Lecture 4

1. Lecture 4

2. Symmetry and group theory

3. Natural symmetry in plants

4. Symmetry in animals

5. Symmetry in the human body

6. Symmetry in modern art M. C. Escher

7. Symmetry in arab architecture La Alhambra, Granada (Spain)

8. Symmetry in baroque art Gianlorenzo Bernini Saint Peter’s Church Rome

9. 7th grade art project Silver Star School Vernon, Canada

10. Re2(CO)10

11. C60 C2F4

12. Symmetry in chemistry • Molecular structures • Wave functions • Description of orbitals and bonds • Reaction pathways • Optical activity • Spectral interpretation (electronic, IR, NMR) • ...

13. Molecular structures A molecule is said to have symmetry if some parts of it may be interchanged by others without altering the identity or the orientation of the molecule

14. C3, 120º Symmetry Operation: Transformation of an object into an equivalent or indistinguishable orientation Symmetry Elements: A point, line or plane about which a symmetry operation is carried out

15. 5 types of symmetry operations/elements Operation 1: Identity Operation, do nothing. Identity: this operation does nothing, symbol: E

16. C2 C3 Operation 2: Cn, Proper Rotation: Rotation about an axis by an angle of 2/n = 360/n NH3 H2O How about: NFO2?

17. The Operation: Proper rotation Cnis the movement(2p/n) The Element: Proper rotation axis Cn is the line 180° (2p/2) Applying C2 twice Returns molecule to original oreintation C22 = E C2

19. C2 Proper Rotation: Cn = Rotation about an axis by an angle of 2/n PtCl4

20. C4 Proper Rotation: Cn = Rotation about an axis by an angle of 2/n PtCl4

21. C2 Proper Rotation: Cn = Rotation about an axis by an angle of 2/n PtCl4

22. C2 Proper Rotation: Cn = Rotation about an axis by an angle of 2/n PtCl4

23. C2 Proper Rotation: Cn = Rotation about an axis by an angle of 2/n PtCl4

24. C2 Proper Rotation: Cn = Rotation about an axis by an angle of 2/n PtCl4

25. Operations can be performed sequentially Can perform operation several times. Means m successive rotations of 2p/n each time. Total rotation is 2mp/n m times Observe

26. C3 axis Iron pentacarbonyl, Fe(CO)5 The highest order rotation axis is the principal axis and it is chosen as the z axis What other rotational axes do we have here?

27. Let’s look at the effect of a rotation on an algebraic function Consider the pz orbital and let’s rotate it CCW by 90 degrees. px proportional to xe-ar where r = sqrt(x2 + y2 + z2) using a coordinate system centered on the nucleus y y C4 x o x o px C4 px How do we express this mathematically? The rotation moves the function as shown. The value of the rotated function, C4 px, at point o is the same as the value of the original function px at the point o . The value of C4 px at the general point (x,y,z) is the value of px at the point (y,-x,z) Moving to a general function f(x,y,z) we have C4 f(x,y,z) = f(y,-x,z) Thus C4 can be expressed as (x,y,z) (y,-x,z). If C4 is a symmetry element for f then f(x,y,z) = f(y,-x,z)

28. According to the pictures we see that C4 px yields py. Let’s do it analytically using C4 f(x,y,z) = f(y,-x,z) We start with px = xe-ar where r = sqrt(x2 + y2 + z2) and make the required substitution to perform C4 y y C4 x o x o px C4 px Thus C4 px (x,y,z) = C4 xe-ar = ye-ar = py And we can say that C4 around the z axis as shown is not a symmetry element for px

29. Operation 3: Reflection and reflection planes (mirrors) s s

30.  (reflection through a mirror plane) s NH3 Only one s?

31. H2O, reflection plane, perp to board s What is the exchange of atoms here?

32. H2O another, different reflection plane s’ What is the exchange of atoms here?

33. Classification of reflection planes If the plane contains the principal axis it is called sv F F F F B B F F If the plane is perpendicular to the principal axis it is called sh sn = E (n = even) sn = s (n = odd) Sequential Application:

34. Operation 4: Inversion: i Center of inversion or center of symmetry (x,y,z)  (-x,-y,-z) in = E (n is even) in = i (n is odd)

35. Inversion not the same as C2 rotation!!

37. Figures with center of inversion Figures without center of inversion

38. Operation 5: Improper rotation (and improper rotation axis): Sn Rotation about an axis by an angle 2/n followed by reflection through perpendicular plane S4 in methane, tetrahedral structure.

39. Some things to ponder: S42 = C2 Also, S44 = E; S2 = i; S1 = s

40. Summary: Symmetry operations and elements

41. Successive operations, Multiplication of Operators Already talked about multiplication of rotational Operators Means m successive rotations of 2p/n each time. Total rotation is 2mp/n But let’s examine some other multiplications of operators C4 1 2 1 4 1 C4 s 2 2 3 4 3 4 3 C4 s’ s We write s x C4 = s’, first done appears to right in this relationship between operators.

42. Translational symmetry not point symmetry

43. Symmetry point groups The set of all possible symmetry operations on a molecule is called the point group (there are 28 point groups) The mathematical treatment of the properties of groups is Group Theory In chemistry, group theory allows the assignment of structures, the definition of orbitals, analysis of vibrations, ... See: Chemical Applications of Group Theory by F. A. Cotton

44. To determine the point group of a molecule

46. Groups of low symmetry