Quantum Mechanics: Density
Functional Theory and Practical
Application to Alloys
Stewart Clark
Condensed Matter Section
Department of Physics
University of Durham
Outline
• Aim: To simulate real materials and
experimental measurements
• Method: Density functional theory and high
performance computing
• Results: Brief summary of capabilities and
performing calculations
Introduction to Computer Simulation:
Edinburgh, May 2010
What would we like to achieve?
• Computers get cheaper and more powerful every year.
• Experiments tend to get more expensive each year.
• IF computer simulation offers acceptable accuracy then
at some point it should become cheaper than
experiment.
• This has already occurred in many branches of science
and engineering.
• Possible to achieve this for properties of alloys?
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Property Prediction
•Property calculation of alloys provided link with
experimental measurements:
-
For analysis
For scientific/technological interest
•To enable interpretation of experimental results
•To predict properties over and above that of
experimental measurements
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Aim of ab initio calculations
Atomic Numbers
Solve quantum mechanics
for the material
Predict physical and chemical
properties of systems
Introduction to Computer Simulation: Edinburgh, May
2010
From first principles
The equipment
Application
Scientific
problemsolving
“Base
Theory”
(DFT)
Implementation
(the algorithms
and program)
Setup model,
run the code
“Analysis
Theory”
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Research
output
Properties of materials
• Whole periodic table.
• Periodic units containing thousands of atoms
(on large enough computers).
• Structural optimisation (where are the
atoms?).
• Finite temperature (atomic motion).
• Lots of others…if experiments can measure it,
we try to calculate it – and then go further…
• Toolbox for material properties
Introduction to Computer Simulation:
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The starting point
ˆ
H  E
As you can see, quantum mechanics is “simply” an eigenvalue problem
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Set up the problem
Let’s start defining various quantities
Assume that the nuclei (Mass Mi) are at:
R1, R2, …, RN
Assume that the electrons (mass me) are at:
r1, r2, …, rm
Now let’s put some details in the SE
Hˆ  R1 , R 2 ,..., R N , r1 , r2 ,..., rm   E R1 , R 2 ,..., R N , r1 , r2 ,..., rm 
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Summary of problem to solve
Hˆ N ,e N ,e
R ,r   E
I
i
N ,e
N ,e
R ,r 
I
i
Where

Hˆ N ,e  TˆN  Tˆe  VˆN  N  VˆN  e  Vˆe  e
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The full problem
2
 h 2   2
 i 
I




2
M
m
 I
I
e 

i


2
2
e
ZIe
1
 1 1



4  2
ri  R I
2
0 
i  j ri  r j
i,I


I J



 R1 , R 2 , ..., R N , r1 , r2 ,..., rm   E  R1 , R 2 , ..., R N , r1 , r2 , ..., rm 
2 
Z I Z J e 

R I  R J 

•Why is this a hard problem?
•Equation is not separable: genuine many-body
problem
•Interactions are all strong – perturbation
won’t work
•Must be Accurate --- Computation
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Model Systems

In this kind of first-principles
calculation
– Are 3D-periodic
– Are small: from one atom to a few
thousand atoms
Bulk alloy



Supercells
Periodic boundaries
Bloch functions
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Slab for surfaces

First simplification
• The electron mass is much smaller than the
nuclear mass
• Electrons remain in a stationary state of the
Hamiltonian wrt nuclear motion
Hˆ N ,e N ,e
R ,r    r  R 
I
i
N ,e
i
N ,e
I
• Nuclear problem is separable (and, as we
know, the nucleus is merely a point charge!)
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Electrons are difficult!
• The mathematical difficulty of solving the
Schrodinger equation increases rapidly with N
• It is an exponentially difficult problem
• The number of computations scales as eN
• With modern supercomputers we can solve
this directly for a very small number of
electrons (maybe 4 or 5 electrons)
• Materials contain of the order of 1026
electrons
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Density functional theory
• Let’s write the Hamiltonian operator in the
following way:
Hˆ  Tˆ  Vˆ  Uˆ
– T is the kinetic energy terms
– V is the potential terms external to the electrons
– U is the electron-electron term
we’ve just classified it into different
• so
‘physical’ terms
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The electron density
• The electronic charge density is given by
n ( r) 

...   r1 , r2 ,..., rn  r1 , r2 ,..., rn dr 2 dr 3 ...dr n
*
• so integrate over n-1 of the dimensions gives
the probability, n(r), of finding an electron at r

