1. Lecture 3

3. Principal rotation axis?

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

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

6. 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 is Dnh and C is Cnh.
D3h D? D?
C? or S4 D?
C2h
C? or S6
C? or S4
C? or S4

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

8. No sh. Any plane containing Cn?
D is Dnh if present; Dn if not. If present (C or S) is Cnv.
Dn or Dnd
Dnd
Cn, Cnv, or S2n
Dn or Dnd D3
Cn, Cnv, or S2n
C3v
Cn, Cnv, or S2n

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

10. 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

11. 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, if present
S2n axis, if present.
If above criteria not met
Dnh
Dnd 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.

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

13. The point groups of common molecular shapesTd
D4h
C2v

14. The point groups of common molecular shapes
C4v
D3h

15. The point groups of common molecular shapesOh
D4h
D3d

16. 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

17. 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

18. 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):

19. 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.

20. 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.

21. 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.

22. 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.

23. 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.

24. 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

25. Atomic Orbitals 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

26. Different ways that objects can behave for a group consisting of E and the reflection plane.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”.

27. 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.

28. 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 h1; h2 becomes h2. = =
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

29. 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 representation derived from the two hybrids can be attached to the table. G
(h1, h2)
Note that G = A’ + A”

30. 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

31. 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

32. 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