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 2011!

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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

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

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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 describers 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 s

 O: 4C3, mutually perpendicular, and 3C2 +

6C2Oh (octahedral group): O + i + 3 sh + 6 sdIcosahedral group:Ih : 6C5, 10C3, 15C2, i, 15 s

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

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C3 s1

s1

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C3

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s2