1.021 , 3.021, 10.333, 22.00 Introduction to Modeling and Simulation
Spring 2012
Part II – Quantum Mechanical Methods
Brief Introduction to Part II
Lecture 0
Jef frey C. Grossman
Department of Materials Science and Engineering
Content overview
I. Particle and continuum methods
1. Atoms, molecules, chemistry
2. Continuum modeling approaches and solution approaches
3. Statistical mechanics
4. Molecular dynamics, Monte Carlo
5. V isualization and data analysis
6. Mechanical properties – application: how things fail (and how to prevent it)
7. Multi-scale modeling paradigm
II. Quantum mechanical methods
1. It’ s A Quantum W orld: The Theory of Quantum Mechanics
2. Quantum Mechanics: Practice Makes Perfect
3. The Many-Body Problem: From Many-Body to Single- Particle
4. Quantum modeling of materials
5. From Atoms to Solids
6. Basic properties of materials
7. Advanced properties of materials
8. What else can we do?
8. Biological systems (simulation in biophysics) – how proteins work and how to model them
Computer Hardware Historical Milestones
1946: Eniac
Op/s: 5000
Sq. ft: 3000
1952: IBM SSEC
Op/s: 2000
Sq. ft: 1000
1951: MIT Whirlwind
Op/s: 200,000
Sq. ft: 3100
1964: CDC 6600
Op/s: 3,000,000
Sq. ft: 3100
1968: Apollo Guide Apollo 7& 1 1 missions
1971: Kenbak-1
First personal computer 256 Bytes of memory
1974: Xerox Alto Built-in mouse Connect to network
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JCG Personal Computer History
T andy TRS-80 (a.k.a. “T rash-80”) Atari 400 (note the stylish keyboard)
Start-up screens
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quantum
modeling
Multi-scale modeling
Courtesy of Elsevier, Inc., http://www.sciencedirect.com . Used with permission.
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uantu mechanists
Werner Heisenberg, I"Iax Planck, Louis de BrogIie,AIbert Einstein, Niels Bohr, Erwin Schrodinger, I*Iax Born, john von Neumann,
Paul Dirac,WoIfgang Pauli
( I 900 - 1930 )
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Why uantu echanics!
Classical mechanics
Newton's laws 1687 ) F
d run )
( d1
Pro J ?
Movies of a Quantum Ball Thrown into a W all
Courtesy of Bernd Thaller. Used with permission.
Example: Diels-Alder Reaction:
1,3-butadiene + ethylene → cyclohexene
Courtesy ChemWiki .
Predicting what these electrons do is what gives us those much- needed energy curves – quantum mechanics is key!
See Lecture 1 video for animation. © James E. Kendall/MSC Caltech . All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/ .
A Little Bit of Schrödinger
W ritten out explicitl y , the Schrödinger equation looks like this:
2 2 2 2 ( r , t )
2 m x 2 y 2 z 2 V ( r , t ) i t
is the wavefunction for the system of particles, which characterizes the particle’ s motion. One can derive all properties of the system of particles from its wavefunction.
W e no longer ask “Where is/are the particle(s)?”, but instead ask “What is the probability distribution governing the positions?”
Solving the Sch r odinger Equation
H G ( → r ) = E G ( → r )
Anal ytic solutions become extr emel y complicated, e v en f or simple systems.
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Y ear 1929 …
The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the dif ficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.
P .A.M. Dirac, Proc. Roy . Soc. 123, 714 (1929)
… and 1963
If there is no complete agreement [ … ] between the results of one’ s work and the experiment, one should not allow oneself to be too discouraged [ … ]
P .A.M. Dirac, Scientific American, May 1963
Density functional theory
Energy
Electron densit y
E 0 E [ n 0 ]
Hohenberg & K ohn, 1964
Inter acting
Non-inter acting
-
-
-
-
-
-
- -
- -
Image of Walter Kohn receiving Nobel prize removed due to copyright restrictions.
Ko h n & S h a m , 1 9 6 5
W alter K ohn (left), receiving the Nobel priz e in chemistry in 1 9 9 8 .
