Symmetry and group theory representation of groups

The main point is the presentation about the vibration symmetry group theory

1. Symmetry and Group Theory
Representations of groups.

2. Matrix Representations of Groups
•Diagrams are cumbersome
•Require numerical method
–Allows mathematical analysis
–Represent using vectors or mathematical functions
–Use position vectors or attach Cartesian vectors to molecule or atoms of
a molecule
–Observe the effect of symmetry operations on these vectors
•Vectors are said to form the basis of the representation
•Effects of symmetry operations generate the
TRANSFORMATION MATRIX
–[new coordinates]=[transformation matrix][old coordinates]

3. Transformation matrices

7. 8
For the specific case of ammonia, C
3:
Consider (anti-clockwise) rotation of T
y and T
x around z:

8. Matrices for symmetry operations

9. Character Tables
Character tables list all of the symmetry elements of the
group, along with a complete set of irreducible
representations.

10. Character Tables for Point Groups
C
2V E C
2 
v (xz)’
v (yz)
A
1 1 1 1 1
A
2 1 1 -1 -1
B
1 1 -1 1 -1
B
2 1 -1 -1 1
Irreducible representation
Point Group LabelSymmetry Operations
Symmetry Representation Labels
(Mulliken symbols)
The Order is the total number of operations
In C
2v the order is 4: 1 E, 1 C
2, 1 
v and 1 ’
v
z x
2
,y
2
,z
2
R
z xy
x, R
y xz
y, R
x yz
Character
Linear and
rotational
functions
Symmetry of Functions
Square and
binary
functions

11. C
2V E C
2
v (xz)’
v (yz)
A
1 1 1 1 1
A
2 1 1 -1 -1
B
1 1 -1 1 -1
B
2 1 -1 -1 1
Dimensionality of
the representation
“1” = “symmetric”.
“-1” = “anti-symmetric”.
There are infinite number of
representations of the C
2v group.
However, these 4 reps are the only
irreducible representations.
All other reps are comprised of these
four and are called Reducible
representations
Totally symmetric representation
A
1 B
1

12. C
2V E C
2
v (xz)’
v (yz)
A
1 1 1 1 1
A
2 1 1 -1 -1
B
1 1 -1 1 -1
B
2 1 -1 -1 1
1)1-D (E=1) Representations:
A: Symmetric to highest order C
n
B: Antisymmetric to highest order C
n
2-D (E=2) Representations: E
3-D (E=3) Representations: T
Mulliken symbols (for non-Linear Groups)
2) Groups with Inversion (i):
Symmetric: subscript g
Antisymmetric: subscript u
3) Non-principal C
n or σ
v:
Symmetric: subscript 1
Antisymmetric: subscript 2
4) If needed, groups σ
h:
Symmetric: ‘
Antisymmetric: ‘’

13. Mulliken symbols (for non-Linear Groups)

14. Character Tables for Point Groups
The effect of symmetry elements on mathematical functions is useful to
us because orbitals are mathematical functions! Analysis of the
symmetry of a molecule will provide us with insight into the orbitals
used in bonding.
Symmetry of Functions
C
2V E C
2 
v (xz)’
v (yz)
A
1 1 1 1 1 z x
2
,y
2
,z
2
A
2 1 1 -1 -1 R
z xy
B
1 1 -1 1 -1 x, R
y xz
B
2 1 -1 -1 1 y, R
x yz
The functions to the right are called basis functions. They represent
mathematical functions such as orbitals, rotations, etc.

15. 3
Consider the C
2v point group
How do the s and p orbitals transform under the
C2v symmetry operations
Orbital EC
2

v (xz)’
v (yz)
2s, 2p
z11 1 1
2p
x 1-1 1 -1
2p
y 1-1-1 1

16. C
2V E C
2 
v (xz)’
v (yz)Orbital
A
1 1 1 1 1 2s, 2p
z
B
1 1 -1 1 -1 2p
x
B
2 1 -1 -1 1 2p
y
“1” = “symmetric”.
“-1” = “anti-symmetric”.
The Characters give numerical factors relating each transformed orbital to its
original form
A
1 B
1
Consider the C
2v point group

