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
1
Part II Topics
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
2
It’s a Quantum World:The Theory of Quantum Mechanics
Quantum Mechanics: Practice Makes Perfect
From Many-Body to Single-Particle; Quantum Modeling of Molecules
Application of Quantum Modeling of Molecules: Solar Thermal Fuels
Application of Quantum Modeling of Molecules: Hydrogen Storage
From Atoms to Solids
Quantum Modeling of Solids: Basic Properties
Advanced Prop. of Materials:What else can we do?
Application of Quantum Modeling of Solids: Solar Cells Part I
Application of Quantum Modeling of Solids: Solar Cells Part II
Application of Quantum Modeling of Solids: Nanotechnology
Motivation
Last time: 1-electron
quantum mechanics
to describe spectral
lines
?
Image of NGC 604 nebula is in the public domain. Source: Hubble
Space Telescope Institute (NASA). Via Wikimedia Commons.
Image adapted from Wikimedia Commons, http://commons.wikimedia.org.
.
Today: many electrons
to describe materials.
?
3
Courtesy of Bernd Thaller. Used with permission.
Lesson outline
• Review
• The Many-body Problem
• Hartree and Hartree-Fock
• Density Functional Theory
• Computational Approaches
• Modeling Software
• PWscf
4
Review: Schrödinger
equation
H time independent:
in
f˙(t)
f (t)
=
1(⌃
r , t) = 1(⌃
r ) · f (t)
H1(r
r)
1(r
r)
H (⇧
r ) = E (⇧
r)
time independent Schrödinger equation
stationary Schrödinger equation
5
= const. = E
/(⌃
r , t) = /(⌃
r) · e
- i Et
Review:The hydrogen
atom
stationary
Schrödinger equation
Hl = El
T +V
2
e
v
l
o
ts
jus
2m
☺
2
2m
6
⇥
2
⇥2 + V
e2
4⇥
0r
⇥
⇥
⇥
=E
(⌃
r ) = E (⌃
r)
⇤(r) = E⇤(r)
Radial Wavefunctions
for a Coulomb V(r)
R10
r2R210
2
0.4
1
Z
0
0.2
0
1
2
3
4
R20
2
6
Thickness dr
y
R30
r
rR
2
0
1
2
3
4
r
2
6
10
r
0
2
6
10
r
0
2
6
10
r
2
21
0.15
0.1
0.05
0
rR
2
0.4
2
30
0.08
0.2
4
0
-0.1
8
12
16
0.04
r
R31
X
0.15
0.1
0.05
0
10
R21
r
0
r2R220
0.6
0.4
0.2
0
-0.2
0.12
0.08
0.04
0
0
r
0
0
4
8
12
16
r
0
4
8
12
16
r
0
4
8
12
16
r
r2R231
0.08
0.8
0.04
4
0
-0.04
8
12
16
r
0.4
0
r2R232
R32
0.04
0.8
0.02
0
0.4
0
4
8
12
16
r
0
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 the
first Bohr radius.
7
Images by MIT OpenCourseWare.
Angular Parts
Image by MIT OpenCourseWare.
8
Review:The hydrogen
atom
quantum numbers
n
1
2
2
2
l
0
0
1
1
ml
0
0
0
±1
a0 =
9
Atomic
Orbital
1s
ψn l m (r, θ, φ)
l
1
-r/a0
e
π a03/2
2s

1
r  -r/2a0
2
e
3/2 
4 2π a0  a0 
2p
1
r -r/2a0
e
cos θ
3/2 a
4 2π a0
0
2p
1
r -r/2a0
±iφ
e
sin
θ
e
8 π a03/2 a0
2
h = .0529 nm = first Bohr radius
2
me
Image by MIT OpenCourseWare.
10
Courtesy of David Manthey. Used with permission. Source: http://www.orbitals.com/orb/orbtable.htm.
Review:The hydrogen atom
11
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license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
Review:The hydrogen atom
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license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
12
Review: Next? Helium
e-
r1
+
r2
r12
e-
H1 = E1
H 1 + H2 + W
T1 + V1 + T2 + V2 + W
2
2m
⇥21
e2
4⇥
0 r1
2
2m
⇥22
cannot be solved analytically
13
e2
4⇥
0 r2
+
⇥
⇥
(⌥
r1 , ⌥
r2 ) = E (⌥
r1 , ⌥
r2 )
(r1 , r2 ) = E (r1 , r2 )
e2
4⇥
0 r12
⇥
⇤(r1 , r2 ) = E⇤(r1 , r2 )
problem!
Review: Spin
new quantum number: spin quantum number
for electrons: spin quantum number can ONLY be
up
14
down
Everything is spinning ...
Stern–Gerlach experiment (1922)
⇧
F
=
⇤E
⇧
= ⇤m
⇧ ·B
Image courtesy of Teresa Knott.
15
Everything is spinning ...
In quantum mechanics particles can have
a magnetic moment and a ”spin”
magnetic
moment
16
m
⇥
spinning
charge
Everything is spinning ...
conclusion from the
Stern-Gerlach experiment
for electrons: spin can ONLY be
up
17
down
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 excluded
from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
18
Pauli’s exclusions
principle
Two electrons in a system cannot have
the same quantum numbers!
hydrogen
quantum numbers:
main n: 1,2,3 ...
orbital l: 0,1,...,n-1
magnetic m: -l,...,l
spin: up, down
19
...
