Chapter 4 Symmetry and Group Theory

4.1 Symmetry Elements and Operations

• Symmetry element – a point, line, or plane about which a symmetry operation is carried out.

• Symmetry operation – A geometry transformation that leaves object in an indistinguishable configuration

• This chapter covers point groups symmetry – at least one point is not moved by the symmetry operation.

Examples of Symmetry

Five Point Group Symmetry Operations

• E – Identity – no movement – all objects have this symmetry

• Cn – Proper rotation – rotation by 360°/n about an axis. Always counter clockwise when viewed down the axis.

• s – a plane of symmetry. Sometimes called a mirror plane

• i – inversion center. There are equivalent points in any direction in opposite directions equidistant from the center.

• Sn – Improper rotation – Rotation by 360°/n about an axis, followed by a reflection perpendicular to the axis.

Rotations

Six-Fold Rotation combinations

Bilateral Symmetry – a Plane of Symmetry in a Human

Multiple Planes of Symmetry in a Pencil

Examples of a Inversion Center, Staggered Ethane, and Lack of an

Inversion Center in Methane

Figures with an Inversion Center

Figures Lacking an Inversion Center

Improper Rotation in Methane

An Excellent Interactive Symmetry Site

• Symmetry at Otterbien

4.2 Point Groups

First Step

• Assign a molecule to a point group

• A point group has a set of symmetry operations for the molecule.

• A flow chart (next page) is used to assign the point group.

Molecule whose Point Groups are Assigned in the Text

4.2.1 Point Groups of Low and High Symmetry

We will go through the flow chart step by step

• MAW tips

• If there is a unique atom (only one atom of an element or one atom connected a specific way), all the symmetry elements must run through that atom

• If there are more than one atom of an element look for operations relating that set of atoms

• Sometime atoms of the same element are not related by symmetry operations

First look for low symmetry or high symmetry point groups

Next Look for Groups of High Symmetry Including Linear Molecules

4.2.2 Other Groups

• For Other Groups There Must be Some Detailed Examination of the Molecules

• First Find the Highest Order RotationAxis

• Examples on Next Slide

The Order of the Rotation Axis will be a Subscript in the Group Name

• Dn Dnh Dnv Cn Cnh Cnv Sn

• D groups – dihedral groups contain C2 axes perpendicular to the highest order axes.

• Note for D2 groups the highest order axis is arbitrary. There are 3 mutually perpendicular axes.

Perpendicular C2 axes? Yes a D group, No a C or S group

Look for a Horizontal Mirror Plane that is Perpendiciular to the Highest Order

RotationAxes Dnh or Cnh

Next Look for A Mirror Plane That Runs along the Highest Order Rotation

Axis Dnd or Cnv

• Any molecules not assigned will be in pure rotation groups Dn or Cn or the very rare Sn if there is additional Sn axes.

• These assignments are on the next slide

Additional Point Group Assignments 1 of 3 slides for Table 4.4

A Nice Summary of Non-High or Non-Low Symmetry Groups

Additional High Symmetry Groups

4.3 Properties and Representations of Groups

Symmetry Operations for NH3

Group

• Needs set of objects (a,b,c,d,e,…) and a combining operation x

• Has the following properties • Identity a X E = E X a = a

• Inverse a X a-1 = E

• Closure a X b = a member of the group

• Associative property (a X b) X c = a X (b X c)

• Note Commutative Property is not required a X b = b X a is not necessarily true

4.3.1 Matrices

Symmetry operations in water may be repersented by matrices

4.3.2 Representations of Point Groups

Review of matrix multiplication

• Matrix i rows and j columns

• Left multiply a i x k by a k x j = i x j resultant

• Example on board

• Symmetry operations as matrices

Character sum of diagonal elements of a matrix

Example water symmetry matrices

Irreducible Reresentations in water from block diagonalization

4.3.3 Character Tables

Character table for water

Matrices for NH3 in C3v

C3v character table

Additional Features of Character table on page 107

•Why are there two C3 axes?

• What do the primes mean? C2’C2”

• What is the difference between σh σv and σd

• What do Rx Ry Rz mean?

• How do the orbitals transform?

• Match symmetry operations with molecule

• What does symmetric ( = 1) vs

antisymmetric ( = -1) mean?

•What do the labels A1 B2 mean?

