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

Previously: Welcome to 2011!. 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. Symmetry may be defined as a feature of an object which is invariant to transformation.

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

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  1. Previously: Welcome to 2011! 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

  2. 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 z y x

  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, Cn 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. 4. Improper Rotation axis, Sn where n = order of rotation Something NEW!!! Cn followed by s z y

  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 C1 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. NOTE: that only symmetry operations form groups, not symmetry elements. Start with an object that has a C3 axis. 1 3 2

  9. Now, observe what the C3 operation does: 2 1 3 C3 C32 1 3 3 2 2 1

  10. A useful way to check the 4 group properties is to make a “multiplication” table: 2 1 3 C3 C32 1 3 3 2 2 1

  11. Now, observe what happens when two symmetry elements exist together: Start with an object that has only a C3 axis. 1 3 2

  12. Now, observe what happens when two symmetry elements exist together: Now add one mirror plane, s1. 1 3 2 s1

  13. Now, observe what happens when two symmetry elements exist together: 1 3 s1 C3 3 2 2 1 s1

  14. Here’s the thing: Do the set of operations, {C3 C32 s1} still form a group? How can you make that decision? 1 3 3 s1 C3 3 2 2 1 1 2 s1

  15. This is the problem, right? How to get from A to C in ONE step! What is needed? A B C 1 3 3 s1 C3 3 2 2 1 1 2 s1

  16. What is needed? Another mirror plane! 1 3 3 s1 C3 3 2 2 1 1 2 s1 1 s2 3 2

  17. And if there’s a 2nd mirror, there must be …. 1 3 3 3 2 1 2 2 1 s2 s3 s1

  18. Does the set of operations {E, C3 C32 s1 s2 s3} form a group? 2 1 3 C3 C32 1 3 3 2 2 1 1 3 3 3 2 1 2 2 1 s2 s3 s1

  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. The set {E, C3 C32 s1 s2 s3} is the operations of the C3v point group. Point Group symmetry assignments for any object can most easily be assigned by following a flowchart.

  20. 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 + nCn ( n operations) Cnh= E + nCn + sh (2n operations) Cnv= E + nCn + nsv( 2n operations)  Dihedral Point Groups or Dn class Dn= Cn + nC2 (^) Dnd= Cn+ nC2 (^) + nsd Dnh= Cn+ nC2 (^) + sh Sn groups: S1 = Cs S2 = Ci S3 = C3h S4 , S6 forms a group S5 = C5h

  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 Icosahedralgroup: Ih: 6C5, 10C3, 15C2, i, 15 s

  22. What’s the difference between: sv and sh sv is parallel to major rotation axis, Cn sh is perpendicular to major rotation axis, Cn 1 3 3 3 2 2 1 1 2 sv sh

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

  24. 1 3 2 1 3 2

  25. 1 s1 C3 3 2 s1

  26. 1 1 C3 3 2 3 2 s2

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