Part II Lesson 3 From many-body to single-particle ... · 12 12 16 16 4 4 8 8 12 12 16 16 0 0 4 8...
Transcript of Part II Lesson 3 From many-body to single-particle ... · 12 12 16 16 4 4 8 8 12 12 16 16 0 0 4 8...
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1.021, 3.021, 10.333, 22.00 : Introduction to Modeling and Simulation : Spring 2012
Part II – Quantum Mechanical Methods : Lecture 3
From Many-Body to Single-Particle: Quantum
Modeling of Molecules Jeffrey C. Grossman
Department of Materials Science and Engineering Massachusetts Institute of Technology
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2. Quantum Mechanics: Practice Makes Perfect
3. From Many-Body to Single-Particle; Quantum Modeling of Molecules
4. Application of Quantum Modeling of Molecules: Solar Thermal Fuels
Part II Topics 1. It’s a Quantum World:The Theory of Quantum Mechanics
5. Application of Quantum Modeling of Molecules: Hydrogen Storage
6. From Atoms to Solids
7. Quantum Modeling of Solids: Basic Properties
8. Advanced Prop. of Materials:What else can we do?
9. Application of Quantum Modeling of Solids: Solar Cells Part I
10. Application of Quantum Modeling of Solids: Solar Cells Part II
11. Application of Quantum Modeling of Solids: Nanotechnology
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Motivation Last time: 1-electron quantum mechanics to describe spectral lines
? Image of NGC 604 nebula is in the public domain. Source: HubbleImage adapted from Wikimedia Commons, http://commons.wikimedia.org. Space Telescope Institute (NASA). Via Wikimedia Commons.
.
Today: many electrons to describe materials.
? Courtesy of Bernd Thaller. Used with permission.3
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Lesson outline • Review
• The Many-body Problem
• Hartree and Hartree-Fock
• Density Functional Theory
• Computational Approaches
• Modeling Software
• PWscf
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Review: Schrödinger equation
H time independent: 1(⌃r, t) = 1(⌃r) · f(t)
f(t) H1(rr)in = = const. = E
f(t) 1(rr)
H (⇧r) = E (⇧r) - �i Et /(⌃r, t) = /(⌃r) · e
time independent Schrödinger equation stationary Schrödinger equation
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Review:The hydrogen atom
stationary Schrödinger equation Hl = El
� T + V
⇥� = E�
� �
just solve
☺
�2� � ⇥2 + V
⇥�(⌃r) = E�(⌃r)
2m
�2 2e⇥2 �
⇥⇤(�r) = E⇤(�r)
2m 4⇥�0r
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Radial Wavefunctions for a Coulomb V(r)
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Z
Thickness dr
y
X
r
2
2
1
10
0 3 4
R10
00
0.2
0.4
1 2 3 4
r2R220
r2R210
r2R221
r2R230
r2R231
r2R232
rr
r
r
r
r
r
r
r
-0.20
0
0
0.20.4
0.040.080.12
0.6
2
r
r
r
6 10
2 6 10
00
0.10.05
0.05
0.15
2 6 10
0
00
0
0
00
02 6 10
R20
R21
R30
R31
R32
0.1
4
4
8
8
12
12
16
16
4
4
8
8
12
12
16
16
00
4 8 12 16
r0 4 8 12 16
-0.1
0.2
0.4
0.04
0.08
-0.04
0.04
0.08
0.02
0.04
0.4
0.8
0
0.4
0.8
Radial functions Rnl(r) and radial distribution functions r2R2nl(r) for
atomic hydrogen. The unit of length is aµ = (m/µ) a0, where a0 is thefirst Bohr radius.
0.15
Images by MIT OpenCourseWare.
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Angular Parts
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Image by MIT OpenCourseWare.
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Review:The hydrogen atom
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ψn l ml
(r, θ, φ)
1 a
e03/2
-r/a0
π
14 2 a
e03/2
-r/2a2- ra0
0
π
14 2 a
e cos θ03/2
-r/2ara0
0
π
18 a
e sin θ e03/2
-r/2ara
0
0
π±iφ
a0 = h2
me2= .0529 nm = first Bohr radius
1
2
2
2
0
0
1
1
0
0
0
1±
1s
2s
2p
2p
n l mlAtomicOrbital
quantum numbers
Image by MIT OpenCourseWare.
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10 Courtesy of David Manthey. Used with permission. Source: http://www.orbitals.com/orb/orbtable.htm.
