1. Lecture 5

2. See: Chemical Applications of Group Theory by F. A. Cotton
Symmetry point groups
The set of all possible symmetry operations on a molecule
is called the point group (there are 28 point groups)
The mathematical treatment of the properties of groups
is Group Theory
In chemistry, group theory allows the assignment of structures,
the definition of orbitals, analysis of vibrations, ...
See: Chemical Applications of Group Theory by F. A. Cotton 2

3. To determine
the point group
of a molecule 3

4. 4

5. Groups of low symmetry 5

6. 6

8. Principal rotation axis?

10. Any C2 axes perpendicular to principal axis?
D Groups
C or S Groups

11. How many C2 axis perpendicular to principal axis, Cn?
There are n C2 axes !!

12. sh ? If yes then done!

13. D3h D? D? C? or S4 D? C2h C? or S6 C? or S4 C? or S4
Is there a horizontal plane, sh , perpendicular to principal axis, Cn?
If so, then can assign D  Dnh and C  Cnh.
D3h D? D?
C? or S4 D?
Recall classified as D vs C/S on basis of perpendicular C2
C2h
C? or S6
C? or S4
C? or S4

14. Recapitulation
C3, 3C2 , sh
C2, no C2 , sh
D3h
C2h

15. s containing the principal axis?

16. We have no sh. Next question: any plane containing Cn?
D  Dnh if present; D  Dn if not.
(C or S)  Cnv. if present
Dn or Dnd
Dnd
Cn, Cnv, or S2n
Dn or Dnd D3
Cn, Cnv, or S2n
C3v
Cn, Cnv, or S2n

17. D3 Point group? C3, C2 , no sh, no sv C3, no C2 , no sh, 3sv
C3v
C3, C2 , no sh, 3sv
D3d

18. S2n ?

19. Cn or S2n? If there is an S2n axis colinear with Cn
then S2n otherwise Cn.
C2, no C2 , no sh, no sv,
S4 colinear with C2 S4
C2, no C2 , no sh, no sv,
no S2n colinear with C2 C2

20. D classifications
C classifications
General Case: Look for nC2 axes perpendicular to the highest order Cn axis
Dnh or Dnd or Dn
Cnh or Cnv or S2n or Cn
Sub categories
horizontal plane of symmetry perpendicular to Cn if present.
n vertical planes, containing principal axis, if present
S2n axis, if present.
If above criteria not met
Dnh
Dnd NA Dn
Cnh
Cnv
S2n Cn
Notes
Vertical planes contain the highest order Cn axis. In the Dnd group the planes are designated dihedral as they lie between the C2 axes.
Simply having a Cn axis does not guarantee that a molecule will be D or C. Other possigilities include Td, Oh, Ih, and related groups.
If doubts persist consult the character tables.

21. The point groups of common molecular shapes
D∞h
C2v
D3h
C2v
C3v

22. The point groups of common molecular shapesTd
D4h
C2v

23. The point groups of common molecular shapes
C4v
D3h

24. The point groups of common molecular shapesOh
D4h
D3d

25. Character Tables Group theory makes use of the properties of matrices
Each symmetry operation may be expressed as a transformation matrix:
[New coordinates] = [transformation matrix][old coordinates]
Example: in Cartesian coordinate system, reflection in x = 0 plane
Changes the value of x to –x, multiplies it by -1
Leaves y unchanged
Leaves z unchanged =
Results of transformation.
Transformation matrix
Original coordinates

26. Matrix multiplication= V’ M V
To get an element of the product vector a row in the operation square matrix is multiplied by the original vector matrix.
For example
V’2 = y’ =
M2,1 * V1
+ M2,2 * V2
+ M2,3 * V3
y’ = 0 * x * y * z = y

27. Character Tables - 2
The matrix representation of the symmetry operations of a point group is
the set of matrices corresponding to all the symmetry operations in that group. The matrices record how the x,y,z coordinates are modified as a result of an operation.
For example, the C2v point group consists of the following operations
E: do nothing. Unchanged.
C2: rotate 180 degrees about the z axis: x becomes –x; y becomes –y and z unchanged.
sv (xz): y becomes –y
sv’ (yz): x becomes -x E C2
sv (xz):
sv’ (yz):

28. Operations Applied to Functions - 1
Consider
f(x) = x2
sv’ (f(x)) = sv(x2) = (-x)2 = x2 = f(x) or
sv’ (f(x)) = 1 * f(x)
f(x) is an eigenfunction of this reflection operator with an eigenvalue of +1. This is called a symmetric eigenfunction.
Similarly
f(x) = x3
sv’ (f(x)) = -1 * f(x)
f(x) is an eigenfunction of this reflection operator with an eigenvalue of -1. This is called a antisymmetric eigenfunction.

