1 or unit matrix of any dimantoin
to each operation
 For example C2h point group
:
1111:
iCE:C
1
1
h22h





104. Trivial representation
 Assigning a number 1 or unit matrix of any dimantoin
to each operation
 For example C2h point group



























10
01
10
01
10
01
10
01
:
1111:
iCE:C
1
1
h22h 

105. Trivial representation
 Assigning a number 1 or unit matrix of any dimantoin
to each operation
 For example C2h point group
 It satisfy



























10
01
10
01
10
01
10
01
:
1111:
iCE:C
1
1
h22h 
iC2 h

106. Trivial representation
 Assigning a number 1 or unit matrix of any dimantoin
to each operation
 For example C2h point group
 It satisfy



























10
01
10
01
10
01
10
01
:
1111:
iCE:C
1
1
h22h 



















10
01
10
01
10
01
111soiC2 h

107. Non trivial representation
 There is non trivial representation too

108. Non trivial representation
 There is non trivial representation too
 It satisfy
:
:
iCE:C
2
2
h22h




iC2 h

109. Non trivial representation
 There is non trivial representation too
 It satisfy
:
11-1-1:
iCE:C
2
2
h22h




iC2 h

110. Non trivial representation
 There is non trivial representation too
 It satisfy



























1-0
01-
1-0
01
1-0
01-
10
01
:
11-1-1:
iCE:C
2
2
h22h 
iC2 h

111. Non trivial representation
 There is non trivial representation too
 It satisfy



























1-0
01-
1-0
01
1-0
01-
10
01
:
11-1-1:
iCE:C
2
2
h22h 



















1-0
01-
10
01-
10
01-
1-11-so1C2 h

112.  Other two possible representation are
1-1-11:
1-11-1:
4
3


iC2 h

113.  Other two possible representation are
 These representation in foam of table for C2h point
group
1-1-11:
1-11-1:
4
3



114.  Other two possible representation are
 These representation in foam of table for C2h point
group
1-1-11:
1-11-1:
4
3


1-11-1
1-1-11
11-1-1
1111
iCEC
4
3
2
1
h22h






115. Bases

116. Bases
Cartesian
coordinates
x,y,z

117. Bases
Cartesian
coordinates
Wave
function
x,y,z px,py,pz,dxy,
dyz, etc

118. Bases
Cartesian
coordinates
Wave
function
Rotational
coordinates
x,y,z px,py,pz,dxy,
dyz, etc
Rx,Ry,Rz

119. Cartesian coordinate
 Consider a molecule belonging to C2h point group

120. Cartesian coordinate
 Consider a molecule belonging to C2h point group
 Concentrate on the effect of four symmetry operation
on coordinate

121. Cartesian coordinate
 Consider a molecule belonging to C2h point group
 Concentrate on the effect of four symmetry operation
on coordinate
 Let z axis be C2 axis

122. Cartesian coordinate
 Consider a molecule belonging to C2h point group
 Concentrate on the effect of four symmetry operation
on coordinate
 Let z axis be C2 axis
C2

123. Cartesian coordinate
 Consider a molecule belonging to C2h point group
 Concentrate on the effect of four symmetry operation
on coordinate
 Let z axis be C2 axis
C2
+Z
-X
+X
+Y
-Z
-Y

124. Cartesian coordinate
 The xy plane is molecular plane

125. Cartesian coordinate
 The xy plane is molecular plane
h

126. Cartesian coordinate
 The xy plane is molecular plane
 When we apply the operation of molecular plane there
only change in z axis x and y remain same as shoan
-Z
+X
-X
-Y
+Z
+Y
h

127. Cartesian coordinate
 Now we are appalling operation i
i

128. Cartesian coordinate
 Now we are appalling operation i
 When we apply the operation i there is change in all 3
axis as shon
+Z
-X
+X
+Y
-Z
-Y
i

129.  Effect on z
 i(z),)(,)(C,E(z) h2 zz 

130.  Effect on z
-zi(z),)(,)(Cz,E(z) h2  zzzz 

131.  Effect on z
 This generate following representation
-zi(z),)(,)(Cz,E(z) h2  zzzz 
(z)
BasesiCE h2 

132.  Effect on z
 This generate following representation
-zi(z),)(,)(Cz,E(z) h2  zzzz 
(z)1-1-11
BasesiCE h2 

133.  Effect on x and y
 i(x),)(,)(C,E(x) h2 xx 
 i(y),)(,)(C,E(y) h2 yy 

134.  Effect on x and y
-xi(x),)(,)(Cx,E(x) h2  xxxx 
 i(y),)(,)(C,E(y) h2 yy 

135.  Effect on x and y
-xi(x),)(,)(Cx,E(x) h2  xxxx 
-yi(y),)(,)(Cx,E(y) h2  yyyy 

136.  Effect on x and y
 This generate following representation
-xi(x),)(,)(Cx,E(x) h2  xxxx 
y)or(x
BasesiCE h2 
-yi(y),)(,)(Cx,E(y) h2  yyyy 

