Calculating the properties of materials from first principles
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Transcript Calculating the properties of materials from first principles
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
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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
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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!
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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:
Edinburgh, May 2010