chapter 3: group theory · 2020. 9. 7. · according to group theory, a raman active mode is...

Chapter 3: Group TheoryChapter 3: Group Theory

2

Review of the Previous Lecture

1. Defined symmetry elements and symmetry operations

2. Applied the flow chart to determine the point group of a molecule

3. Determined molecular orientation

3

1. Character Tables

Each point group has an associated character table which lists all of the symmetry elements/operations that members of the group possess.

The table is mathematically derived using matrices.

C2V

C2V E C2 σv (xz) σv '(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

Symmetry Elements/Operations

IrreducibleRepresentations

BasisFunctions

Characters

4

1A. Irreducible Representations

Each point group can be decomposed into basic symmetry properties known as irreducible representations.

An irreducible representation is the simplest way to describe a molecular property.

Let’s deconstruct this soccer ball into irreduciblerepresentations:

1. Shape

2. Color

3. Pattern

5


The irreducible representations are defined by a set of characters.

Positive value (i.e. +1): Symmetric behavior

Negative value (i.e. -1): Antisymmetric behavior


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

IrreducibleRepresentations

Characters

6


Total # of irreducible representations = # of classes of symmetry elements/operations in a group


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

4

4

7


The Mullikin labels give us information about degeneracies as follows:

A and B labels indicate a non-degenerate state

E label indicates a double degeneracy

T label indicates a triply degeneracy


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

Mullikin (Symmetry)Labels

8

1B. Basis Functions

The basis functions represent certain types of properties and their symmetry will be defined by a specific irreducible representation.


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

BasisFunctions

9

1B. Basis Functions

I. Rx, Ry, Rz : Define rotations around the x, y, or z axes

II. x, y, z : Define translations (movement) along the x, y, or z axes


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

BasisFunctions

10

1B. Basis Functions

III. Orbital Type

Central Atom

Atomic OrbitalBasis Function

sThe first row

(entirely symmetric row)

px x

py y

pz z

dz2 z2 or 2z2 - x2 - y2

dx2-y2 x2 - y2

dxy xy

dxz xz

dyz yz

Binary

Functions

11

The d-orbitals

+z

+y

+x

12

The d-orbitals

+z

+y

+z

+x

+y

+x

+y

+x

+z

+y

+x

dz2 dyz dxz

dxy dx2-y2

13

1B. Basis Functions


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

BasisFunctions

Central Atom

Atomic OrbitalBasis Function

sThe first row

(entirely symmetric row)

pxx

pyy

pzz

dz2 z2 or 2z2 - x2 -y2

dx2-y2 x2 -y2

dxyxy

dxzxz

dyzyz

Note:

Orbitals of the same type will not necessarily be represented by the same irreducible representation.

i.e. H2O molecule

pz : A1 ; px : B1 ; py : B2

14

1C. Application of the character table


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

Which irreducible representation represents a translation along the y-axis?

+z

+y

+x

15


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


+z

+y

+xLet us formally determine this.

Approach 1- Draw out all operations

I. Draw the molecule in its new location

II. Apply and draw out each symmetry operation from the original position

Determine if the molecule looks unmoved or moved

Label the operation with the characters

Unmoved: +1

Moved: -1

16


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


+z

+y


+z

+y

+z

+y

+z

+y

+z

+y

y translation

(Redrawn 4 times)

E+z

+y

+1

C2

σv (xz)

σv'(yz)

+z

+y

-1

+z

+y

-1

+z

+y

+1

17


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


+z

+y


B2 1 -1 -1 1

+z

+y

+z

+y

+z

+y

+z

+y

y translation

(Redrawn 4 times)

E+z

+y

+1

C2

σv (xz)

σv'(yz)

+z

+y

-1

+z

+y

-1

+z

+y

+1

18


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


+z

+y

+xLet us formally determine this.

Approach 2- Determine an appropriate basis set

I. The basis set will represent the y-translation

II. Observe how the vectors transform according to the symmetry operations

Determine if the vectors look unmoved or moved

Label the operation with the characters

Unmoved: +1

Moved: -1

19


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


+z

+y


B2 1 -1 -1 1

+z

+y

+z

+y

+z

+y

+z

+y

Basis Set fory-translation

(Redrawn 4 times)

E

+1

C2

σv (xz)

σv'(yz)

-1

-1

+1

+z

+y

+z

+y

+z

+y

+z

+y

20

2. Reducible Representations

Molecular structure and dynamics can be defined by a net sum of irreducible representations.

Reducible Representations = Σ Irreducible representations

21

3. Molecular motion

The motion of a molecule containing N atoms can be described in terms of the 3 Cartesian axes (x,y,z).

The molecule has 3N degrees of freedom, which describe the translational, rotational, and vibrational motions of the molecule.

3N- 3 due to translations in the x, y, z directions- 3 due to rotations around the x, y, z axes (Rx, Ry, Rz)

3N-6 : # of vibrational modes for nonlinear molecules

3N-5 : # of vibrational modes for linear moleculesNormal vibrational modes

22

3A. Molecular vibration and spectroscopy

Infrared (IR): Molecular vibrations due to excitation

Raman: Molecular vibrations that result in differences in scattered light due to relaxation to different lower energy vibrational levels.

E

ν = 0

ν = 1

ν = 2

ν = Very high

states

Rayleigh

Raman

Scattering

Stokes

Raman

Scattering

Anti-stokes

Raman

Scattering

IR

Absorption

23

E

ν = 0

ν = 1

ν = 2

ν = Very high

states

IR

Absorption

24

E

ν = 0

ν = 1

ν = 2

ν = Very high

states

Rayleigh

Raman

Scattering

Stokes

Raman

Scattering

Anti-stokes

Raman

ScatteringIn

ten

sity

ννs < νl νs = νl νs > νl

Taken from Edinburgh Instruments

25

3B. Vibrational spectroscopy

3N- 3 due to translations in the x, y, z directions- 3 due to rotations around the x, y, z axes (Rx, Ry, Rz)

3N-6 : # of vibrational modes for nonlinear molecules

3N-5 : # of vibrational modes for linear moleculesNormal vibrational modes

Is concerned with the observation of the degrees of vibrational freedom.

Each normal mode of vibration will transform as an irreducible representation of the molecule’s point group.

26

3C. Determining if a mode is IR and/or Raman active

I. IR : Results in a change in dipole moment. According to group theory, an IR active mode is symmetric with the x, y, z basis functions.

No Δdm Yes Δdm; IR Active

D∞h C∞v

Dipole Moment

Irred. rep. = Σg+

for the vibrationIrred. rep. = A1 ; transforms like zfor the vibration

Symmetric Stretch

27

3C. Determining if a mode is IR and/or Raman active

II. Raman : Results in a change in polarizability or distortion of the electron cloud. According to group theory, a Raman active mode is symmetric with the binary basis functions (i.e. x2, xy, etc.)

Both are Raman active

Symmetric Stretch

28

3D. Rule of mutual exclusion

For centrosymmetric molecules, the rule of mutual exclusion applies. IR active vibrations are Raman inactive and vice versa.

Identify a centrosymmetric molecule by identifying the presence of an inversion center (i)

The Raman active vibrationis IR inactive

i No i

chapter 3: group theory · 2020. 9. 7. · according to group theory, a raman active mode is...

Documents