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

Previously: Welcome to a new academic year!. 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. Learn how to see differently….

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

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  1. Previously: Welcome to a new academic year! 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 Learn how to see differently…..

  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 3 2 2 1 1 2

  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 1 1 3 2 s2 3 2 3 2 s3 s1

  18. Previously in Inorganic Chemistry ….. 1. Symmetry elements and operations 2. Properties of Groups 3. Symmetry Groups, i.e., Point Groups Today in Inorganic…. 1. How to Assign Point Groups “the flowchart” 2. Classes of Point Groups 3. Inhuman Transformations 4. Symmetry and Chirality And as always, Learning how to see differently…..

  19. 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

  20. 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. 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.

  21. 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

  22. 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 + 6sv O: 3C4 and 4C3, many C2 Oh(octahedral group): O + i + 3sh+ 6sv Icosahedralgroup: Ih: 6C5, 10C3, 15C2, i, 15sv

  23. See any repeating relationship among the Cubic groups ? T: 4C3 and 3C2, mutually perpendicular Td(tetrahedral group): T + 3S4 axes + 6sv O: 3C4 and 4C3, many C2 Oh (octahedral group): O + i + 3 sh + 6 sv Icosahedral group: Ih: 6C5, 10C3, 15C2, i, 15sv

  24. See any repeating relationship among the Cubic groups ? T: 4C3and 3C2, mutually perpendicular Td(tetrahedral group): T + 3S4 axes + 6sv O: 3C4 and 4C3, many C2 Oh: 3C4 and 4C3, many C2+ i + 3 sh + 6 sv Icosahedralgroup: Ih: 6C5, 10C3, 15C2, i, 15sv How is the point symmetry of a cube related to an octahedron? How is the symmetry of an octahedron related to a tetrahedron? …. Let’s see!

  25. 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

  26. 5 types of symmetry operations. Which one(s) can you do?? Rotation Reflection Inversion Improper rotation Identity

  27. Previously in Inorganic Chemistry ….. 1. How to Assign Point Groups “the flowchart” 2. Classes of Point Groups 3. Inhuman Transformations Today in Inorganic…. Symmetry and Chirality Introducing: Character Tables 3. Symmetry and Vibrational Spectroscopy Still learning how to see differently…..

  28. First, some housekeeping 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 Point Group (or Symmetry Group) Assignments: checking in 3. 1st introspection due Friday Sept. 16 and Problems set #2 due next Tuesday.

  29. Chirality What is it?? How do you look for it? Is this molecule chiral? It’s mirror image…

  30. Chirality: dissymmetric vs. asymmetric

  31. Chirality: Dissymmetric: having a non-superimposible mirror image (dissymmetric = chiral) vs. Asymmetric: without any symmetry (has C1 point symmetry)

  32. 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 plane a S2 axis = inversion center

  33. 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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