Today in Inorganic…. Symmetry elements and operations Properties of Groups

Today in Inorganic….

Symmetry elements and operations

Properties of Groups

Symmetry Groups, i.e., Point Groups

Classes of Point Groups

How to Assign Point Groups

Previously:

Welcome to a new academic year!

Learn how to see differently…..

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Symmetry may be defined as a feature of an object which is invariant to transformationSymmetry 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 transformationThere 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 transformationThere 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 transformationThere 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 transformationThere 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:1

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C3 C32

Now, observe what happens when two 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, {C3 C3

2 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


1. How to Assign Point Groups “the flowchart”

2. Classes of Point Groups

3. Inhuman Transformations

4. Symmetry and Chirality

Previously in Inorganic Chemistry …..

1. Symmetry elements and operations

2. Properties of Groups

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

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

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Does the set of operations {E, C3 C32 s1 s2

s3}form a group?

s3s1

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s2

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C3 C32

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} is the

operations of the C3v point group.

The Types of point groupsIf 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 = CsS2 = CiS3 = C3hS4 , 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 C2Oh (octahedral group): O + i + 3 sh + 6 svIcosahedral group:Ih : 6C5, 10C3, 15C2, i, 15 sv

See any repeating relationship among the Cubic groups ?


O: 3C4 and 4C3, many C2Oh (octahedral group): O + i + 3 sh + 6 svIcosahedral group:Ih : 6C5, 10C3, 15C2, i, 15 sv

See any repeating relationship among the Cubic groups ?


O: 3C4 and 4C3, many C2Oh : 3C4 and 4C3, many C2 + i + 3 sh + 6 svIcosahedral 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?

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

5 types of symmetry operations.

Which one(s) can you do??

RotationReflectionInversionImproper rotationIdentity


1. Symmetry and Chirality

2. Introducing: Character Tables

3. Symmetry and Vibrational Spectroscopy

Previously in Inorganic Chemistry …..

1. How to Assign Point Groups “the flowchart”

2. Classes of Point Groups

3. Inhuman Transformations

Still learning how to see differently…..

First, some housekeeping

1. What sections of Chapter 4 are we covering? (in Housecroft) In Chapter 4: 4.1 - .7 first part, through p.104 (not pp.105-110) and 4.8

2. Point Group (or Symmetry Group) Assignments: checking in

3. 1st introspection due Friday Sept. 16 and Problems set #2 due next Tuesday.

Chirality

What is it??

How do you look for it?

Is this molecule chiral? It’s mirror image…

Chirality:

dissymmetric

vs.

asymmetric

Chirality:

Dissymmetric: having a non-superimposible mirror image (dissymmetric = chiral)

vs.

Asymmetric: without any symmetry(has C1

point symmetry)

Chirality as defined through Symmetry:

A Dissymmetric molecule has no Sn axis.

Is this contradictory to what you learned in Organic Chemistry?

NO because:a S1 axis = mirror planea S2 axis = inversion

center

Chirality as defined through Symmetry:

A Dissymmetric molecule has no Sn axis.

These molecules:• do not have mirror symmetry• do not have an inversion

BUT they are not chiral because they have a S4 axis.

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Today in Inorganic…. Symmetry elements and operations Properties of Groups