Functional Theory and Practical Application to Alloys

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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?

Introduction to Computer Simulation:
Edinburgh, May 2010

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

Introduction to Computer Simulation:
Edinburgh, May 2010

Aim of
ab initio
calculations

Introduction to Computer Simulation: Edinburgh, May
2010

Atomic Numbers

Solve
quantum
mechanics

for
the

material

Predict physical and chemical

properties of systems

From first principles

Research

output

The equipment

Application

“Base

Theory”

(DFT)

Implementation

(the algorithms

and program)

Setup model,

run the code

Scientific

problem
-

solving

“Analysis

Theory”

Introduction to Computer Simulation:
Edinburgh, May 2010

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:
Edinburgh, May 2010

The starting point

As you can see, quantum mechanics is “simply” an eigenvalue problem

Introduction to Computer Simulation:
Edinburgh, May 2010

Set up the problem

Let’s start defining various quantities

Assume that the nuclei (Mass M
i
) are at:


R
1
, R
2
, …, R
N

Assume that the electrons (mass m
e
) are at:

r
1
, r
2
, …,
r
m

Now let’s put some details in the SE

Introduction to Computer Simulation:
Edinburgh, May 2010

Summary of problem to solve

Where

Introduction to Computer Simulation:
Edinburgh, May 2010

The full problem


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

Introduction to Computer Simulation:
Edinburgh, May 2010

Model Systems

Introduction to Computer Simulation:
Edinburgh, May 2010


In this kind of first
-
principles
calculation


Are 3D
-
periodic


Are small: from one atom to a few

thousand
atoms


Supercells


Periodic boundaries


Bloch functions

Bulk

alloy

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




Nuclear problem is separable (and, as we
know, the nucleus is merely a point charge!)


Introduction to Computer Simulation:
Edinburgh, May 2010

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
e
N


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 10
26

electrons

Introduction to Computer Simulation:
Edinburgh, May 2010

Density functional theory


Let’s write the Hamiltonian operator in the
following way:



T is the kinetic energy terms


V is the potential terms external to the electrons


U is the electron
-
electron term


so we’ve just classified it into different
‘physical’ terms


Introduction to Computer Simulation:
Edinburgh, May 2010

The electron density


The electronic charge density is given by



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
Q

and
then get
n(r
).


Introduction to Computer Simulation:
Edinburgh, May 2010

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?

Introduction to Computer Simulation:
Edinburgh, May 2010

Method behind DFT


Assume two potentials V and V’ lead to the
same ground state density:






We can do the same again interchanging the
dashed and
undashed

quantities thus:

Introduction to Computer Simulation:
Edinburgh, May 2010

Unique potential


If we add these two final equations we are left
with the contradiction



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
).

Introduction to Computer Simulation:
Edinburgh, May 2010

Change of emphasis in QM


But by the definition of the lowest energy
state we must have






And so the ‘variational principle’ tells us how
to solve the problem

Introduction to Computer Simulation:
Edinburgh, May 2010

Don’t bother with the wavefunction!


Express the problem as an energy




And solve
variationally

with respect to the
density





Degrees of freedom

in the density,
n
,

versus energy
E[n
]

Introduction to Computer Simulation:
Edinburgh, May 2010

QM using DFT

N
-
body Schrödinger Equation

Density functional theory (Kohn
-
Sham equations)

Both equations

Introduction to Computer Simulation:
Edinburgh, May 2010

Kohn
-
Sham Equations


Let’s collect all the terms into one to simplify




Where


i

labels each particle in the system


V
i
KS

is the potential felt by particle
i

due to
n(r
)


n(r
) is the charge density

Introduction to Computer Simulation:
Edinburgh, May 2010

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:
Edinburgh, May 2010

Variational Method

Introduction to Computer Simulation:
Edinburgh, May 2010


Schrödinger’s Equation


And the of use the Variational Principle


Minimising
this

within DFT

Solve this

by

DFT: The XC approximation

Introduction to Computer Simulation:
Edinburgh, May 2010


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 ???

Definition of XC

Introduction to Computer Simulation:
Edinburgh, May 2010

Exact XC interaction is unknown

This would be excellent if only we knew what n
xc

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
n
xc
(r,r’)

at
r’
.

