Chem IV - Symmetry and Group Theory

1. Chem IV- Symmetry and Group Theory Chapter 7

2. Part 1 - Introduction to symmetry elements, symmetry operations and point groups

4. Symmetry in Nature

7. The Platonic Solids

8. Christian Art

9. Islamic Art

10. Art Deco

11. Architecture

12. Photography

13. Ndebele

14. Zulu

16. Symmetry analysis • A symmetry operation is an action that leaves a molecule apparently unchanged • Each symmetry operation is associated with a symmetry element

17. H2O Symmetry operation: rotation Point, line, or plane Symmetry element: axis of rotation

18. A symmetry operation leaves at least one point in the molecule unmoved – they are operations of point group symmetry

19. The identity operation, E – do nothing. All molecules have at least E, and some have only the symmetry element E

20. An n-fold rotation is a symmetry operation that leaves a molecule apparently unchanged after rotation by 360o/n. The symmetry element is an n-fold axis of rotation, Cn BH3 Note that C33 = E

21. BrF5 Note that C42 = C2 Associated with the symmetry element C4 we have the symmetry operations C4 C42 ( = C2) C43 C44 ( = E)

22. XeF4 The axis with the highest order, here the C4 axis, is called the principal axis The principal axis defines the z axis When assigning axis of the same order, we give preference to those that go through atoms, C2’, followed by those bisecting the bond angle, C2’’.

23. Identify the axes of rotation of this snowflake. List the symmetry operations associated with each of these symmetry elements

24. Symmetry operation reflection through the symmetry element mirror plane, σ The mirror planes contain the principal C2 axis. They are therefore vertical mirror planes (subscript “v”). z

25. This mirror plane is perpendicular to the principal C4 axis. It is therefore a horizontal mirror (subscript “h”)

26. This mirror plane is perpendicular to the principal C4 axis. It is therefore a vertical mirror (subscript “v”). Vertical mirror planes are those that usually go through atoms, and again, you give those that go through atoms as higher priority, i.e. sv. Those mirror planes that do not go through atoms are sometimes more accurately called dihedral mirror planes.

27. This mirror plane is parallel to (contains) the principal C4 axis and bisects the two C2' axes. It is a dihedral mirror (subscript “d”).

28. Mirror planes in molecules • σh(horizontal): plane perpendicular to principal axis • σd(dihedral), σv(vertical): plane colinear with principal axis • σd: σ parallel to Cn and bisecting two C2' axes or two σv • σv: σ parallel to Cn and are often coincident with lower rotation axes of high priority.

29. Examples of difference between vertical and dihedral planes. The vertical planes lie parallel with the C2’ axes. The dihedral planes are bisecting the C2’ axes and in this case also contain the C2’’ axes.

30. Examples of dihedral planes in the absence of C2 axes. Compare staggered ethane with the complex ML4ClBr. In the former molecule, there are only the three "horizontal" C2 axes and the planes bisect the angle between them. In the Newman projection the reason for calling these planes "dihedral" is clear. Finally we consider the complex in figure 1.15c where there are no "horizontal" C2 axes but there are two sets of planes containing the principal axis. Now the planes on the atoms take priority and are labelled v while the planes between the ligands bisect the angle between the vertical planes and become dihedral. Again which way round you do this is arbitrary but having the atoms as the highest priority pleases most chemists

31. Identify the mirror planes (σh, σd,σv) in the following examples acetylene

32. Identify the mirror planes (σh, σd,σv) in the following examples (Don’t forget the double bonds are delocalised)

33. Identify the mirror planes (σh, σd,σv) in the following examples Os(cp)2

34. The symmetry operation inversion, i, involves projecting each atom through a point, the centre of inversion i,that is located at the centre of the molecule

35. i Staggered form of ethane

36. Do these molecules have a centre of inversion?

37. The symmetry operations i and C2 should not be confused

38. The symmetry operation improper rotation occurs about the symmetry element Improper Axis, Sn This is a compound operation combining a rotation (Cn) with a reflection through a plane perpendicular to the Cn axis σh: Cn followed by σh or σhCn= Sn Read: Do Cn followed by σh Neither the 90o rotation nor the reflection is itself a symmetry operation for a tetrahedral molecule, but their combined effect is the symmetry operation S4

39. Do you see that S1 = σ and that S2 = i?

40. Use a CH4 molecule to verify that 2S4 = C2

41. Identify a S3 symmetry operation in BF3. What is it equal to in this molecule?

42. Identify all the symmetry elements in the (i) eclipsed (ii) staggered conformation of ethane E, C3, C2, σh, σv, S3 E, C3, C2, σd, i, S6

43. Sketch the S4 axis of NH4+. Is there a C4 axis? How many S4 axes are there in the ion?

44. The Point Groups of Molecules The set of symmetry elements of a molecule constitute a group. Since all symmetry operations leave at least one point in the molecule unchanged, the group is called a point group The point group is identified by it Schoenflies symbol

45. The process used to assign a molecule to a point group is straightforward with a few exceptions. Use this schematic to guide you.

46. CO2 OCS

49. Linear – No 2 of Cn, n>2 – No Cn – No σh – No i - No So… SiIBrClF

50. Determine the point groups of the following molecules CH2CBrCl C2H2Cl2Br2 H2O2 H2O S8 1,3,5,7-tetrafluoro- cyclooctatetrane [Ni(en)3]2+ naphthalene XeF4