3s
2s
1s
... ...
3p 3d
2p
...
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 Creative
Commons
license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
20
Periodic table
This image is in the public domain. Source: Wikimedia Commons.
21
The many-body problem
helium: 2e
iron: 26e
er1 r 12
+
er2
=
(⇧
r1 , . . . , ⇧
rn )
Image © Greg Robson on Wikimedia Commons. License: CC-BY-SA. This content is excluded from
our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
22
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
The Multi-Electron
Hamiltonian
Hl = El
Remember
the good old days of the
1-electron H-atom??
2
2m
e2
⇥2
4⇥
0r
They’re
over!
2
⇥
⇤(r) = E⇤(r)
1
h
Zi e
1
e
h
2
2
∇ Ri + ∑∑
−
∇ ri − ∑∑
+ ∑∑
H = −∑
∑
2M i
2 i=1 j=1 ri − r j
2 i=1 j =1 Ri − R j 2m i=1
i=1
i=1 j=1 R i − r j
N
N N
2
Zi Z j e
2
n
N
n
n
2
i≠ j
kinetic energy of ions
n
2
i≠ j
kinetic energy of electrons
potential energy of ions
electron-ion interaction
electron-electron interaction
Multi-Atom-Multi-Electron Schrödinger Equation
H (R1 ,..., R N ;r1 ,..., rn ) Ψ (R1 ,..., R N ;r1 ,..., rn ) = E Ψ (R1 ,..., R N ;r1 ,..., rn )
Born-Oppenheimer Approximation
Source:
Discover
Magazine
Online
25
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Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
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
classically
• significantly simplifies the
Hamiltonian for the electrons:
Images in public domain.
Born
Oppenheimer
This term is just an external potential V(rj)
Zi e
1
e
h
2
∇ ri − ∑∑
+ ∑∑
H =−
∑
2m i=1
2 i=1 j=1 ri − r j
i=1 j=1 R i − r j
2
26
n
N
n
2
n
n
i≠ j
2
Solutions
quantum chemistry
Moller-Plesset
perturbation theory
MP2
27
density
functional
theory
coupled cluster
theory CCSD(T)
Hartree Approach
Write wavefunction as a simple product of single
particle states:
Ψ(r1,...,rn ) = ψ1 (r1 )ψ 2 (r2 )...ψ n (rn )
Hard
Product of Easy
Leads to an equation we can solve on a computer!
28
(
,
2
2
n
2
e ψ j (r ) *
* h 2
∇ +Vext (r) + ∑ ∫ dr
)−
-ψi (r) = εiψi (r )
rj − r *
* 2m
j=1
j≠i
+
.
Hartree Approach
(
,
2
n
e2 ψ j (r ) *
* h2 2
∇ +Vext (r) + ∑ ∫ dr
)−
-ψi (r) = εiψi (r )
rj − r *
* 2m
j=1
j≠i
+
.
The solution for each state depends on all the other
states€ (through the Coulomb term).
29
• 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
in(r) } = { Ψ out(r) } and repeat
if
different
set
{
Ψ
i
i
- if same, you are done
• obtain the self-consistent solution
Simple Picture...But...
Interacting
- -
Non-Interacting
-
After all this work, there is still one major problem:
the solution is fundamentally wrong
The fix brings us back to spin!
30
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.
31
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
Mathematically
• define the exchange operator:
χ12ψ1 (r1 )ψ2 (r2 ) = ψ1 (r2 )ψ2 (r1 )
• exchange operator eigenvalues are ±1:
€
suppose χ 12 φ = χφ
^!
χ 12 χ 12 φ = χ φ = φ
^!
^!
2
χ = 1, or χ = ±1.
2
33
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
Hartree-Fock
• Employing Hartree's approach, but
- enforcing the anti-symmetry condition
- accounting for spin
• Leads to a remarkable result:
(
2,
2
n
n
e ψ j (r ) *
* h2 2
∇ +Vext (r) + ∑ ∫ dr
)−
-ψi (r) − ∑δsi ,s j
rj − r *
* 2m
j=1
j=1
j≠i
j≠i
+
.
e2 *
∫ dr0 r0 − r ψ j (r0)ψi (r0)ψ j (r) = εiψi (r)
• Hartree-Fock theory is the foundation of molecular
35
orbital theory.
• It is based upon a choice of wavefunction that
guarantees antisymmetry between electrons.
it’s an emotional moment....
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
36
Solutions
quantum chemistry
Moller-Plesset
perturbation theory
MP2
37
density
functional
theory
coupled cluster
theory CCSD(T)
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?
38
Solving the Schrodinger Eq.
N
divide 3D space into
NxNxN=2x2x2 grid
N
N points
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Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
Ψ= Ψ(r1, ... , rn)
39
# points = N3n
n=# electrons
Ψ(N3n)
ρ(N3)
1
8
8
10
109
8
100
1090
8
1,000
10900
8
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]
40
Walter Kohn (left), receiving
the Nobel prize in chemistry
in 1998.