Within DFT we can write the
exact

XC interaction as

Exchange
-
Correlation Approximations

Introduction to Computer Simulation:
Edinburgh, May 2010

A simple
, but effective approximation to the exchange
-
correlation
interaction is

The
generalised gradient approximation

contains the next term in a
derivative expansion of the charge density:

Hierarchy of XC
appoximations

Introduction to Computer Simulation:
Edinburgh, May 2010


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.

Zoo of XC approximations

Introduction to Computer Simulation:
Edinburgh, May 2010

LDA

Semi
-
Empirical

RPBE

WC

WDA

CI

PW91

PBE

EXX

sX

CC

B3LYP

PBE0

Meta
-
GGA

HF

MP2

MP4

SDA

OEP

Structure Determination

Introduction to Computer Simulation:
Edinburgh, May 2010


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


Nuclear Positions?


Up until now we assume we know nuclear
positions, {
R
i
}


What if we don’t?


Guess them or take hints from experiment


Get zero of force
wrt

{
R
i
}:


Introduction to Computer Simulation:
Edinburgh, May 2010

Forces


If we take the derivative then:

Product rule

Product rule

Introduction to Computer Simulation:
Edinburgh, May 2010

Forces II


In DFT we have




But only
V
n
-
e

and
V
n
-
n

depends on R and
derivative are taken analytically


We get forces for free!


Optimise under F to obtain {
R
i
}


Add in nuclear KE to obtain finite temperature

Introduction to Computer Simulation:
Edinburgh, May 2010

Example: Lattice Parameters

Introduction to Computer Simulation:
Edinburgh, May 2010

Volume

Energy

Phase I

Phase II

Common tangent gives
transition pressure:

P=
-
dE/dV

V
II

V
I


KS equations can be solved to give energy, E


What does that tell us?

Structures without experiment?

Introduction to Computer Simulation:
Edinburgh, May 2010

U(x)

x

start

stop

Relative energies of structures: examine phase
stability

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

Introduction to Computer Simulation:
Edinburgh, May 2010

Alloys


Alloys are complicated!

Introduction to Computer Simulation:
Edinburgh, May 2010

Phase separated

Ordered

Random

Ordered


Ordered alloys are “easy” (usually!)


The have a repeated unit cell

Introduction to Computer Simulation:
Edinburgh, May 2010

Can perform calculation on
this unit cell:


Electronic structure


Band Structure


Density of States


Etc…

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 A
x
B
1
-
x


Encapsulates only average quantities

Introduction to Computer Simulation:
Edinburgh, May 2010

Supercell Approach


Need large unit cell


Computationally
expensive


A lot of atoms


Require check on
statistics (how many
possible random
configurations?)


Introduction to Computer Simulation:
Edinburgh, May 2010

Virtual Crystal Approximation


What is an “average” atom?








Put
x

of one atom and
1
-
x

of the other atom at
every site

Introduction to Computer Simulation:
Edinburgh, May 2010

VCA Example


NbC
1
-
x
N
x


C and N are
disordered


How does
electronic
structure vary
with
x
?

Introduction to Computer Simulation:
Edinburgh, May 2010

NbC
1
-
x
N
x

Electronic
DoS

Introduction to Computer Simulation:
Edinburgh, May 2010

Electron by electron

Introduction to Computer Simulation:
Edinburgh, May 2010

Relation to experiment


We solve the problem and get energy, E and
density
n(r
)


experiments don’t measure these!


An experiment:

Radiation or

Particle (
k
)

Material (
q
)

“Different”

Radiation or

Particle

(
k
-
q
)

Introduction to Computer Simulation:
Edinburgh, May 2010

Perturbation Theory


Based on compute how the total energy
responds to a perturbation, usually of the DFT
external potential
v


Expand quantities (
E,
n
,
y
,
v
)




Properties given by the derivatives

Introduction to Computer Simulation:
Edinburgh, May 2010

The Perturbations


Perturb the external potential (from the nuclei and
any external field):


Nuclear

positions




phonons


Cell vectors





elastic constants


Electric fields





dielectric response


Magnetic fields





NMR



But not only the potential, any perturbation to the
Hamiltonian:


d/d
k






a
tomic

charges


d/d(PSP
)





alchemical perturbation


Introduction to Computer Simulation:
Edinburgh, May 2010

Example: Phonons


Perturb with respect to nuclear coordinates:





This is equivalent to

Introduction to Computer Simulation:
Edinburgh, May 2010

Atomic Motion

Eigenvectors of 2
nd

order energy give nuclear motion under phonon excitations

Introduction to Computer Simulation:
Edinburgh, May 2010

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