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Review:The hydrogen atom
© R. Nave. All rights reserved. This content is excluded from our Creative Commonslicense. For more information, see http://ocw.mit.edu/help/faq-fair-use/.11
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Review:The hydrogen atom
© unknown. All rights reserved. This content is excluded from our Creative Commonslicense. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
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Review: Next? Helium
+
e -
r1 H1 = E1
e- �H1 + H2 + W
⇥�(⌥r1, ⌥r2) = E�(⌥r1, ⌥r2)
�T1 + V1 + T2 + V2 + W
⇥�(�r1, �r2) = E�(�r1, �r2)
�2 e2 �2 e2� � ⇥2 � � ⇥2 � +
e2
4⇥�0r12
⇥⇤(r1, r2) = E⇤(r1, r2)1 22m 4⇥�0r1 2m 4⇥�0r2
cannot be solved analytically problem!
r2
r12
e
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Review: Spin
new quantum number: spin quantum number for electrons: spin quantum number can ONLY be
up down 14
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Everything is spinning ...
⇧F = �⇤E ⇧= ⇤ ⇧ Bm ·
Stern–Gerlach experiment (1922)
Image courtesy of Teresa Knott.
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Everything is spinning ...
In quantum mechanics particles can have a magnetic moment and a ”spin”
magnetic ⇥m moment
spinning charge
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Everything is spinning ... conclusion from the
Stern-Gerlach experiment
for electrons: spin can ONLY be
up down 17
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Spin History Discovered in 1926 by
Goudsmit and Uhlenbeck
Part of a letter by L.H. Thomas to Goudsmit on March 25 1926 (source: Wikipedia). © Niels Bohr Library & Archives, American Institute of Physics. All rights reserved. This content is excludedfrom our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
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Pauli’s exclusions principle
Two electrons in a system cannot have the same quantum numbers!
hydrogen ... ... ... ...
quantum numbers: 3s 3p 3dmain n: 1,2,3 ... 2s 2p
orbital l: 0,1,...,n-1 magnetic m: -l,...,l
spin: up, down 1s
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Pauli Exclusion Principle “Already in my original paper I stressed the
circumstance that I was unable to give a logical reason for the exclusion principle or to deduce it from more general assumptions. I had always the feeling, and I still
have it today, that this is a deficiency.”
W. Pauli, Exclusion Principle and Quantum Mechanics, Nobel prize acceptance lecture, Stockholm (1946).
Image via Wikimedia Commons. License: CC-BY-SA. This content is excluded from our CreativeCommons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.20
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Periodic table
This image is in the public domain. Source: Wikimedia Commons.
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The many-body problem
helium: 2e- iron: 26e-
e-
e-
r2
+
Image © Greg Robson on Wikimedia Commons. License: CC-BY-SA. This content is excluded fromour Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
� = �(⇧r1, . . . , ⇧rn)
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r1 r12
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Dirac Quotes Year 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 difficulty 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 in 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 23
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€
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��
��
�� �
The Multi-Electron Hamiltonian
2 2eRememberthe good old days of the
1-electron H-atom??
They’re over!
Hl = El ⇥2 ⇥⇤(r) = E⇤(r)
2m 4⇥ 0r
N N N n N n n nh2 2 1 ZiZ je
2 h22 Zie
2 1 e2 H = −∑ ∇Ri
+ ∑∑ − ∑∇ri −∑∑ + ∑∑2m 2Ri − rj ri − rji=1 2Mi 2 i=1
ij≠ =1 j
Ri − R j i=1 i=1 j=1 i=1 j=1 i≠ j
kinetic energy of ions kinetic energy of electrons electron-ion interaction
potential energy of ions electron-electron interaction
Multi-Atom-Multi-Electron Schrödinger Equation H (R1,...,RN ;r1,...,rn ) Ψ(R1,...,RN ;r1,...,rn ) = E Ψ(R1,...,RN ;r1,...,rn )
��
� ��
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Born-Oppenheimer Approximation
Source: Discover Magazine
Online
© Discover Magazine Online. All rights reserved. This content is excluded from our CreativeCommons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
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Born-Oppenheimer Approximation (skinless version) • mass of nuclei exceeds that of the
electrons by a factor of 1000 or more • we can neglect the kinetic energy
of the nuclei • treat the ion-ion interaction
Images in public domain.classically Born Oppenheimer• significantly simplifies the
Hamiltonian for the electrons: n N nh2 Zie
2 e2 H = −
2m ∑∇r2 i −∑∑ + ∑∑2Ri − rj ri − rji=1 i=1 j=1 i=1 j=1
i≠ j
1 nn
This term is just an external potential V(rj)
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Solutions
Moller-Plessetperturbation theory
MP2
coupled clustertheory CCSD(T)
quantum chemistrydensity
functionaltheory
quantum chemistry
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Hartree Approach Write wavefunction as a simple product of single particle states:
Ψ(r1,...,rn ) = ψ1 r1 ( )...ψn rn( )ψ2 r2 ( ) Hard Product of Easy
Leads to an equation we can solve on a computer! ( ,2n* h2 e2 ψ j r( ) * )− ∇2 r dr -ψi ( )r = εiψi r( ) +Vext ( ) + ∑∫2m rj − r ** j=1 + j≠i .