137.  Effect on x and y
 This generate following representation
-xi(x),)(,)(Cx,E(x) h2  xxxx 
y)or(x1-11-1
BasesiCE h2 
-yi(y),)(,)(Cx,E(y) h2  yyyy 

138. Wave function as Bases
 S orbital is spherical symmetrical i.e totally symmetric
 It is un effected by any operation

139. Wave function as Bases
 S orbital is spherical symmetrical i.e totally symmetric
 It is un effected by any operation
(s)
BasesiCEC h22h 

140. Wave function as Bases
 S orbital is spherical symmetrical i.e totally symmetric
 It is un effected by any operation
(s)1111
BasesiCEC h22h 

141. Wave function as Bases
 S orbital is spherical symmetrical i.e totally symmetric
 It is un effected by any operation
 px,py,pz orbital just as x,y,z
(s)1111
BasesiCEC h22h 

142. Wave function as Bases
 S orbital is spherical symmetrical i.e totally symmetric
 It is un effected by any operation
 px,py,pz orbital just as x,y,z
)(p1-1-11
)por(p1-11-1
BasesiCEC
z
yx
h22h 
(s)1111
BasesiCEC h22h 

143. Exercise
 Use the 3d orbitals as bases to generate representation
of the C2h point group
yzxz
xy
h22h
d,d
d
BasesiCEC 

144. Exercise
 Use the 3d orbitals as bases to generate representation
of the C2h point grop
yzxz
xy
h22h
d,d
d1111
BasesiCEC 

145. Exercise
 Use the 3d orbitals as bases to generate representation
of the C2h point group
yzxz
xy
h22h
d,d11-1-1
d1111
BasesiCEC 

146. Rotational coordinates as bases
 Effect of symmetry operation on the rotation at axis
can represented by using rotational coordinates as
bases

147. Rotational coordinates as bases
 Effect of symmetry operation on the rotation at axis
can represented by using rotational coordinates as
bases
 The effect of E,C2, and i on Rx, Ry and Rz can be
seen as follows
 )R(,)(R,)R(C,)E(R xxx2x ih
v

148. Rotational coordinates as bases
 Effect of symmetry operation on the rotation at axis
can represented by using rotational coordinates as
bases
 The effect of E,C2, and i on Rx, Ry and Rz can be
seen as follows
xxxxxx2x R)R(,-R)(R,-R)R(C,R)E(R  ihx 
v

149. Rotational coordinates as bases
 Effect of symmetry operation on the rotation at axis
can represented by using rotational coordinates as
bases
 The effect of E,C2, and on Rx, Ry and Rz can be
seen as follows


)R(,)R(,)R(C)E(R
R)R(,-R)(R,-R)R(C,R)E(R
yyy2,Y
xxxxxx2x
i
i
h
hx


v

150. Rotational coordinates as bases
 Effect of symmetry operation on the rotation at axis
can represented by using rotational coordinates as
bases
 The effect of E,C2, and i on Rx, Ry and Rz can be
seen as follows
Yyyyyy2y,Y
xxxxxx2x
R)R(,-R)R(,-R)R(CR)E(R
R)R(,-R)(R,-R)R(C,R)E(R


i
i
h
hx


v

151. Rotational coordinates as bases
 Effect of symmetry operation on the rotation at axis
can represented by using rotational coordinates as
bases
 The effect of E,C2, and i on Rx, Ry and Rz can be
seen as follows



)(,)R(,)R(C,)E(R
R)R(,-R)R(,-R)R(CR)E(R
R)R(,-R)(R,-R)R(C,R)E(R
zhz2Z
Yyyyyy2y,Y
xxxxxx2x
z
h
hx
Ri
i
i



v

152. Rotational coordinates as bases
 Effect of symmetry operation on the rotation at axis
can represented by using rotational coordinates as
bases
 The effect of E,C2, and i on Rx, Ry and Rz can be
seen as follows
Zzzhzz2zZ
Yyyyyy2y,Y
xxxxxx2x
R)(,R)R(,R)R(C,R)E(R
R)R(,-R)R(,-R)R(CR)E(R
R)R(,-R)(R,-R)R(C,R)E(R



z
h
hx
Ri
i
i



v

153. Rotational coordinates as bases
 The following representation are genrated
)R,(R11-1-1
)(R1111
BasesiCEC
yx
z
h22h 

154. Matrix representation
 Matrix for reflection operation
 When xy is plane of reflection

155. Matrix representation
 Matrix for reflection operation
 When xy is plane of reflection











1-00
010
001
xy

156. Matrix representation
 Matrix for reflection operation
 When xy is plane of reflection
When yz or zx is plane of reflection











