Today in Inorganic….

Uses of Symmetry in Chemistry

Symmetry elements and operations

Properties of Groups

Symmetry Groups, i.e., Point Groups

Previously:

Welcome to a new academic year!

Learn how to see differently…..

Why is Symmetry useful?

~ for Spectroscopy (identifying equivalent parts of molecule)

~ for Classifying Structure

~ for Bonding

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Symmetry may be defined as a feature of an object which is invariant to transformation

Symmetry elements are geometrical items about which symmetry transformations—or symmetry operations—occur.

There are 5 types of symmetry elements.1. Mirror plane of reflection, s

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Symmetry may be defined as a feature of an object which is invariant to transformation

There are 5 types of symmetry elements.2. Inversion center, i

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Symmetry may be defined as a feature of an object which is invariant to transformation

There are 5 types of symmetry elements.3. Proper Rotation axis, Cn

where n = order of rotation

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Symmetry may be defined as a feature of an object which is invariant to transformation

There are 5 types of symmetry elements.

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4. Improper Rotation axis, Sn

where n = order of rotationSomething NEW!!! Cn followed by s

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Symmetry may be defined as a feature of an object which is invariant to transformation

There are 5 types of symmetry elements.5. Identity, E, same as a C1 axis

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When all the Symmetry of an item are taken together, magical things happen.

The set of symmetry operations (NOT elements)in an object can form a Group

A “group” is a mathematical construct that has four criteria (‘properties”)

A Group is a set of things that:1) has closure property2) demonstrates

associativity3) possesses an

identity 4) possesses an

inversion for each operation

Let’s see how this works with symmetry operations.

Start with an object that has a C3 axis.

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NOTE: that only symmetry operations form groups, not symmetry elements.

Now, observe what the C3 operation does:

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C3 C32

A useful way to check the 4 group properties is to make a “multiplication” table:

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C3 C32

Now, observe what happens when two distinct symmetry elements exist together:

Start with an object that has only a C3 axis.

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Now, observe what happens when two symmetry elements exist together:

Now add one mirror plane, s1.

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s1

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Now, observe what happens when two symmetry elements exist together:

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C3 s1

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Here’s the thing:

Do the set of operations, {E, C3 C32 s1}

still form a group?

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How can you make that decision?

C3 s1

s1

This is the problem, right?How to get from A to C in ONE step!

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What is needed?

C3 s1

s1

A CB

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What is needed? Another mirror plane!

C3 s1

s1

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s2

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And if there’s a 2nd mirror, there must be ….

s3s1

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s2

“Multiplication” Table for set of Symmetry Operations {E, C3, C3

2, s1, s2, s3}

  E C3 C32 s1 s2 s3  

E E C3 C32 s1 s2 s3

C3 C3 C32 E s3 s1 s2

C32 C3

2 E C3 s2 s3 s1

s1 s1 s2 s3 E C3 C32

s2 s2 s3 s1 C32 E C3

s3 s3 s1 s2 C3 C32 E

               Note that the standard convention is that you perform the row operation first then the column operation. So the result illustrated earlier in the pink box was obtained by doing C3, then s1 written as (s1 x C3 ).

Next….

1. How to Assign Point Groups “the flowchart”

2. Classes of Point Groups

3. Inhuman Transformations

4. Symmetry and Chirality

So far…..

1. Symmetry elements and operations

2. Properties of Groups

3. Symmetry Groups, i.e., Point Groups

And as always,Learning how to see differently…..

First, some housekeeping

1. What sections of Chapter 3 are we covering? (in Housecroft) In Chapter 3: 3.1 - .7 (to p.76) and 3.8

2. 1st Problems set due Thursday.

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We answered this question: Does the set of operations {E, C3 C3

2 s1 s2 s3} form a group?

s3

s1

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s2

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C3 C32

“Multiplication” Table for set of Symmetry Operations {E, C3, C3

2, s1, s2, s3}

  E C3 C32 s1 s2 s3  

E E C3 C32 s1 s2 s3

C3 C3 C32 E s3 s1 s2

C32 C3

2 E C3 s2 s3 s1

s1 s1 s2 s3 E C32 C3

s2 s2 s3 s1 C3 E C32

s3 s3 s1 s2 C32 C3 E

               

The set of symmetry operations that forms aGroup is call a Point Group—it describes completely the symmetry of an object around a point.

Point Group symmetry assignments for any object can most easily be assigned by following a flowchart.

The set {E, C3 C32 s1 s2 s3} includes the

operations of the C3v point group.

What’s the difference between: sv and sh

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sh is perpendicular to major rotation axis, Cn

sv

sv is parallel to major rotation axis, Cn

sh

The Types of point groups

If an object has no symmetry (only the identity E) it belongs to group C1

Axial Point groups or Cn class Cn = E + n Cn ( n operations)Cnh= E + n Cn + sh (2n operations)Cnv = E + n Cn + n sv ( 2n operations) 

 Dihedral Point Groups or Dn class Dn = Cn + nC2 (^)

Dnd = Cn + nC2 (^) + n sd

Dnh = Cn + nC2 (^) + sh

 Sn groups:

S1 = Cs

S2 = Ci

S3 = C3h

S4 , S6 forms a groupS5 = C5h

 Linear Groups or cylindrical class

C∞v and D∞h= C∞ + infinite sv

= D∞ + infinite sh

 

Cubic groups or the Platonic solids..

T: 4C3 and 3C2, mutually perpendicularTd (tetrahedral group): T + 3S4 axes + 6 sv

 O: 3C4 and 4C3, many C2

Oh (octahedral group): O + i + 3 sh + 6 sv

Icosahedral group:Ih : 6C5, 10C3, 15C2, i, 15 sv

See any repeating relationship among the Cubic groups ?

T: 4C3 and 3C2, mutually perpendicularTd (tetrahedral group): T + 3S4 axes + 6 sv

 O: 3C4 and 4C3, many C2

Oh (octahedral group): O + i + 3 sh + 6 sv

Icosahedral group:Ih : 6C5, 10C3, 15C2, i, 15 sv

See any repeating relationship among the Cubic groups ?

T: 4C3 and 3C2, mutually perpendicularTd (tetrahedral group): T + 3S4 axes + 6 sv

 O: 3C4 and 4C3, many C2

Oh : 3C4 and 4C3, many C2 + i + 3 sh + 6 sv

Icosahedral group:Ih : 6C5, 10C3, 15C2, i, 15 sv

How is the point symmetry of a cube related to an octahedron?

…. Let’s see!

How is the symmetry of an octahedron related to a tetrahedron?

C4

C4

C4 C3

C3

C3

C4

C4

C3

C3

C4

C4

C4 C3

C3

C3

C4 is now destroyed!

Oh

5 types of symmetry operations.

Which one(s) can you do??

RotationReflectionInversionImproper rotationIdentity