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Hartree Approach ( ,2n* h2 e2 ψ j r( ) * )− ∇2 +Vext ( ) + ∫r ∑ dr -ψi r = ( ) ( ) εiψi r
** 2m j=1 rj − r + j≠i .
The solution for each state depends on all the other states (through the Coulomb term). • we don’t know these solutions a priori • must be solved iteratively: - guess form for { Ψiin(r) } - compute single particle Hamiltonians - generate { Ψiout(r) } - compare with old - if different set { Ψiin(r) } = { Ψiout(r) } and repeat - if same, you are done
• obtain the self-consistent solution 29
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Simple Picture...But...
After all this work, there is still one major problem:the solution is fundamentally wrong
The fix brings us back to spin!
- - - - -
- - - -
Interacting Non-Interacting
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Symmetry Holds the Key
Speculation: everything we know with scientific certainty is somehow dictated by symmetry.
The relationship between symmetry and quantum mechanics is particularly striking.
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Exchange Symmetry •all electrons are indistinguishable
•electrons that made da Vinci, Newton, and Einstein who they were, are identical to those within our molecules a bit humbling...
•so if • I show you a system containing electrons•you look away• I exchange two electrons in the system•you resume looking at system• there is no experiment that you can conduct that will
indicate that I have switched the two electrons 32
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Mathematically
•define the exchange operator: ψ1 r2 ( ) χ12ψ1( )r1 ψ2 ( ) r2 = ( )ψ2 r1
•exchange operator eigenvalues are ±1: suppose χ^! 12 φ = χφ ^! ^!χ12 χ12 φ = χ 2φ = φ
χ 2 = 1, or χ = ±1.
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Empirically • all quantum mechanical states are also eigenfunctions of
exchange operators - those with eigenvalue 1 (symmetric) are known as
Bosons - those with eigenvalue -1 (antisymmetric) are known
as Fermions • profound implications for materials properties
- wavefunctions for our many electron problem must be anti-symmetric under exchange
- implies Pauli exclusion principle 34
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€
Hartree-Fock • Employing Hartree's approach, but
- enforcing the anti-symmetry condition - accounting for spin
• Leads to a remarkable result: ( ,2n n* h2 e2 ψ j r( ) * e2 − ∇2 +Vext r ∑∫ dr -ψi r ∑δsi ,s j ∫ dr0 * ( )ψi ( )ψ j r = ( ) ψ j r0 r0 ( ) εiψi r) ( ) + ( ) −
2m r0 − r* j=1 rj − r * j=1 + j≠i . j≠i
•Hartree-Fock theory is the foundation of molecular orbital theory.
• It is based upon a choice of wavefunction that guarantees antisymmetry between electrons.
it’s an emotional moment.... 35
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But...Hartree-Fock
•neglects important contribution to electron energy (called “correlation” energy)
•difficult to deal with: integral operator makes solution complicated
•superceded by another approach: density functional theory
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Solutions
Moller-Plessetperturbation theory
MP2
coupled clustertheory CCSD(T)
quantum chemistrydensity
functionaltheory
density functional
theory
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Solving the Schrodinger Equation
No matter how you slice it, the wavefunction is a beast of an entity to have to deal with.
For example: consider that we have n electrons populating a 3D space.
Let’s divide 3D space into NxNxN=2x2x2 grid points.
To reconstruct Ψ(r), how many points must we keep track of?
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ρ(N3)
8
8
8
8
Solving the Schrodinger Eq.
N N
N
divide 3D space into NxNxN=2x2x2 grid points
Ψ= Ψ(r1, ... , rn) # points = N3n
n=# electrons Ψ(N3n) 1 8
10 109
100 1090
1,000 10900 39
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Working with the Density The electron density seems to be a more manageable quantity.
Wouldn’t it be nice if we could reformulate our problem in terms of the density, rather than the wavefunction?
Energy Electron Density
E0=E[n0] Walter Kohn (left), receiving the Nobel prize in chemistry
in 1998. 40
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Why DFT? computational expense
for system size N: Quantum Chemistry
O(N5-N7)methods; MP2, CCSD(T)
O(N3); O(N) Density Functional Theory
Example
years!!! MP2: 0.1 sec MP2: 1 year
DFT: 0.1 sec CCSD(T): 0.1 sec DFT: 5 hours CCSD(T): 2000
silicon silicon 2 atoms/unit cell 100 atoms/unit cell
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Why DFT?