Why DFT?
computational expense
for system size N:
Quantum Chemistry
methods; MP2, CCSD(T)
O(N5-N7)
Density Functional Theory
silicon
2 atoms/unit cell
DFT: 0.1 sec
41
Example
CCSD(T): 0.1 sec
MP2: 0.1 sec
O(N3); O(N)
silicon
100 atoms/unit cell
DFT: 5 hours
CCSD(T): 2000
years!!!
MP2: 1 year
Why DFT?
100,000
10,000
e
Number of Atoms
n
Li
c
s
ar
1000
g
n
ali
T
F
D
DFT
MP2
100
QMC
CCSD(T)
10
1
42
Exact treatment
2003
2007
Year
2011
2015
Image by MIT OpenCourseWare.
Density functional
theory
=
(⇧
r1 , ⇧
r2 , . . . , ⇧
rN )
n = n(⇤
r)
:
n
o
i
t
c
n
u
f
e
wav licated!
comp
ele
de ctron
nsi
eas ty:
y!
Walter
Kohn
DFT
1964
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license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
43
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-ion
44
ion-electron
electron-electron
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Density functional theory
E[n] = T [n] + Vii + Vie [n] + Vee [n]
kinetic
ion-ion
ion-electron
electron density n(⌅r) =
i
electron-electron
r )|2
|ci (⌅
Eground state = min E[n]
c
Find the wave functions that minimize the
energy using a functional derivative.
45
Density Functional Theory
Finding the minimum leads to
Kohn-Sham equations
ion potential
Hartree potential
exchange-correlation
potential
equations for non-interacting electrons
46
Density functional theory
Only one problem: vxc not known!
approximations necessary
local density
approximation
LDA
47
general gradient
approximation
GGA
Self-consistent cycle
Kohn-Sham equations
Iterations to selfconsistency
sc
f lo
n(⌅
r) =
i
|ci (⌅
r )|
2
3.021j Introduction to Modeling and Simulation - N. Marzari (MIT, May 2003)
48
op
Modeling software
name
49
license
ABINIT
free
ONETEP
pay
Wien2k
pay
VASP
pay
PWscf
free
basis functions
pro/con
plane
very
waves
structured
Wannier
linear
functions
scaling
Ylm +
very
plane waves accurate
plane
fast
waves
plane
fast
waves
Basis functions
⇥=
Matrix eigenvalue equation:
i
expansion in
orthonormalized basis
functions
H1 = E1
ci ci = E
H
⇥
i
d⌥
r
jH
ci
i
i
=E
ci ci
⇥i
Hji ci = Ecj
i
H⇤
c = E⇤
c
50
ci ci
d⌥
r
ci
j
i
i
Plane waves as basis
functions
plane wave expansion:
⇧ j ·⇧
iG
r
(⇧
r) =
cj e
j
plane wave
Cutoff for a maximum G is necessary and results in a finite basis set.
Plane waves are periodic,
thus the wave function is periodic!
periodic crystals:
Perfect!!! (next lecture)
51
atoms, molecules:
be careful!!!
Image by MIT OpenCourseWare.
Put molecule in a big box
unit cell
Make sure the separation is big
enough, so that we do not include
artificial interaction!
Images by MIT OpenCourseWare.
52
scf loop
DFT calculations
total
total
total
total
total
total
total
energy
energy
energy
energy
energy
energy
energy
=
=
=
=
=
=
=
-84.80957141
-84.80938034
-84.81157880
-84.81278531
-84.81312816
-84.81322862
-84.81323129
At the end we get:
53
Ry
Ry
Ry
Ry
Ry
Ry
Ry
exiting loop;
result precise enough
1) electronic charge density
2) total energy
• structure
• bulk modulus
• shear modulus
• elastic constants
• vibrational properties
• sound velocity
54
binding energies
reaction paths
forces
pressure
stress
...
Convergence
g
i
b
x
o
b
y
m
s
a
?
h
W
g
u
o
en
?
Was m
y basis
big
enoug
h?
f
c
s
e
h
t
t
i
x
e
I
t
d
h
i
g
i
D
r
e
h
t
t
a
loop
?
t
n
i
po
55
PWscf 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.
56
water.input
&control
pseudo_dir
= ''”
/
&system
ibrav
=1
celldm(1)
= 15.0
nat
=3
ntyp
=2
occupations
= 'fixed'
ecutwfc
= 60.0
/
&electrons
conv_thr
= 1.0d-8
/
ATOMIC_SPECIES
H 1.00794 hydrogen.UPF
O 15.9994 oxygen.UPF
ATOMIC_POSITIONS {bohr}
O 0.0 0.0 0.0
H 2.0 0.0 0.0
H 0.0 3.0 0.0
K_POINTS {gamma}
Review
• Review
• The Many-body Problem
• Hartree and Hartree-Fock
• Density Functional Theory
• Computational Approaches
• Modeling Software
• PWscf
57
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”, ...
58
MIT OpenCourseWare
http://ocw.mit.edu
3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modelling and Simulation
Spring 2012
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