• What do gerade (g) vs ungerade (u) mean

• Dimensions of the irreducible representations

Transformations of orbitals

4.4 Examples of Applications of Symmetry

4.4.1 Chirality

• Chiral objects are not superimposable on their mirror images

Each Enantiomer Rotates Polarized light in Opposite Directions

Criteria for chirality •The point group the molecule is in must have only proper rotations

•C1 = E, C2, C3, C4, etc.

•If the point group has improper rotations, the molecule can not be chiral

•S1 = σ, S2 = i, S3, S4, S5, S6, etc.

•In organic you used a partial definition, looking for σ or i. Which works in 99.9999999% of cases.

• Point group S4 has no σ or i but is not chiral.

A common chiral structure in transition metal chemistry

• Three rings are like a propeller. The point group is D3 . Molecule has only E, 2C3 a 3C2

• Look for a D3 molecule on the Otterbein site

4.4.2 Molecular Vibrations

• Molecules have 3N degrees of freedom. N = number of atoms. N x 3-Dimensions of space.

• 6 of these degrees are translations (x,y, and z) and Rotations (Rx, Ry, and Rz). Linear molecules have one less rotation.

• So there are 3N-6 vibration modes possible. (3N-5 for a linear molecule)

Coordinate system for water

Vibrational analysis

• Create a coordinate system

• Carry out the symmetry operations for the molecule to create a reducible representation, Γ .

• If a vector is the same : it contributes +1 to the character for that operation

• If a vector goes to the negative of itself: it contributes -1 to the chaacter for that operation.

• If the atom moves, It contribute zero to the character for that operation.

• Reduce the reducible representation (Γ) to a sum of irreducible representations of the point group. This is the symmetry of all degrees of freedom.

Water generation of Γ on the board

Math interlude: Formula to reduce reducible representations

• A reducible representation may be expresses as a sum on irreducible representation of the point group in only one way.

• # an irred. rep. occurs in the sum =

• 1/order[Σ (# symm. ops.)(irred)(red) ]

• Symm. Ops. = 3 of symmetry operation in the class

• irred = Character of the irreducible representation from the chartacter table

• red = Character of the reducible representation ,Γ

• The sum is over all the classes of the point group and must be done for each irreducible representation.

Examples

Water

• Reduction

• #A1 = 1/4[(9)(1) +(-1)(1)+(3)(1)+(1)(1)] = 12/4 = 3

• #A2 = 1/4[(9)(1) +(-1)(1)+(3)(-1)+(1)(-1)] = 4/4 = 1

• #B1 = 1/4[(9)(1) +(-1)(-1)+(3)(1)+(1)(-1)] = 12/4 = 3

• #B2 = 1/4[(9)(1) +(-1)(-1)+(3)(-1)+(1)(1)] = 8/4 = 2

Proof

• E C2(z) σ(xz) σ(xz)

• 4 A1 3 3 3 3

• 1A2 1 1 -1 -1

• 3B1 3 -3 3 -3

• 2B2 2 -2 -2 2

• Γ 9 -1 3 1

Subtraction of Translations and Vibrations

Irreducible representations for x,y,z and Rx, Ry, Rz are found in the

character tables

Vibrational modes of water Demonstration of symmetry on the board and illustration of

vibrations on Models 360

A more complicated molecule XeF4 3(5) – 6 = 9 vibrations

Reducible Representation for XeF4

Reduction is cumbersome, skip to the Results

Translations of XeF4

Rotations in XeF4

Selected Vibrations in XeF4

Vibrational analysis of NH3 in C3v

Reduces to

3A1 + A2 + 4E

- A1(x) E(x,y)

- A2(Rz) E(Rx, Ry)

2A1 2E

A1 symmetric stretch, symmetric bend IR and Raman active

(z) (x2+y2, z2)

E asymmetric stretch, asymmetric bend IR and Raman active

(x,y) ((x2-y2, xy) (xz, yz))

Selected Vibration Modes

C2v

D2h

Reducible Representation cis isomer

CO stretching modes cis isomer

Note both bands are IR and Raman active.

Reducible Representation for the trans isomer

Note: B3u is IR active only, and Ag is Raman active only.

Octahedral M(CO)3 facial complex C3v

Reduces to A1 + E

A1 z one band IR active and Raman

E (x,y) one band IR active and Raman

XeO4 tetrahedral

Raman only

IR and Raman