1. 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…..

2. Why is Symmetry useful? ~ for Spectroscopy (identifying equivalent parts of molecule) ~ for Classifying Structure ~ for Bonding

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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.

10. 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

11. 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

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

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

14. 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 3 3 2 2 1 1

15. Here’s the thing: Do the set of operations, {E, 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

16. 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

17. 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

18. 1 1 2 2 3 3 1 1 2 2 3 3 And if there’s a 2 nd mirror, there must be …. 33 11 1 1 2 2 3 3 22

19. “Multiplication” Table for set of Symmetry Operations {E, C 3, C 3 2,  1,  2,  3 } EC3C3 C32C32 11 22 33 EEC3C3 C32C32 11 22 33 C3C3 C3C3 C32C32 E 33 11 22 C32C32 C32C32 EC3C3 22 33 11 11 11 22 33 EC3C3 C32C32 22 22 33 11 C32C32 EC3C3 33 33 11 22 C3C3 C32C32 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 C 3, then  1 written as (  1 x C 3 ).

20. 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…..

21. 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. 1 st Problems set due Thursday.

22. 3 3 1 1 2 2 3 3 2 2 1 1 We answered this question: 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

23. “Multiplication” Table for set of Symmetry Operations {E, C 3, C 3 2,  1,  2,  3 } EC3C3 C32C32 11 22 33 EEC3C3 C32C32 11 22 33 C3C3 C3C3 C32C32 E 33 11 22 C32C32 C32C32 EC3C3 22 33 11 11 11 22 33 EC32C32 C3C3 22 22 33 11 C3C3 EC32C32 33 33 11 22 C32C32 C3C3 E

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

25. 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

26. 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

27. Linear Groups or cylindrical class C∞v and D∞h = C∞ + infinite  v = D∞ + infinite  h Cubic groups or the Platonic solids.. T: 4C 3 and 3C 2, mutually perpendicular T d (tetrahedral group): T + 3S 4 axes + 6  v O: 3C 4 and 4C 3, many C 2 O h (octahedral group): O + i + 3  h + 6  v Icosahedral group: I h : 6C 5, 10C 3, 15C 2, i, 15  v

28. See any repeating relationship among the Cubic groups ? T: 4C 3 and 3C 2, mutually perpendicular T d (tetrahedral group): T + 3S 4 axes + 6  v O: 3C 4 and 4C 3, many C 2 O h (octahedral group): O + i + 3  h + 6  v Icosahedral group: I h : 6C 5, 10C 3, 15C 2, i, 15  v

29. See any repeating relationship among the Cubic groups ? T: 4C 3 and 3C 2, mutually perpendicular T d (tetrahedral group): T + 3S 4 axes + 6  v O: 3C 4 and 4C 3, many C 2 O h : 3C 4 and 4C 3, many C 2 + i + 3  h + 6  v Icosahedral group: I h : 6C 5, 10C 3, 15C 2, i, 15  v 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?

31. C4C4 C4C4 C4C4 C3C3 C3C3

33. C3C3

39. C4C4 C4C4 C3C3 C3C3

40. C4C4 C4C4 C4C4 C3C3 C3C3

44. C3C3 C 4 is now destroyed!

45. OhOh

46. 5 types of symmetry operations. Which one(s) can you do?? Rotation Reflection Inversion Improper rotation Identity