1. Lecture 4

2. Symmetry and group theory2

3. Natural symmetry in plants3

4. Symmetry
in animals 4

5. Symmetry in the human body5

6. The platonic solids 6

7. Symmetry in modern art
M. C. Escher 7

8. Symmetry in arab architecture La Alhambra, Granada (Spain)8

9. Symmetry in baroque art Gianlorenzo Bernini Saint Peter’s Church Rome9

10. Native American crafts
Symmetry in
Native American crafts 10

11. 7th grade art project
Silver Star School
Vernon, Canada 11

12. Re2(CO)10 12

13. C60
C2F4 13

14. Symmetry in chemistry Molecular structures Wave functions
Description of orbitals and bonds
Reaction pathways
Optical activity
Spectral interpretation (electronic, IR, NMR)
... 14

15. 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 15

16. Transformation of an object into an equivalent or indistinguishable
Symmetry Operation:
Transformation of an object into an equivalent or indistinguishable
orientation
C3, 120º
Symmetry Elements:
A point, line or plane about which a symmetry operation is
carried out 16

17. 5 types of symmetry operations/elements
Operation 1: Identity Operation, do nothing.
Identity: this operation does nothing, symbol: E 17

18. Operation 2: Cn, Proper Rotation:
Rotation about an axis by an angle of 2/n = 360/n C2 C3
NH3
H2O
How about:
NFO2? 18

19. Returns molecule to original oreintation
The Operation: Proper rotation Cn is 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

20. Proper rotation axes
C3, 120º
C2 180º
NH3
H2O
How about:
NFO2? 20

21. Rotation angle Symmetry operation 60º C6 120º C3 (= C62) 180º
240º
C32(= C64)
300º
C65
360º
E (= C66) 21

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

23. Cn = Rotation about an axis by an angle of 2/n
Proper Rotation:
Cn = Rotation about an axis by an angle of 2/n
PtCl4 C4 23

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

25. Cn = Rotation about an axis by an angle of 2/n
Proper Rotation:
Cn = Rotation about an axis by an angle of 2/n
PtCl4 C2 25

26. Cn = Rotation about an axis by an angle of 2/n
Proper Rotation:
Cn = Rotation about an axis by an angle of 2/n
PtCl4 C2 26

27. Cn = Rotation about an axis by an angle of 2/n
Proper Rotation:
Cn = Rotation about an axis by an angle of 2/n
PtCl4 C2 27

28. Cn = Rotation about an axis by an angle of 2/n
Proper Rotation:
Cn = Rotation about an axis by an angle of 2/n
PtCl4
Can perform operation several times.
C2, C4 C2 C2 C2 C2
Rotation 2m/n 28

29. 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

30. The highest order rotation axis is the principal axis
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? 30

31. 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)

32. Thus C4 px (x,y,z) = C4 xe-ar = ye-ar = py
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

33. Operation 3: Reflection and reflection planes
(mirrors) s s 33

34.  (reflection through a mirror plane)s
NH3
Only one s? 34

35. H2O, reflection plane, perp to boards
What is the exchange of atoms here? 35

36. H2O another, different reflection planes’
What is the exchange of atoms here? 36

37. the principal axis it is called sv
Classification of reflection planes B F
If the plane contains
the principal axis it is called sv B F
If the plane is perpendicular
to the principal axis
it is called sh
Sequential Application:
sn = E (n = even)
sn = s (n = odd) 37

38. 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) 38

39. Inversion not the same as C2 rotation !!39

40. 40

41. Figures with center of inversion
Figures without center of inversion 41

42. 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. 42

43. Some things to ponder: S42 = C2
Also, S44 = E; S2 = i; S1 = s 43

44. Summary: Symmetry operations and elements
proper rotation
axis (Cn)
improper rotation
axis (Sn)
reflection
plane (s)
inversion
center (i)
Identity
(E) 44

45. 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 2 1 1 1 4 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.

46. Translational symmetry not point symmetry

47. See: Chemical Applications of Group Theory by F. A. Cotton
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 47

48. To determine
the point group
of a molecule 48

49. 49

50. Groups of low symmetry 50

51. 51