• This is (clearly!) a unique functional of the
external potential, V
• That is, fix V, solve SE (somehow) for  and
then get n(r).
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DFT
• Let’s consider the reverse question: for a given
n(r), does this come from a unique V?
• Can two different external potentials, V and V’,
give rise to the same electronic density?
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Method behind DFT
• Assume two potentials V and V’ lead to the
same ground state density:
E   |H |
 ' | H | '  ' | H ' V  V ' | '
 ' | H ' | '  ' | V  V ' | '
 E '  ' | V  V ' | '
• We can do the same again interchanging the
dashed and undashed quantities thus:

E '  E   | V ' V | 
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Unique potential
• If we add these two final equations we are left
with the contradiction
E ' E  E  E '
• so our initial assumption must be incorrect
• That is, there cannot be two different external
potentials that lead to the same density
• 
We have a one-to-one correspondence
between density, n(r), and external potential,
V(r).
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Change of emphasis in QM
• But by the definition of the lowest energy
state we must have
' | H | '   | H | 
' | F | '  ' | V | '   | F |    | V | 
 n ' ( r)V ( r) dr  F n ' ( r)    n ( r)V ( r) dr  F n ( r) 
E  n ' ( r)   E  n ( r) 
• And so the ‘variational principle’ tells us how
to solve the problem
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
Don’t bother with the wavefunction!
• Express the problem as an energy
E  n   Ts n   E H  n   V n   E xc n 
• And solve variationally with respect to the
density
Degrees of freedom
in the density, n,
versus energy E[n]
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QM using DFT
N-body Schrödinger Equation

2
2
h
1
e
2

 
 2 m
2 4  0
e


i j
n
1
ri  r j
exact

r  
e
2
4  0
ZI


ri  R I
i,I
1
e
2

2 4  0
I J

 r , r ,...   E r , r ,... 
1 2
1 2
R I  R J 

ZIZJ
 ...   r , r , r ,...  r , r , r ,...dr
*
1
2
3
1
2
3
2
dr 3 ...
Density functional theory (Kohn-Sham equations)
2
 h
1 e
2
i 


2 4  0
 2 m e

n ( r' )
r  r'
dr ' 
e
2
4  0
n
DFT

I
Z I n (r ' )
r'  R I
dr ' 
1
e
2
2 4  0

I J
r     i* r  i r 
i
Both equations

n
exact
r  
n
DFT
r 
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ZIZJ
RI  RJ

  xc n r  i r    i  i  r 

Kohn-Sham Equations
• Let’s collect all the terms into one to simplify
 1 2

KS
  i  V i n r  i  r    i i  r 
 2

• Where
– i labels each particle in the system
KS is the potential felt by particle i due to n(r)
–
V

i
– n(r) is the charge density
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Kohn-Sham Equations
• The Kohn-Sham (KS) equations are formally
exact
• The KS particle density is equal to the exact
particle density
• We have reduced the 1 N-particle problem to
N (coupled) 1-particle problems
• We can solve 1-particle problems!
Introduction to Computer Simulation:
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Variational Method


Schrödinger’s Equation
And the of use the Variational Principle
Solve this
Hˆ   E 
by
Minimising this
within DFT
E 
 Hˆ 
 
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DFT: The XC approximation
• Basically comes from our attempt to map 1 N-body
QM problem onto N 1-body QM problems
• Attempt to extract single-electron properties from
interacting N-electron system
• These are quasi-particles
“DFT cannot do…” : This statement is dangerous and usually ends incorrectly (in
many publications!)
Should read:
“DFT using the ??? XC-functional can be used to calculate ???, but that particular
functional introduces and error of ??? because of ???
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Definition of XC
Exact XC interaction is unknown
Within DFT we can write the exact XC interaction as
E xc [ n ] 
1
n(r )

2
n xc ( r , r ' )
| r  r '|
drdr '
This would be excellent if only we knew what nxc was!
This relation defines the XC energy.
It is simply the Coulomb interaction between an electron an r and the value
of its XC hole nxc(r,r’) at r’.
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Exchange-Correlation Approximations
A simple, but effective approximation to the exchange-correlation
interaction is
E
LDA
xc
[ n ( r )] 
 n ( r )
hom
xc
[ n ( r )] dr
The generalised gradient approximation contains the next term in a
derivative expansion of the charge density:
GGA
E xc
[ n ( r )] 
n
(
r
)

