Ch 4 Lecture2 Applications of Symmetry

I. MatricesA. Why Matrices? The matrix representations of the point group’s operations will

generate a character table. We can use this table to predict properties.

B. Definitions and Rules

1) Matrix = ordered array of numbers

2) Multiplying Matrices

a) The number of columns of matrix #1 must = number of rows of matrix #2

b) Fill in answer matrix from left to right and top to bottom

c) The first answer number comes from the sum of [(row 1 elements of matrix #1) X (column 1 elements of matrix #2)]

d) The answer matrix has same number of rows as matrix #1

The answer matrix has same number of columns as matrix #2

5243or 17

23

5438

4327

4862414

403207

84

37

62

51

e) Relevant example:

f) Exercise 4-4

II. Representations of Point GroupsA. Matrix Representations of C2v

1) Choose set of x,y,z axes

a) z is usually the Cn axis

b) xz plane is usually the plane of the molecule

2) Examine what happens after the molecule undergoes each symmetry operation in the point group (E, C2, 2)

321

100

010

001

321

3) Transformation Matrix = matrix expressing the effect of a symmetry operation on the x,y,z axes to give x’,y’,z’

4) E Transformation Matrix

a. x,y,z x,y,z

b. What matrix times x,y,z doesn’t change anything?

z'

y'

x'

z

y

x

???

???

???

transformationmatrix

z

y

x

z

y

x

100

010

001

E Transformation Matrix

O

H H

zy

x

5) C2 Transformation Matrix

a. x,y,z -x, -y, z

b. Correct matrix is:

6) v(xz) Transformation Matrix

a. x,y,z x,-y,z

b. Correct matrix is:

7) v(yz) Transformation Matrix

a. x,y,z -x,y,z

b. Correct matrix is:

z

y-

x-

z

y

x

100

010

001

z

y-

x

z

y

x

100

010

001

z

y

x-

z

y

x

100

010

001

zy

x

zy

x

zy

x

C2

v

d

O

H H

O

H H

O

H H

8) These 4 matrices are the “Matrix Representation” of the C2v point group

a. All point group properties transfer to the matrices as well

b. Example: Ev(xz) = v(xz)

B. Reducible and Irreducible Representations

1) Character = sum of diagonal from upper left to lower right (only defined for square matrices)

a) The set of characters = a reducible representation () or shorthand version of the matrix representation

a) For C2v Point Group:

100

010

001

100

010

001

100

010

001

E C2 v(xz) v(yz)

3 -1 1 1

2) Reducible and Irreducible Representations

a. Each matrix in the C2v matrix representation can be block diagonalized

b. To block diagonalize, make each nonzero element into a 1x1 matrix

c. When you do this, the x,y, and z axes can be treated independently

i. Positions 1,1 always describe x-axis

ii. Positions 2,2 always describe y-axis

iii. Positions 3,3 always describe z-axis

d. Generate a partial character table from this treatment

100

010

001

100

010

001

100

010

001

100

010

001

E C2 v(xz) v(yz)

Axis used E C2 v(xz) v(yz)

x 1 -1 1 -1

y 1 -1 -1 1

z 1 1 1 1

3 -1 1 1

IrreducibleRepresentations

Reducible Repr.

III. Character TablesA. The C2v Character Table

1) We have found three of the irreducible representations of the character table through matrix math

2) One more (A2) irreducible representation is derived from the first three due to the properties of character tables (below)

3) Rx, Ry, Rz stand for rotation about the x, y, z axes respectivelyx’s are p and d orbitals

4) Other symbols we need to knowa) R = any symmetry operationb) = character (#)

c) i,j = different representations (A1, B2, etc…)

d) h = order of the group (4 total operations in the C2v case)

C2v E C2 v(xz) v(yz)

A1 1 1 1 1 z x2, y2, z2

A2 1 1 -1 -1 Rz xy

B1 1 -1 1 -1 x, Ry xz

B2 1 -1 -1 1 y, Rx yz

B. The C3v Character Table for NH3

1) The threefold symmetry of NH3 makes for complex transformation matrices

2) Though more complex, the C3v Character Table can be generated similarly to that of the C2v group

C. Notes on Character Tables

1) Multiple operations in the same class are listed together

2) Different C2 axes are listed separately with primes (‘)

a) Those through outer atoms are ‘

b) Those not through outer atoms are ”

3) Symmetry of orbitals are listed except for s orbitals, which are always in the first listed A irreducible representation