17. y
x
d orbital functions can also be treated in a similar way (e.g., d
xy)
E
C
2
No change
 symmetric
 1’s in table
y
x
y
x

v (xz)
’
v (yz)
Opposite
 anti-symmetric
 -1’s in table
y
x
Direct Products
The z axis is pointing
out of the screen!
C
2V E C
2 
v (xz)’
v (yz)Orbital
A
1 1 1 1 1 2s, 2p
z
A
2 1 1 -1 -1 3d
xy
B
1 1 -1 1 -1 2p
x
B
2 1 -1 -1 1 2p
y
A
2 = B
1 x B
2

18. C
2V 1 -1 1 -1
1 1 -1 1 -1
-1 -1 1 -1 1
1 1 -1 1 -1
-1 -1 1 -1 1
B
1
C
2VEC
2
v (xz)’
v (yz)
A
111 1 1
A
211 -1 -1
B
11-1 1 -1
B
21-1 -1 1

19. C
2V E C
2 
v (xz) ’
v (yz)
A
1 1 1 1 1 z x
2
,y
2
,z
2
A
2 1 1 -1 -1 R
z xy
B
1 1 -1 1 -1 x, R
y xz
B
2 1 -1 -1 1 y, R
x yz
’
v (yz)

E
z
y
x
C
2

v (xz)
z
y
x
z
y
x
z
y
x
1 1 1 1

z
y
x
z
y
x
z
y
x
z
y
x
1 -1 1 -1

z
y
x
z
y
x
z
y
x
z
y
x1 -1 -1 1

20. C
2V E C
2 
v (xz) ’
v (yz)
A
1 1 1 1 1 z x
2
,y
2
,z
2
A
2 1 1 -1 -1 R
z xy
B
1 1 -1 1 -1 x, R
y xz
B
2 1 -1 -1 1 y, R
x yz
A
2
B
2
B
1

21. C
2V E C
2 
v (xz)’
v (yz)
A
1 1 1 1 1 z x
2
,y
2
,z
2
A
2 1 1 -1 -1 R
z xy
B
1 1 -1 1 -1 x, R
y xz
B
2 1 -1 -1 1 y, R
x yz
Square and Binary Product Functions
x → B
1
z → A
1
xz → B
1
x → B
1
y → B
2
xy → A
2
Γ
(dxz) = 1 -1 1 -1 = B
1 Γ
(dxy) = 1 1 -1 -1 = A
2

22. C
2V E C
2 
v (xz)’
v (yz)
A
1 1 1 1 1 z x
2
,y
2
,z
2
A
2 1 1 -1 -1 R
z xy
B
1 1 -1 1 -1 x, R
y xz
B
2 1 -1 -1 1 y, R
x yz
y
x
d orbital functions can also be treated in a similar way
E
C
2
No change
 symmetric
 1’s in table
y
x
y
x

v (xz)
’
v (yz)
Opposite
 anti-symmetric
 -1’s in table
y
x
Symmetry of orbitals and functions
The z axis is pointing
out of the screen!

23. C
2V E C
2 
v (xz)’
v (yz)
A
1 1 1 1 1 z x
2
,y
2
,z
2
A
2 1 1 -1 -1 R
z xy
B
1 1 -1 1 -1 x, R
y xz
B
2 1 -1 -1 1 y, R
x yz
y
x
d orbital functions can also be treated in a similar way
E
C
2

v (xz)
’
v (yz)
No change
 symmetric
 1’s in table
y
x
Symmetry of orbitals and functions
The z axis is pointing
out of the screen!
So these are
representations of the
view of the d
z
2 orbital
and d
x
2
-y
2 orbital down
the z-axis.
y
x
y
x

24. C
2V E C
2 
v (xz)’
v (yz)
A
1 1 1 1 1 z x
2
,y
2
,z
2
A
2 1 1 -1 -1 R
z xy
B
1 1 -1 1 -1 x, R
y xz
B
2 1 -1 -1 1 y, R
x yz
Note that the representation of orbital functions changes depending on the point
group – thus it is important to be able to identify the point group correctly.
Symmetry of orbitals and functions
D
3h E2 C
33 C
2
h2 S
33 
v
A’
1 1 1 1 1 1 1 x
2
+ y
2
, z
2
A’
2 1 1 -1 1 1 -1 R
z
E’ 2-1 0 2-1 0 (x,y)(x
2
- y
2
, xy)
A’’
1 1 1 1 -1-1 -1
A’’
2 1 1 -1 -1-1 1 z
E’’ 2-1 0 -21 0(R
x, R
y)(xz, yz)