xc

GGA
[ n ( r ),  n ( r )] dr
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Hierarchy of XC appoximations
• LDA depends only on one variable (the density).
• GGA’s require knowledge of 2 variables (the density and
its gradient).
• In principle one can continue with this expansion.
• If quickly convergent, it would characterise a class of
many-body systems with increasing accuracy by functions
of 1,2,6,…variables.
• How fruitful is this? As yet, unknown, but it will always be
semi-local.
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Zoo of XC approximations
LDA
B3LYP
RPBE
WC
SDA
Meta-GGA
PW91
Semi-Empirical
sX
EXX
PBE0
PBE
WDA
HF
OEP
MP2
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CI
MP4
CC
Structure Determination
–
–
–
–
Minimum energy corresponds to zero force
Much more efficient than just using energy alone
Equilibrium bond lengths, angles, etc.
Minimum enthalpy corresponds to zero force and
stress
– Can therefore minimise enthalpy w.r.t. supercell
shape due to internal stress and external
pressure
– Pressure-driven phase transitions
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Nuclear Positions?
• Up until now we assume we know nuclear
positions, {Ri}
• What if we don’t?
• Guess them or take hints from experiment
• Get zero of force wrt {Ri}:
F   R E (R)
E  H
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Forces
• If we take the derivative then:
 E
R



H
R

R
H  
H
R
  H

R
 
 
H


 E
|   |
 

 R 
R
  R
 E
 

R
H
R
 |  
H
R


H
R

  E 
Product rule


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Product rule
Const   0

Forces II
• In DFT we have
Hˆ ( r, R )  
h
2
2m
 r  V e  e n r   V n  e n r, R   V n  n n R   V xc n r 
• But only Vn-e and Vn-n depends on R and
derivative are taken analytically
• We get forces for free!
• Optimise under F to obtain {Ri}
• Add in nuclear KE to obtain finite temperature
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Example: Lattice Parameters
Energy
•KS equations can be solved to give energy, E
•What does that tell us?
Common tangent gives
transition pressure:
Phase II
P=-dE/dV
Phase I
VII
VI
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Volume
Structures without experiment?
U(x)
start
stop
x
Relative energies of structures: examine phase
stability
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Summary so far
• Can get electronic density and energy
• Can use forces (and stresses) to optimise
structure from an “intelligent” initial guess
• Minima of energy gives structural phase
information
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Alloys
• Alloys are complicated!
Phase separated
Ordered
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Random
Ordered
• Ordered alloys are “easy” (usually!)
• The have a repeated unit cell
Can perform calculation on
this unit cell:
•Electronic structure
•Band Structure
•Density of States
•Etc…
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Disordered
• There are two main approaches:
• The Supercell approach
– Make a large unit cell with species randomly
distributed as required
– Characterises microscopic quantities
• The Virtual Crystal Approximation (VCA)
– Make each atom behave as if it were an average of
various species AxB1-x
– Encapsulates only average quantities
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Supercell Approach
• Need large unit cell
• Computationally
expensive
• A lot of atoms
• Require check on
statistics (how many
possible random
configurations?)
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Virtual Crystal Approximation
• What is an “average” atom?
• Put x of one atom and 1-x of the other atom at
every site
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VCA Example
• NbC1-xNx
• C and N are
disordered
• How does
electronic
structure vary
with x?
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NbC1-xNx Electronic DoS
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Electron by electron
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Relation to experiment
• We solve the problem and get energy, E and
density n(r) – experiments don’t measure these!
• An experiment:
“Different”
Radiation or
Particle
(k-q)
Radiation or
Particle (k)
Material (q)
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Perturbation Theory


Based on compute how the total energy
responds to a perturbation, usually of the DFT
external potential v
Expand quantities (E, n, , v)
E  E
(0)
 E
(1 )
 E
2
(2)
 ...
• Properties given by the derivatives
1  E
n
E
(n)

n! 
n
1 E
2
and in particular
E
(2)
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
2 
2
The Perturbations

Perturb the external potential (from the nuclei and
any external field):
• Nuclear positions
• Cell vectors
• Electric fields
• Magnetic fields





phonons
elastic constants
dielectric response
NMR
But not only the potential, any perturbation to the
Hamiltonian:
• d/dk
• d/d(PSP)


atomic charges
alchemical perturbation
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Example: Phonons
• Perturb with respect to nuclear coordinates:
E  E0 
E
lk 
u lk  
l ,k , 
1
2!
E
l ,k , 
lk  l ' k ' 
u lk  u l ' k '   K
l ',k ',
• This is equivalent to

f ( x )  f x 0  

2
df
dx
x
x0
1 d f
2! dx
x K
2
2
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x0
Atomic Motion
Eigenvectors of 2nd order energy give nuclear motion under phonon excitations
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Summary
• First principles electronic structure calculations:
 Extremely powerful technique in condensed matter
• Applicable to many different sciences






Condensed Matter Physics
Chemistry
Biology
Material Science
Surfaces
Geology
• Simulate experimental measurements
• Computationally possible, but still requires good
computer resources
Introduction to Computer Simulation:
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