42
100,000
10,000
1000
100
10
2003 2007 2011 20151
Year
Num
ber o
f Ato
ms
Linear
scalin
g DFT
DFT
QMC
CCSD(T)
Exact treatment
MP2
Image by MIT OpenCourseWare.
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wave function:
complicated!
electrdensity:on
easy!
Density functional theory
� = �(⇧r1, ⇧r2, . . . , ⇧rN )
n = n(⇤r)
Walter DFT Kohn 1964
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Density functional theory electron density n
ion Total energy is a functional
of the electron density.
E[n] = T [n] + Vii + Vie[n] + Vee[n] kinetic ion-electron
ion-ion electron-electron
© source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse..44
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Density functional theory E[n] = T [n] + Vii + Vie[n] + Vee[n]
kinetic ion-ion ion-electron electron-electron
electron density n(⌅r) = �
|ci(⌅r)|2
i
Eground state = min E[n] c
Find the wave functions that minimize the energy using a functional derivative.
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Density Functional Theory Finding the minimum leads to
Kohn-Sham equations
ion potential Hartree potential exchange-correlation potential
equations for non-interacting electrons 46
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Density functional theory
Only one problem: vxc not known! approximations necessary
local density general gradient approximation approximation
LDA GGA
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3.021j Introduction to Modeling and Simulation N. Marzari (MIT, May 2003)
Iterations to selfconsistency
�
-
n(⌅r) = i
|ci(⌅r)|2
scf loop
Self-consistent cycle Kohn-Sham equations
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Modeling software name license basis functions pro/con
ABINIT free plane waves
very structured
ONETEP pay Wannier functions
linear scaling
Wien2k pay Ylm + plane waves
very accurate
VASP pay plane waves
fast
PWscf free plane waves
fast
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�
Basis functions ⇥ =
� ciciMatrix eigenvalue equation:
i expansion inH1 = E1
orthonormalized basis functions
H �
cici = E �
cici i i ⇥ ⇥
d⌥r ��H �
ci�i = E d⌥r �� �
ci�ij j i i
Hjici = Ecj i
H⇤c = E⇤c 50
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Plane waves as basis functions
plane wave expansion: �(⇧r) = �
j Cutoff for a maximum G is necessary and results in a finite basis set.
Plane waves are periodic, thus the wave function is periodic!
plane wave
cj e i ⇧Gj ·⇧r
periodic crystals: atoms, molecules: Perfect!!! (next lecture) be careful!!!
51
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Put molecule in a big box
unit cell
Make sure the separation is big enough, so that we do not include
artificial interaction!
52
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DFT calculations sc
f loo
p total energy = -84.80957141 Rytotal energy = -84.80938034 Rytotal energy = -84.81157880 Rytotal energy = -84.81278531 Rytotal energy = -84.81312816 Ry exiting loop; total energy = -84.81322862 Ry result precise enough total energy = -84.81323129 Ry
1) electronic charge density At the end we get:
2) total energy
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• structure
• bulk modulus binding energies
reaction paths • shear modulus forces
• elastic constants pressure
stress • vibrational properties
... • sound velocity
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point?
Convergence
Was my box big
enough?
Was my basis bigenough?
Did I exit the scf
loop at the right
?55
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PWscf input water.input
What atoms are involved? Where are the atoms sitting? How big is the unit cell? At what point do we cut the basis off? When to exit the scf loop?
All possible parameters are described in
INPUT_PW.
&controlpseudo_dir = ''”/&systemibrav = 1 celldm(1) = 15.0nat = 3ntyp = 2occupations = 'fixed' ecutwfc = 60.0/&electronsconv_thr = 1.0d-8/ATOMIC_SPECIESH 1.00794 hydrogen.UPFO 15.9994 oxygen.UPFATOMIC_POSITIONS {bohr}O 0.0 0.0 0.0H 2.0 0.0 0.0H 0.0 3.0 0.0 K_POINTS {gamma}
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Review • Review
• The Many-body Problem
• Hartree and Hartree-Fock
• Density Functional Theory
• Computational Approaches
• Modeling Software
• PWscf
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Literature
• Richard M. Martin, Electronic Structure
• Kieron Burke,The ABC of DFT chem.ps.uci.edu/~kieron/dft/
• wikipedia,“many-body physics”,“density functional theory”,“pwscf”, “pseudopotentials”, ...
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