1. 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!

2. x Symmetry may be defined as a feature of an object which is invariant to transformation There are 5 types of symmetry elements. 1. Mirror plane of reflection,  z y

3. 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 z y x

4. 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, C n where n = order of rotation z y x

5. Symmetry may be defined as a feature of an object which is invariant to transformation There are 5 types of symmetry elements. y 4. Improper Rotation axis, S n where n = order of rotation Something NEW!!! C n followed by  z

6. 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 C 1 axis z y x

7. 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 property 2) demonstrates associativity 3) possesses an identity 4) possesses an inversion for each operation

8. Let’s see how this works with symmetry operations. Start with an object that has a C 3 axis. 1 1 2 2 3 3 NOTE: that only symmetry operations form groups, not symmetry elements.

9. Now, observe what the C 3 operation does: 1 1 2 2 3 3 3 3 1 1 2 2 2 2 3 3 1 1 C3C3 C32C32

10. A useful way to check the 4 group properties is to make a “multiplication” table: 1 1 2 2 3 3 3 3 1 1 2 2 2 2 3 3 1 1 C3C3 C32C32

11. Now, observe what happens when two symmetry elements exist together: Start with an object that has only a C 3 axis. 1 1 2 2 3 3

12. Now, observe what happens when two symmetry elements exist together: Now add one mirror plane,  1. 1 1 3 3 11 2 2

13. Now, observe what happens when two symmetry elements exist together: 1 1 2 2 3 3 3 3 2 2 C3C3 11 11 1 1

14. Here’s the thing: Do the set of operations, {C 3 C 3 2  1 } still form a group? 1 1 2 2 3 3 3 3 1 1 2 2 3 3 2 2 1 1 How can you make that decision? C3C3 11 11

15. This is the problem, right? How to get from A to C in ONE step! 1 1 2 2 3 3 3 3 1 1 2 2 3 3 2 2 1 1 What is needed? C3C3 11 11 ACB

16. 1 1 2 2 3 3 3 3 1 1 2 2 3 3 2 2 1 1 What is needed? Another mirror plane! C3C3 11 11 1 1 2 2 3 3 22

17. 3 3 1 1 2 2 3 3 2 2 1 1 And if there’s a 2 nd mirror, there must be …. 33 11 1 1 2 2 3 3 22

18. 3 3 1 1 2 2 3 3 2 2 1 1 Does the set of operations {E, C 3 C 3 2  1  2  3 } form a group? 33 11 1 1 2 2 3 3 22 1 1 2 2 3 3 3 3 1 1 2 2 2 2 3 3 1 1 C3C3 C32C32

19. The set of symmetry operations that forms a Group 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, C 3 C 3 2  1  2  3 } is the operations of the C 3v point group.

20. The Types of point groups If an object has no symmetry (only the identity E) it belongs to group C 1 Axial Point groups or C n class C n = E + n C n ( n operations) C nh = E + n C n +  h (2n operations) C nv = E + n C n + n  v ( 2n operations) Dihedral Point Groups or Dn class D n = C n + nC2 (^) D nd = C n + nC2 (^) + n  d D nh = C n + nC2 (^) +  h Sn groups: S 1 = C s S 2 = C i S 3 = C 3h S 4, S 6 forms a group S 5 = C 5h

21. 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 perpendicular Td (tetrahedral group): T + 3S4 axes + 6 s O: 4C3, mutually perpendicular, and 3C2 + 6C2 Oh (octahedral group): O + i + 3 sh + 6 sd Icosahedral group: Ih : 6C5, 10C3, 15C2, i, 15 s

22. What’s the difference between:  v and  h 1 1 2 2 3 3 3 3 1 1 2 2 3 3 2 2 1 1  h is perpendicular to major rotation axis, C n vv  v is parallel to major rotation axis, C n hh

23. 5 types of symmetry operations. Which one(s) can you do?? Rotation Reflection Inversion Improper rotation Identity

24. 1 1 2 2 3 3 1 1 2 2 3 3

25. 1 1 2 2 3 3 C3C3 11 11

26. 1 1 2 2 3 3 C3C3 1 1 2 2 3 3 22