Lecture 3

Principal rotation axis?

Any C2 axes perpendicular to principal axis?

C or S Groups

D Groups

How many C2 axis perpendicular to principal axis, Cn?

There are n C2 axes !!

Is there a horizontal plane, h , perpendicular to principal axis, Cn? If so, then can assign D is Dnh and C is Cnh.

C? or S6

D? D? C? or S4 D?

C? or S4 C? or S4

D3h

C2h

Recapitulation

C3, 3C2 , h C2, no C2 , h

D3h C2h

No h. Any plane containing Cn? D is Dnh if present; Dn if not. If present (C or S) is Cnv.

Dn or Dnd Cn, Cnv, or S2n Dn or Dnd

Cn, Cnv, or S2nCn, Cnv, or S2n

Dnd

C3v

D3

Point group?

C3, C2 , no h, 3v

D3d

C3, no C2 , no h, 3v

C3v

C3, C2 , no h, no v

D3

C2, no C2 , no h, no v,S4 colinear with C2

S4

S4

C2, no C2 , no h, no v,no S2n colinear with C2

C2

Cn or S2n? If there is an S2n axis colinear with Cn then S2n otherwise Cn.

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

1. horizontal plane of symmetry perpendicular to Cn if present.

2. n vertical planes, if present

3. S2n axis, if present.

4. If above criteria not met

1. Dnh

2. Dnd

3. ----------

4. Dn

1. Cnh

2. Cnv

3. S2n

4. Cn

Notes

1. Vertical planes contain the highest order Cn axis. In the Dnd group the planes are designated dihedral as they lie between the C2 axes.

2. 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.

3. If doubts persist consult the character tables.

D∞h C2v

D3h C2vC3v

The point groups of common molecular shapes

Td

D4h

C2v

The point groups of common molecular shapes

C4v

D3h

The point groups of common molecular shapes

The point groups of common molecular shapes

Oh

D4h

D3d

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

100

010

001

'

'

'

z

y

x

z

y

x

Original coordinates

Transformation matrix

Results of transformation.

=

Matrix multiplication

To get an element of the product vector a row in the operation square matrix is multiplied by the original vector matrix.

100

010

001

'

'

'

z

y

x

z

y

x=

V’ M V

jj

jii VMV ,'

For example

V’2 = y’ = M2,1 * V1 + M2,2 * V2 + M2,3 * V3

y’ = 0 * x + 1 * y + 0 * z = y

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.

v (xz): y becomes –y

v’ (yz): x becomes -x

100

010

001

100

010

001

E

100

010

001

C2

100

010

001

v (xz): v’ (yz):

Operations Applied to Functions - 1

Consider

f(x) = x2

v’ (f(x)) = v(x2) = (-x)2 = x2 = f(x)

v’ (f(x)) = 1 * f(x)

or

f(x) is an eigenfunction of this reflection operator with an eigenvalue of +1. This is called a symmetric eigenfunction.Similarly

f(x) = x3

v’ (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.

Plots of Functions

0

5

10

15

20

25

30

-5

-4.6

-4.2

-3.8

-3.4 -3

-2.6

-2.2

-1.8

-1.4 -1

-0.6

-0.2 0.2

0.6 1

1.4

1.8

2.2

2.6 3

3.4

3.8

4.2

4.6 5

Series1

Here f(x) is x2. It can be seen to be a symmetric function for reflection at x = 0 because of mirror plane.

0

5

10

15

20

25

30

-5

-4.6

-4.2

-3.8

-3.4 -3

-2.6

-2.2

-1.8

-1.4 -1

-0.6

-0.2 0.2

0.6 1

1.4

1.8

2.2

2.6 3

3.4

3.8

4.2

4.6 5

Series1

Reflection yields.

The reflection carries out the mapping shown with the red arrows.

-150

-100

-50

0

50

100

150

-5

-4.6

-4.2

-3.8

-3.4 -3

-2.6

-2.2

-1.8

-1.4 -1

-0.6

-0.2 0.2

0.6 1

1.4

1.8

2.2

2.6 3

3.4

3.8

4.2

4.6 5

Series1

Plots of Functions - 2

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.

-150

-100

-50

0

50

100

150

-5

-4.6

-4.2

-3.8

-3.4 -3

-2.6

-2.2

-1.8

-1.4 -1

-0.6

-0.2 0.2

0.6 1

1.4

1.8

2.2

2.6 3

3.4

3.8

4.2

4.6 5

Series1

Reflection yields.

-150

-100

-50

0

50

100

150

-5

-4.6

-4.2

-3.8

-3.4 -3

-2.6

-2.2

-1.8

-1.4 -1

-0.6

-0.2 0.2

0.6 1

1.4

1.8

2.2

2.6 3

3.4

3.8

4.2

4.6 5

Series1

Plots of Functions - 2

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.

-150

-100

-50

0

50

100

150

-5

-4.6

-4.2

-3.8

-3.4 -3

-2.6

-2.2

-1.8

-1.4 -1

-0.6

-0.2 0.2

0.6 1

1.4

1.8

2.2

2.6 3

3.4

3.8

4.2

4.6 5

Series1

Reflection yields.

Operations Applied to Functions - 2

Now consider

f(x) = (x-2)2 = x2 – 4x + 4

v’ (f(x)) = v(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).

0

10

20

30

40

50

60

-5

-4.6

-4.2

-3.8

-3.4 -3

-2.6

-2.2

-1.8

-1.4 -1

-0.6

-0.2 0.2

0.6 1

1.4

1.8

2.2

2.6 3

3.4

3.8

4.2

4.6 5

Series1

0

10

20

30

40

50

60

-5

-4.6

-4.2

-3.8

-3.4 -3

-2.6

-2.2

-1.8

-1.4 -1

-0.6

-0.2 0.2

0.6 1

1.4

1.8

2.2

2.6 3

3.4

3.8

4.2

4.6 5

Series1

Reflection yields this function, not an eigenfunction.

Neither symmetric nor antisymmetric for reflection thru x = 0.

Atomic Orbitals

s orbital

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

z

Atomic Orbitals

p orbital

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

z

Different ways that objects can behave for a group consisting of E

and the reflection plane.

Cs E h

A’

A”

1 1

1 -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”.

Hybrids

hybrid

Reflection

Do not get the same hybrid back.

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.

z The two hybrids form the basis of a reducible representation of the operation

Reducible RepresentationsUse the two spz hybrids as the basis of a representation

h1 h2

10

01

01

10

E operation. h operation.

2

1

h

h=

2

1

h

h

2

1

h

h=

1

2

h

h

The hybrids are unaffected by the E operation.

The reflection operation interchanges the two hybrids.

Proceed using the trace of the matrix representation.

1 + 1 = 2 0 + 0 = 0

h1 becomes h1; h2 becomes h2. h1 becomes h1; h2 becomes h2.

RepresentationsCs E h

A’

A”

1 1

1 -1

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.

2 0 (h1, h2)

Note that = A’ + A”

h1h2

, ,h1 - h2 h1 + h2

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

Point group Symmetry operations

Characters+1 symmetric behavior

-1 antisymmetricMülliken symbols

Each row is an irreducible representation

Character Table

x, y, zSymmetry of translations (p orbitals)

Rx, Ry, Rz: rotations

Classes of operations

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