4) Irreducible Representation Labels

a) Degeneracy (dimension) is determined by the character of E operation

i. A if E = 1 and of Cn = 1

ii. B if E = 1 and of Cn = -1

iii. E if E = 2 (doubly degenerate)

iv. T if E = 3 (triply degenerate)

b) Subscripts

i. 1 if symmetric to perpendicular C2 axis (or v)

ii. 2 if antisymmetric to perpendicular C2 axis (or v)

iii. g if symmetric to i

iv. u if antisymmetric to i

c) Primes

i. ‘ if symmetric to h

ii. “ if antisymmetric to h

IV. Applications of SymmetryA. Chiral Molecules

1) Molecules not superimposable with their mirror images are called chiral or dissymetric

2) They may still have some symmetry operations: E, Cn

3) Chiral molecules cannot have i, , or Sn symmetry operations

B. Molecular Vibrations

1. To use symmetry, we must assign axes

to each atom of the molecule

a. The z-axis is usually the Cn axis

b. The x-axis is in the molecular plane

c. The y-axis is perpendicular to the molecular plane

2. Degrees of Freedom = possible atomic movements in the molecule

a. 3N degrees of freedom for a molecule of N atoms

b. Nonlinear molecules

i. 3 translations (along x, y, z)

ii. 3 rotations (around x, y, z)

iii. 3N – 6 vibrations

c. Linear molecules

i. Only 2 rotations change the molecule

ii. 3N – 5 vibrations

3. We will use group theory to determine the symmetry of all nine motions and then assign them to translation, rotation, and vibration

a) Look at the C2v character table

b) Add up how many vectors stay the same after an operation

i. If the atom moves, none of its vectors stay the same

ii. If the atom stays and the vector is unchanged = +1

iii. If the atom stays and the vector is reversed = -1

c) Reduce the reducible representation to its irreducible components

C2v E C2 v(xz) v(yz)

A1 1 1 1 1 z x2, y2, z2

A2 1 1 -1 -1 Rz xy

B1 1 -1 1 -1 x, Ry xz

B2 1 -1 -1 1 y, Rx yz

C2v E C2 v(xz) v(yz)

9 -1 3 1

tionrepresenta eirreducibl

theofcharacter

tionrepresenta reducible

theofcharacter

class in the

operations ofnumber 1

given type a of

tionsrepresenta eirreducibl

ofnumber The

order

i. nA1 = ¼[(1x9x1)+(1x-1x1)+(1x3x1)+(1x1x1)] = 3 A1

ii. nA2 = ¼[(1x9x1)+(1x-1x1)+(1x3x-1)+(1x1x-1)] = 1 A2

iii. nA1 = ¼[(1x9x1)+(1x-1x-1)+(1x3x1)+(1x-1x1)] = 3 B1

iv. nA1 = ¼[(1x9x1)+(1x-1x-1)+(1x3x-1)+(1x1x1)] = 2 B2

d) All motions of water match 3A1 + A2 + 3B1 + 2B2

e) Use the character table to remove translations

x, y, z = A1 + B1 + B2

f) Use the character table to remove rotations

Rx, Ry, Rz = A2 + B1 + B2

g) The motions remaining are the vibrations = 2A1 + B1

i. A1 = totally symmetric

ii. B1 = antisymmetric to C2 and to reflection in yz plane

4) Symmetry and IR

a. IR only “sees” a vibration if the vibration changes the molecule’s dipole

b. Motion along the x, y, z axes creates a changed dipole

a. Infrared Active vibrations match up with x, y, z on character table

b. Infrared Inactive vibrations don’t

c. For water, all three vibrations are infrared active

5) Examples and Exercises pages 113-116

C. Molecular Vibrations of ML2(CO)2 complexes

1) The symmetry of cis- ML2(CO)2 complexes is C2v

a) The C=O stretch has only one possible direction of motion

b) Instead of using xyz vectors at each atom, we can use a single vector

c) Reducible representation from the 2 vectors

d) 2 possible vibrations from reduction formula: A1 + B1 (see both)

C2v E C2 v(xz) v(yz)

2 0 2 0

2) The symmetry of trans- ML2(CO)2 complexes is D2h

a. Symmetry operations on the vectors generate a reducible representation

b. Reduction formula give 2 irreducible representations

c. Only the B3u representation is IR Active

d. We can tell cis from trans by the number of C=O IR bands