Quantum Dots:
Particle-on-a-sphere model for
Buckminsterfullerene
Chem. Phys. Lett. 1993, 205, 200-206
quantum chemistry • particle-in-a-box • nanotechnology
The road from organic chemistry to quantum dots, tiny
molecular clusters whose electronic wavefunctions mimic
those of atoms, turns out to pass briefly through a “galaxy far,
far away.” For decades it was thought that elemental carbon
existed in only two pure forms: graphite and diamond. About
30 years ago, radioastronomers and astrochemists discovered
that red giant stars eject substantial amounts of carbon as they
switch from “burning” hydrogen to helium. In 1985, Harold
Kroto convinced Richard Smalley at Rice University to
explore the chemistry using laser vaporization and molecular
beams techniques that Smalley’s group had developed. In the
course of these experiments Smalley’s group identified and
characterized C60, a new form of elemental carbon, quickly
shifting the focus of the research problem from one of the
universe’s largest structures, to one of its smallest.
Smalley and his co-workers deduced that C60 was a spheroid,
in part by assembling paper cut-outs of pentagons and
hexagons at the kitchen table while having a beer! In fact,
Smalley was not the first to propose that such a carbon
structure might be stable. In fact, researchers had already
done theoretical calculations to show that C60 would be stable.
Smalley’s paper in Nature [Nature 1985 318, 162], however,
showing that these molecules self-assembled and providing a
mechanism for their synthesis and isolation catalyzed the
development of a new field – nanotechnology. Theory and
experiment both continue to provide key insights in this
emerging area.
Before calculators…
"The Curta is a precision
calculating machine for all
arithmetical operations.
Curta adds, subtracts,
multiplies, divides, square
and cube roots... and every
other computation arising in
science and commerce...
Available on a trial basis.
Price $125.”
From an advertisement in the
back pages of Scientific
American in the 1960s. This is
about $700 in 2002 dollars –
about the same price as
Mathematica. Curta’s sell on
e-Bay for thousands of dollars
these days.
© 2004 Michelle M. Francl. May be reproduced for use in an individual classroom. May not be sold or used in other collections
without the express permission of the author. These materials were produced as part of “P-Chem with a Purpose,” funded by the
National Science Foundation, grant DUE-0340873.
Quantum Dots
Chem. Phys. Lett. 1993, 205, 200-206
Practical Nanotech
Today lasers are a
ubiquitous example of
quantum mechanics at
work, in fifty years,
quantum dots may be the
poster child for practical
applications of quantum
phenomena. Quantum
dots composed of
fluorescent
semiconductor
nanocrystals of cadmium
and selenium have been
used as fluorescent
“tags” for biological
molecules in vivo.
Current dyes used in
high-resolution
fluorescence microscopy
pose technical problems,
including limitations on
the number of targets that
can be highlighted and
the photo-instability of
the dyes themselves. The
quantum dot tags can be
tailored to the system of
interest and are highly
stable. Given the toxicity
of cadmium and
selenium, however, there
are clearly still barriers to
overcome before these
can see use in humans.
Want to know more? Read B.
Dubertret et al., "In vivo
imaging of quantum dots
encapsulated in phospholipid
micelles," Science 2002, 298,
1759-62.
A particle-on-a-sphere model for C60
Michael R. Savina, Lawrence L. Lohr and Anthony H.
Francis
Questions and Problems
The questions and problems below are based on the paper cited. They
are meant to encourage you to read the paper critically, you may need to
consult other articles in the literature to answer these question. If you
were the editor of the journal, what questions might you have for the
authors?
1. Convince yourself that a buckyball really is nearly
spherical by building the scale paper model of
buckminsterfullerene attached.
2. The authors note that “perimeter models” have been
developed for cata-condensed hydrocarbons. What is a
cata-condensed hydrocarbon? Give an example that is
not given in the paper, draw its molecular structure.
3. Write down the Schrödinger equation for the perimeter
model. What are the solutions for the wavefunctions and
energies?
