Chemistry reaching for 3D

1. Chemistry reaching for 3D

3. Lapis lazuli = ultramarine Egyptian blue

9. Hemoglobin C2954H4516N780O806S12Fe4

11. van’t Hoff, LeBel 1874 Pasteur “Leçons de Chimie” 1860

12. J. H. van’t Hoff Voorstel tot Uitbreiding der Tegenwoordige in de Scheikunde gebruikte Structuurformules in de Ruimte, 1874

15. chirality = handedness

17. carvone

19. Heinrich Wölfflin

21. brass, an alloy of Cu and Zn

22. Alfred Werner

23. Cu5Zn8γ-brass Zn Cu Cu Zn

24. 3, 2, 1, 0 …..

25. Low (less than three ) dimensional models have a special attraction for theorists, in chemistry or physics. • Reasons: • Sometimes the higher dimensional problems are more difficult • to solve, if not impossible. Witness the one-dimensional Ising model (say of interacting spins) – easily solved in one-dimension in 1925 by Ernst Ising. Devilishly hard in 2-D, solved by Lars Onsager in 1944. Laziness, or a coming to terms with reality? • (2) A desire for simplicity. One way in to beauty, but… also a dangerous weakness of the human mind. • (2) Modeling: Sometimes the simpler problem reveals the underlying physical essence that is obscured in the more complicated problem. So the lower dimension problem may be the road to understanding.

26. Examples of problems that are unique to a dimension or a subset of dimensions • A first-order phase transition cannot take place in one dimension, if short-range forces are assumed.

28. Examples of problems that are unique to a dimension or a subset of dimensions • A first-order phase transition cannot take place in one dimension, if short-range forces are assumed. • The ideal Bose-Einstein gas does not undergo its quantum mechanical condensation in D=1 or 2, only in D greater than 2. • In three-dimensional translated arrays you can’t have 5-fold, 7-fold or higher rotational axes. To put it another way, you can’t tile your bathroom floor with only perfect pentagons. • 4. You can’t pack 3-D space with perfect tetrahedra.

29. The 17 wallpaper groups Magdolna and Istvan Hargittai Symmetry Through the Eyes of a Chemist

30. Percolation threshhold is a function of dimensionality

31. Element Lines: Bonding in the Ternary Gold Polyphosphides, Au2MP2 with M = Pb, Tl, or Hg, X.-D. Wen, T. J. Cahill, and R. Hoffmann, J. Am. Chem. Soc., 131, 2199-2207 (2009). Eschen and Jeitschko (Au+)2M0(P-)2 M = Hg, Tl, Hg M-M ~ 3.20 A

32. Element Lines: Bonding in the Ternary Gold Polyphosphides, Au2MP2 with M = Pb, Tl, or Hg, X.-D. Wen, T. J. Cahill, and R. Hoffmann, J. Am. Chem. Soc., 131, 2199-2207 (2009). Eschen and Jeitschko (Au+)2M0(P-)2 M = Hg, Tl, Hg M-M ~ 3.20 A

33. Chemistry in more than 3 dimensions?

34. Some ionic and intermetallic crystal structures have really complicated geometries β-Mg2Al3 NaCd2 Cd3Cu4 1124 atoms, a=25.9Å 1832 atoms, a=28.2Å 1192 atoms, a=30.6Å

35. Sm11Cd45 Mg44Rh7 Li21Si5 Many of these structures are made up of slightly irregular tetrahedra…. Work of Stephen Lee, Danny Fredrickson, Rob Berger 416 atoms, a=18.71Å 448 atoms, a=21.70Å 408 atoms, a=20.15Å

36. Dimensions impose limitations Pentagons can’t tile a 2-D surface… Tetrahedra can’t tile a 3-D space…

37. 1884

38. Constrained to 2-D… Allowed to move in 3-D… …can’t get into triangle …can get into triangle You can do in higher dimensions what can’t be done in lower dimensions

39. Constrained to 2-D… Allowed to move in 3-D… …can’t get into triangle …can get into triangle You can do in higher dimensions what can’t be done in lower dimensions A four-dimensional creature could tickle you from the inside…..

40. You can do in higher dimensions what can’t be done in lower dimensions Pentagons can’t tile a 2-D surface… …unless the surface curves into 3-D Dodecahedron

41. Packing in Higher Dimensions Tetrahedra can’t tile a 3-D space… …unless the space curves into 4-D ?

42. Shadow Photograph wikipedia.org What Is a Projection? The image that results from “collapsing” an object to a lower-dimensional space Technically: multiplying a set of (n x 1) vectors by an (m x n) matrix to get a set of (m x 1) vectors, m<n

43. [Socrates:]  Behold! human beings living in a underground den, which has a mouth open towards the light and reaching all along the den; here they have been from their childhood, and have their legs and necks chained so that they cannot move, and can only see before them, being prevented by the chains from turning round their heads. Above and behind them a fire is blazing at a distance, and between the fire and the prisoners there is a raised way; and you will see, if you look, a low wall built along the way, like the screen which marionette players have in front of them, over which they show the puppets. And do you see, I said, men passing along the wall carrying all sorts of vessels, and statues and figures of animals made of wood and stone and various materials, which appear over the wall? Some of them are talking, others silent. [Glaucon] You have shown me a strange image, and they are strange prisoners. Like ourselves, I replied; and they see only their own shadows, or the shadows of one another, which the fire throws on the opposite wall of the cave? True, he said; how could they see anything but the shadows if they were never allowed to move their heads? And of the objects which are being carried in like manner they would only see the shadows? Yes, he said. Plato, The Republic

44. Features of Projection • Objects can be projected in an infinite number of ways • Projection can take symmetry away from a highly symmetric object

45. by projection, a complex (in a lower dimension) arrangement may be derived from a simpler higher dimensional object Simple 2-D square lattice Projection Non-repeating 1-D structure

46. Packing in Higher Dimensions Tetrahedra can’t tile a 3-D space… …unless the space curves into 4-D ? “600-Cell”

48. 600-cell: 120 vertices; 720 edges, 1200 edges, 600 ideal polyhedra

49. 600-cell Dodecahedron

50. The 54-Cluster • Td projection of half of the 600-cell • Packing of nearly regular tetrahedra • Pseudo-fivefold axes along ‹110›