29. Plots of Functions Reflection yields.
Here f(x) is x2. It can be seen to be a symmetric function for reflection at x = 0 because of mirror plane.
The reflection carries out the mapping shown with the red arrows.
X2 is an eigenfunction of s with eigenvalue 1

30. Plots of Functions - 2 Reflection yields.
Here f(x) is x3. It can be seen to be a antisymmetric function for reflection at x = 0.
The reflection carries out the mapping shown with the red arrows.
X3 is an eigenfunction of s with eigenvalue -1

31. Plots of Functions - 2 Reflection yields.
Here f(x) is x3. It can be seen to be a antisymmetric function for reflection at x = 0.
The reflection carries out the mapping shown with the red arrows.

32. Operations Applied to Functions - 2
Now consider
f(x) = (x-2)2 = x2 – 4x + 4
sv’ (f(x)) = sv(x-2)2 = (-x-2)2 = x2 + 4x + 4
f(x) = (x-2)2 is not an eigenfunction of this reflection operator because it does not return a constant times f(x).
Reflection yields this function, not an eigenfunction.
Neither symmetric nor antisymmetric for reflection thru x = 0.

33. Atomic Orbitals Reflection
Get the same orbital back, multiplied by +1, an eigenfunction of the reflection, symmetric with respect to the reflection. The s orbital forms the basis of an irreducible representation of the operation
s orbital z

34. Atomic Orbitals s Reflection
Get the same orbital back, multiplied by -1, an eigenfunction of the reflection, antisymmetric with respect to the reflection. The p orbital behaves differently from the s orbital and forms the basis of a different irreducible representation of the operation
p orbital z

35. Different ways that objects can behave for a group consisting of E and the reflection plane.
Basis of Irreducible Representations. Cs
E sh A’ A” 1
x, y,Rz
z, Rx,Ry
x2,y2,z2,xy
yz, xz
s orbital is spherical behaves as x2 + y2 + z2. s orbital is A’
pz orbital has a multiplicative factor of z times a spherical factor. Behaves as A”.

36. sp Hybrids Reflection Do not get the same hybrid back. hybrid
The two hybrids form the basis of a reducible representation of the operation z
Recall: the hybrid can be expressed as the sum of an s orbital and a p orbital. = +
Reduction: expressing a reducible representation as a combination of irreducible representations.

37. Reducible Representations
Use the two spz hybrids as the basis of a representation h1 h2
sh operation.
E operation.
h1 becomes h1; h2 becomes h2.
h1 becomes h2; h2 becomes h1. = =
The reflection operation interchanges the two hybrids.
The hybrids are unaffected by the E operation.
Proceed using the trace of the matrix representation.
= 0
= 2

38. Representations Cs E sh A’ A” 1 1 -1 x, y,Rz z, Rx,Ry x2,y2,z2,xy
x, y,Rz
z, Rx,Ry
x2,y2,z2,xy
yz, xz
The reducible representation derived from the two hybrids can be attached to the table. G
(h1, h2)
Note that G = A’ + A”

39. Return to polynomials:
f(x) = (x-2)2 = x2 – 4x + 4
sv (f(x)) = sv(x-2)2 = (-x-2)2 = x2 + 4x + 4 =g(x)
Neither f nor g is an eigenfunction of s but, taken together, they do form an irreducible representation since they show what the s operator does.
Both x2 and 4 are unchanged by the s operator.
sx2 = x2 and s4 = Eigenfunctions with eigenvalue +1
4x becomes multiplied by -1 as a result of the reflection
sx = -x eigenvalue =1
We can reduce the reducible representation to a combination of irreducible reps.
f(x) = x2 + 4 – 4x
g(x) = x x
F and g are reducible,
x2,4 are irreducible of sym A,
x is irreducible of sym A’

40. Character Tables - 3
Irreducible representations are not linear combinations of other representation
(Reducible representations are)
# of irreducible representations = # of classes of symmetry operations
Instead of the matrices, the characters are used (traces of matrices)
A character Table is the complete set of irreducible representations of a point group

41. Each row is an irreducible representation
Character Table
Symmetry operations
Point group
Characters
+1 symmetric behavior
-1 antisymmetric
Mülliken symbols
Each row is an irreducible representation

42. Symmetry of translations (p orbitals)
Classes of operations
x, y, z
Symmetry of translations (p orbitals)
Rx, Ry, Rz: rotations
dxy, dxz, dyz, as xy, xz, yz
dx2- y2 behaves as x2 – y2
dz2 behaves as 2z2 - (x2 + y2)
px, py, pz behave as x, y, z
s behaves as x2 + y2 + z2