Chemistry reaching for 3D
Lapis lazuli = ultramarine
Egyptian blue
Hemoglobin
C2954H4516N780O806S12Fe4
van’t Hoff, LeBel 1874
Pasteur
“Leçons de Chimie” 1860
J. H. van’t Hoff
Voorstel tot Uitbreiding der Tegenwoordige in de Scheikunde gebruikte
Structuurformules in de Ruimte, 1874
chirality = handedness
carvone
Heinrich Wölfflin
brass, an alloy of
Cu and Zn
Alfred Werner
Cu5Zn8 γ-brass
Zn
Cu
Cu
Zn
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 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.
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 5fold, 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.
The 17 wallpaper groups
Magdolna and Istvan Hargittai
Symmetry Through the Eyes of a Chemist
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, 21992207 (2009).
Eschen and Jeitschko
(Au+)2M0(P-)2 M = Hg, Tl, Hg
M-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, 21992207 (2009).
Eschen and Jeitschko
(Au+)2M0(P-)2 M = Hg, Tl, Hg
M-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
416 atoms, a=18.71Å
Sm11Cd45
Mg44Rh7
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
Dimensions impose limitations
Pentagons can’t
tile a 2-D surface…
Tetrahedra can’t
tile a 3-D space…
1884
You can do in higher dimensions what can’t be
done in lower dimensions
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
Constrained to 2-D…
Allowed to move in 3-D…
…can’t get into triangle
…can get into triangle
A four-dimensional creature could tickle you from the inside…..
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
Packing in Higher Dimensions
Tetrahedra can’t
tile a 3-D space…
…unless the space
curves into 4-D
?
What Is a Projection?
Shadow
Photograph
wikipedia.org
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
[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
Tetrahedra can’t
tile a 3-D space…
…unless the space
curves into 4-D
?
“600-Cell”
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
Mg44Rh7
Berger, RF; Lee, S; Johnson, J; Nebgen, B; Sha, F;
Xu, J. Chem. Eur. J. 2008, 14, 3908-3930.
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
Elser, V; Sloane, NJA. J. Phys. A:
Math. Gen. 1987, 20, 6161-6168.
4-D
3-D
Our work
Projections from EE8: to 3-D


8


 0 0  4 0 1 1  1  1 the 8 - D   3 - D 





0  4 1  1 1  1 closest -   atomic 
0 0


0  4 0

 positions 
0
1

1

1
1
packed








 lattice 
Li21Si5
Sm11Cd45
Mg44Rh7
320 of 416 atoms
correctly projected
312 of 448 atoms
correctly projected
288 of 408 atoms
correctly projected
Berger, RF; Lee, S; Johnson, J; Nebgen, B; So,
Chem. Eur. J. 2008, 14, 6627-6639.
a restricted range of electron counts
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
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