Why do we need quantum mechanics ?
Example: Bonding and Structure
Figures by Nicola Marzari. Photo courtesy of Nicola Marzari and David Vanderbilt/Rutgers University.
Paraelectric (cubic) and ferroelectric (tetragonal) phases of PbT iO 3
Example: Electronic, optical, magnetic properties
Courtesy of Felice Frankel. Used with permission.
Example: Nanotechnology
Nanotechnology Scientist in Nanotechnology created The Spiderman Hulk
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How small is nano?
10 0 m = 1m
10 -9 = 1 nanometer
10 9 m = 1 million km
10 0 m = 1m
Images from Eames Power of Ten film removed due to copyright restrictions. See http://www.powersof10.com/ .
Nanotechnology Definition
“How Super-Cows and Nanotechnology will Make Ice Cream Healthy”
telegr aph.co .uk , A ugust 21, 2005
In a field somewhere in Count y D own, Northern Ireland, is a herd of 40 super -cows that could tak e all the poisonous guilt out of bingeing on ice cream. Unilev e r , the manufacturer of P ersil and PG Tips, is sponsoring a secret research project b y a leading B ritish agricultur al science institution into how to reduce the lev els of satur ated fat in cow's milk.
It is also experimenting with nanotechnolog y , or the science of in visibly tin y things. Unilev er believ es that b y halving the siz e of particles that mak e up the emulsion - or fatt y oil - that it uses to mak e ice cream, it could use 90 per cent less of the emulsion.
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New Optical Properties
Quantum Dots
optically boring
Courtesy of Felice Frankel. Used with permission.
optically exciting
+ quantum dot
bulk semiconductor
nano scooper
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Quantum Confinement
laser
quantu m confinemen t
hole electron
quantum dot
Physical confinement of excited state leads to unique quantum effect : light emitted depends on size .
bulk semiconductor
In order to keep CO 2 emissions in check, we will need to consume half of our electricity through renewable sources by the year 2050.
Need major improvements in ef ficiency and cost in order to take advantage of these resources.
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Example: make solar cells cheaper and more ef ficient
Four Basic Steps:
Understand, predict, and tailor these key fundamental processes.
1) Convert a photon into an electron and a hole
2) Electron and hole “thermalize”
3) The electron-hole pair dif fuses
4) Electron and hole are separated and taken out
Amorphous vs. Crystalline Silicon Solar Cells
• Gap: 1.12 eV
• Lower absorption coef ficient, device ~100 m
• Mobilities:
• electron: 250 m in 1 ns
• hole: 250 m in 3 ns
• Gap: 1.6-1.8 eV
• High absorption coef ficient, device ~1 m
• Mobilities:
• electron: 0.25 m in 1 ns
• hole: 0.25 m in 200 ns
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Example: Energy Storage Materials
solar fuels, hydrogen storage
Fuel
Solar
Quantum chemisty methods, reaCion path eneqetics, excited states.
Image is in the public domain.
Tunable thermoQnamia
for H deso@on.
MgH
Ru
CD
samd neat Energy.
fi - l * )
LgRu
Run
Pushing electrons up a hill takes quantum mechanics
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Example: Concr ete Science
• Cement accounts for 5-‐10% of global CO 2 Emissions
• Cement is mainly made of synthetic rock : Clinker
• 4 major phases: Alite (Ca3SiO5), Belite (Ca2SiO4), Aluminate (Ca3Al2O6), Ferrite (Ca2AlFeO5)
• Many different polymorphisms, not fully understood
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Surface Reactivity is K e y P r oblem
Alite (C3S)
7 p o l ymo rp h s
Constitutes 50-‐70 %
Reaction with water : Fast
Belite (C2S)
3 p o l ymo rp h s
Constitutes 15-‐30 %
Reaction with water : Slow
Adapted from Cement Microscopy , Halliburton Services, Duncan, OK.
Courtesy Andrew R. Barron . L icense : CC-BY .
• Wh y? Ho w?
• Can w e tune the Belite r eaction rate?
It ’ s a quantum w orld!
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