4. Draw, to scale, an energy level diagram for benzene
based on the perimeter model. Based on what you know
from organic chemistry, is this a good model? How well
does the first excitation energy calculated from this
model match the experimental value (which occurs at
roughly 200 nm)?
5. What concerns might you have about using the perimeter
model for larger condensed hydrocarbons, such as
anthracene or phenacene?
6. Equation (1) in this paper gives the energy for an electron
moving on the surface of a sphere. The authors note that
there are 60 π electrons in buckminsterfullerene, and that
the HOMO for this model is l=5. Is l the only quantum
number in this system? Sketch the energy levels and
label them with all appropriate quantum numbers to show
that the HOMO is l=5.
7. In equation (6) the authors show a matrix element
Yl'm' Ylm Yl m" "
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Quantum Dots
and note that one can make use of parity arguments
(properties of odd and even functions) to determine when
it is zero. They imply that if l and l’ don’t differ by an
even integer the integral will vanish. Explain, using what
you know about parity. Write out an example of this
matrix element that should vanish. Evaluate it explicitly
to show that it is equal to zero.
Further Reading
•
•
•
•
"Molecular Machines", Accts. Chem. Res., vol. 34, no. 6
(2001). A special issue devoted to nanotechnology.
Phillip Ball, Designing the Molecular World, Princeton
University Press, 1994. pp. 38-53; 186-215.
"C60: Buckminsterfullerene", H.W. Kroto, J.R. Heath,
S.C. O'Brien, R.F. Curl, R.E. Smalley, Nature 1985 318,
162. The paper that began it all.
Smalley’s Nobel address:
http://www.nobel.se/chemistry/laureates/1996/smalleylecture.html
More information about paper models of molecules can
be found in Molecular Origami [Robert Hanson,
University Science Books, 1995]. See also Bob Hanson's
web site at http://www.stolaf.edu/people/hansonr/mo/.
An alternative paper model for a buckyball can be made
by
following
the
instructions
at
http://www.merrimack.edu/~thull/combgeom/bucky/buc
kynotes.html. A good introduction to the geometry of
these types of structures is also given here.
Acknowledgements
With thanks to Prof. Hanson of St. Olaf’s for permission
to reproduce the origami buckyballs.
What do Cheetos have to do
with quantum mechanics and
flamingos?
Simple quantum mechanical
models have been applied to
large systems successfully in
the past. Using th eparticle in
a box model for dyes is a
common physical chemistry
lab. Carotenoid dyes, based on
a linear conjugated diene
skeleton, provide nature with
some colorful accents.
Canthaxanthin, for example, is
fed to captive flamingos to
produce their characteristic
pink color (a similar pigment
found in brine shrimp does the
same favor for wild
flamingos). Canaries, whose
signature color is a greenish
yellow, can be turned red if
they are fed paprika during
their molt. If you're tired of
only changing the color of your
hair, you can try for a pumpkin
look for fall. The compound
that gives this class of
vegetable pigments its name —
β-carotene — when consumed
in large quantities by humans,
will turn them orange. [It was
observed clinically in Britain
during WW II when food
shortages led some people to
include large amounts of
carrots in their diets.] And if
you thought the bright color of
Cheez-Whiz and Cheetos was
artificial -- it's not. Bixin or
annatto, a natural pigment used
for centuries is the source of
that unforgettable orange.
Researchers have recently
elucidated the biochemical
pathway for the synthesis of
bixin and are pursuing genetic
engineering approaches to its
bulk synthesis in tomatoes
[Florence Bouvier in Science,
300:2089-2091, June 27,
2003].
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Quantum Dots
To create a scale paper model of a bucky ball, cut out the figures on this page and the
next. Fold first on the solid lines, then use transparent tape to make two halves, finally
connect the halves. Sturdier models can be made by using copying these templates onto
heavier paper. Reprinted from Molecular Origami by Robert Hanson, University Science
Books, 1995. Used with permission.
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Quantum Dots
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