25. cos(120°) = -0.5
sin(120°) = 0.87
Symmetry of orbitals and functions
D
3h E2 C
33 C
2
h2 S
33 
v
A’
1 1 1 1 1 1 1 x
2
+ y
2
, z
2
A’
2 1 1 -1 1 1 -1 R
z
E’ 2-1 0 2-1 0 (x,y)(x
2
- y
2
, xy)
A’’
1 1 1 1 -1-1 -1
A’’
2 1 1 -1 -1-1 1 z
E’’ 2-1 0 -21 0(R
x, R
y)(xz, yz)
More notes about symmetry labels and characters:
-“E” indicates that the representation is doubly-degenerate – this means that the
functions grouped in parentheses must be treated as a pair and can not be
considered individually.
-The prime (‘) and (“) double prime in the symmetry representation label indicates
“symmetric” or “anti-symmetric” with respect to the 
h.
y
x
C
3
y
-0.5
-0.5
x
y
x
C
2 (x)
(+0.87) + (-0.87) + (-0.5) + (-0.5) = -1
1
-1
(-1) + (1) = 0
+0.87
-0.87

26. Symmetry of orbitals and functions
O
hE8 C
36 C
26 C
43 C
2
(C
4
2
)
i6 S
48 S
63 
h6 
d
A
1g11 1 1 1 1 1 1 1 1 x
2
+ y
2
+ z
2
A
2g11 -1 -1 1 1-1 1 1 -1
E
g2-1 0 0 2 2 0 -1 2 0 (2z
2
- x
2
- y
2
,
x
2
- y
2
)
T
1g30 -1 1 -1 3 1 0 -1 -1(R
x, R
y, R
z)
T
2g30 1 -1 -1 3-1 0 -1 1 (xz, yz, xy)
A
1u11 1 1 1 -1-1 -1 -1 -1
A
2u11 -1 -1 1 -11 -1 -1 1
E
u2-1 0 0 2 -20 1 -2 0
T
1u30 -1 1 -1 -3-1 0 1 1 (x, y, z)
T
2u30 1 -1 -1 -31 0 1 -1
More notes about symmetry labels and characters:
-“T” indicates that the representation is triply-degenerate – this means that the
functions grouped in parentheses must be treated as a threesome and can not be
considered individually.
-The subscripts g (gerade) and u (ungerade) in the symmetry representation label
indicates “symmetric” or “anti-symmetric” with respect to the inversion center, i.

27. Consider an element A of a group G
Classes of a Group
If we take another element of G, X and its inverse X
-1
, and perform the
following transformation:
X A X
-1
= B
We say that B is the similarity transform of A by X
Or the elements A and B are conjugate to one
another
The collection of all similarity transforms of A by all elements X form a
class
Or all conjugate elements of a group form a class of the group.

28. Example: Character Table of a C
3v Group

29. Determine C
3v Classes

30. Classes in Symmetry Point Groups
You can test all possible similarity transforms to find the conjugate elements using
X
-1
AX = B , however this is tedious
Use C
3v as a example

31. C
3V E C
3
1
C
3
2

v (1)
v (2)
v (3)
A
1 1 1 1 1 1 1 T
z
A
2 1 1 1 -1 -1 -1 R
z
E 2 -1 -1 0 0 0 (Tx, Ty)
(Rx,Ry)
C
3V E 2C
3 3
v
A
1 1 1 1 T
z
A
2 1 1 -1 R
z
E 2 -1 0 (Tx, Ty)
(Rx,Ry)
Example: Character Table of a C
3v Group
Classes of a Group

32. C
3V E C
3
1
C
3
2

v (1)
v (2)
v (3)
A
1 1 1 1 1 1 1 T
z
A
2 1 1 1 -1 -1 -1 R
z
E
The transition along, and the rotation about, z (T
z, R
z) generate
irreducible representations, as follows
Consider ammonia - the C
3v
point group