1-00
010
001
xy

157. Matrix representation
 Matrix for reflection operation
 When xy is plane of reflection
When yz or zx is plane of reflection











1-00
010
001
xy











100
010
001-
yz

158. Matrix representation
 Matrix for reflection operation
 When xy is plane of reflection
When yz or zx is plane of reflection











1-00
010
001
xy











100
010
001-
yz











100
01-0
001
zx

159. Matrix representation
 Matrix for inversion operation
 After inversion operation x,y and z transform in to -x,-
y,-z

160. Matrix representation
 Matrix for inversion operation
 After inversion operation x,y and z transform in to -x,-
y,-z
I rr 

161. Matrix representation
 Matrix for inversion operation
 After inversion operation x,y and z transform in to -x,-
y,-z
1-00
01-0
001-
I











rr

162. Matrix representation
 Matrix for rotation operation
cossin-
sincos









rCr n

163. Matrix representation
 Matrix for rotation operation
 For three dimension z is rotation axis
cossin-
sincos









rCr n











100
0cossin-
0sincos
)( 

zCn

164. Matrix representation
 Matrix for improper rotation
hCS 
Z is rotation axis
so plane is xy

165. Matrix representation
 Matrix for improper rotation
1-00
010
001
100
0cossin-
0sincos
CS h


























S
Z is rotation axis
so plane is xy

166. Matrix representation
 Matrix for improper rotation


































1-00
0cossin-
0sincos
1-00
010
001
100
0cossin-
0sincos
CS h





S
Z is rotation axis
so plane is xy

167. Matrix representation
 Matrix representation for identity element











100
010
001
E

168. Matrix representation
 Matrix representation for identity element
Trace The sum of the diagonal matrix is called trace
or character of matrix











100
010
001
E











100
010
001
E

169. Matrix representation
 Matrix representation for identity element
Trace The sum of the diagonal matrix is called trace
or character of matrix











100
010
001
E











100
010
001
E
Trace =3

170. Reducible representation
 The matrix representation which can be block
dignailised i.e reducing the dimension of matrix
 example
55
30000
07200
06500
00010
00001


















171. Reducible representation
 The matrix representation which can be block
dignailised i.e reducing the dimension of matrix
 example
55
30000
07200
06500
00010
00001


















172.  Matrix representation for H2O is
 So its 3D representation is
Reducible representation








































100
010
001-
100
01-0
001
100
01-0
001-
100
010
001
)()(C v2 yzxzE v
zandyx,
BasesCEC vv22v



173.  Matrix representation for H2O is
 So its 3D representation is
Reducible representation








































100
010
001-
100
01-0
001
100
01-0
001-
100
010
001
)()(C v2 yzxzE v
zandyx,
BasesCEC vv22v



174.  Matrix representation for H2O is
 So its 3D representation is
Reducible representation








































100
010
001-
100
01-0
001
100
01-0
001-
100
010
001
)()(C v2 yzxzE v
zandyx,111-3
BasesCEC vv22v



175.  Matrix representation for H2O is
 But we can reduce the dimension of these matrix so it
can reduced in to three 1 dimension representation
Reducible representation








































100
010
001-
100
01-0
001
100
01-0
001-
100
010
001
)()(C v2 yzxzE v

176.  Matrix representation for H2O is
 First representation
Reducible representation








































100
010
001-
100
01-0
001
100
01-0
001-
100
010
001
)()(C v2 yzxzE v
BasesCEC vv22v



177.  Matrix representation for H2O is
 First representation
Reducible representation








































100
010
001-
100
01-0
001
100
01-0
001-
100
010
001
)()(C v2 yzxzE v
x1-11-1
BasesCEC vv22v



178.  Matrix representation for H2O is
 second representation
Reducible representation








































100
010
001-
100
01-0
001
100
01-0
001-
100
010
001
)()(C v2 yzxzE v
y11-1-1
BasesCEC vv22v



179.  Matrix representation for H2O is
 third representation
Reducible representation








































100
010
001-
100
01-0
001
100
01-0
001-
100
010
001
)()(C v2 yzxzE v
z1111
BasesCEC vv22v



182. The Grand orthogonality theorem
 nnmmij
ji
mnjmni
R ll
h
RR )()(
jiif0
jiif1

ij




mmif0
mmif1mm





nnif0
nnif1nn
