Chemistry reaching for 3D

Lapis lazuli = ultramarine Egyptian blue

Hemoglobin

C2954H4516N780O806S12Fe4

“Leçons de Chimie” 1860

van’t Hoff, LeBel 1874

Pasteur

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

chirality = handedness

carvone

Heinrich Wölfflin

http://upload.wikimedia.org/wikipedia/commons/3/3d/1GZX_Haemoglobin.png

brass, an alloy of

Cu and Zn

Alfred Werner

Zn Cu Cu Zn

Cu5Zn8 γ-brass

3, 2, 1, 0 …..

Low (less than three ) dimensional models have a special attraction for theorists, in chemistry or physics.Reasons: (1)Sometimes the higher dimensional problems are more difficultto 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.

Examples of problems that are unique to a dimension or a subset of dimensions

1.A first-order phase transition cannot take place in one dimension, if short-range forces are assumed.

Examples of problems that are unique to a dimension or a subset of dimensions

1.A first-order phase transition cannot take place in one dimension, if short-range forces are assumed.

2.The ideal Bose-Einstein gas does not undergo its quantum mechanical condensation in D=1 or 2, only in D greater than 2.

3.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.

Magdolna and Istvan HargittaiSymmetry Through the Eyes of a Chemist

The 17 wallpaper groups

Percolation threshhold is a function of dimensionality

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, HgM-M ~ 3.20 A

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, HgM-M ~ 3.20 A

Chemistry in more than 3 dimensions?

• Some ionic and intermetallic crystal structures have really complicated geometries

β-Mg2Al3 NaCd2 Cd3Cu4

1832 atoms, a=28.2Å 1192 atoms, a=30.6Å 1124 atoms, a=25.9Å

Li21Si5 Sm11Cd45 Mg44Rh7

416 atoms, a=18.71Å 448 atoms, a=21.70Å 408 atoms, a=20.15Å

Many of these structures are made up of slightly irregular tetrahedra….

Work of Stephen Lee, Danny Fredrickson, Rob Berger

Pentagons can’t tile a 2-D surface…

Tetrahedra can’t tile a 3-D space…

Dimensions impose limitations

1884

Constrained to 2-D…

…can’t get into triangle

Allowed to move in 3-D…

…can get into triangle

You can do in higher dimensions what can’t be done in lower dimensions

Constrained to 2-D…

…can’t get into triangle

Allowed to move in 3-D…

…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…..

You can do in higher dimensions what can’t be done in lower dimensions

Dodecahedron

…unless the surface curves into 3-D

Pentagons can’t tile a 2-D surface…

Packing in Higher Dimensions

…unless the space curves into 4-D

?Tetrahedra can’t tile a 3-D space…

What Is a Projection?

wikipedia.org

Shadow Photograph

The image that results from “collapsing” an object to a lower-dimensional spaceTechnically: 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

[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

Features of Projection• Objects can be projected in an infinite number of

ways• Projection can take symmetry away from a highly

symmetric object

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

Packing in Higher Dimensions

…unless the space curves into 4-D

“600-Cell”

?Tetrahedra can’t tile a 3-D space…

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

Dodecahedron 600-cell

The 54-Cluster

• Td projection of half of the 600-cell– Packing of nearly regular tetrahedra– Pseudo-fivefold axes along ‹110›

54-Clusters of Different Sizes

• Ubiquitous in complex structures– Overlapping, on different length scales

Berger, RF; Lee, S; Johnson, J; Nebgen, B; Sha, F; Xu, J. Chem. Eur. J. 2008, 14, 3908-3930.

Mg44Rh7

The Limits of 4-D• We want a crystal with 600-cell point group• The E8 lattice is such; an 8-dimensional closest-packed lattice

– every lattice point is equivalent– beloved by mathematicians (and string theorists!) for its

packing and topology

(n1, n2, n3, n4, n5, n6, n7, n8)

(n1+½, n2+½, n3+½, n4+½, n5+½, n6+½, n7+½, n8+½)

ni are integers with even sum

E8 4-D 3-DElser, V; Sloane, NJA. J. Phys. A: Math. Gen. 1987, 20, 6161-6168.

Our work

Projections from E8 to 3-D

positions

atomic

D-3

lattice

packed

-closest

D-8 the

:

11110040

11114000

111104008E

320 of 416 atoms correctly projected

312 of 448 atoms correctly projected

288 of 408 atoms correctly projected

Li21Si5 Sm11Cd45 Mg44Rh7

Berger, RF; Lee, S; Johnson, J; Nebgen, B; So, Chem. Eur. J. 2008, 14, 6627-6639.

Compound Valence e- per atom

Li21Si5 1.58

Mg44Rh7 1.59

Mg44Ir7 1.59

Zn21Pt5 1.62

Cu5Zn8 1.62

Cu41Sn11 1.63

Mg29Ir4 1.64

a restricted range of electron counts

Could this be a message from higher dimensions?

„Kunst gibt nicht das Sichtbare wieder, sondern Kunst macht sichtbar.“

Paul Klee

„Art does not reproduce what we see, it makes it visible“

Hemoglobin

C2954H4516N780O806S12Fe4

Mirror images

Structure and dimensionality in some covalently and ionically bonded arrays,

and its consequences

Roald Hoffmann and Xiao-Dong Wen, with Taeghwan Hyeon, Zhongwu Wang, and Thomas Cahill