Instructor :
Class :
Grade :
Prof. Rampi Ramprasad
Wednesday, 5:00 pm – 8:00 pm, Gentry 119E
Homework (50%), Midterm (25%), Final term presentation/paper (25%)
Course Objectives :
This course is intended to provide an overview of the theory and practices underlying modern electronic structure materials computations, primarily density functional theory (DFT). Students involved primarily/partially in materials computations, as well as those focused on experimental materials research wishing to learn about DFT techniques will benefit from this course . Several case studies will be presented. The course will culminate in a term project that will provide the students with an opportunity to address a problem close to their research using an appropriate computational method. Prior programming experience is not essential. This course is not about curve fitting, data analysis or data visualization!
Primary Text :
Density Functional Theory: A Practical Introduction , D. Sholl & J. Steckel, Wiley (2009).
Supplementary Texts :
Methods of Electronic-Structure Calculations: From Molecules to Solids , M. Springborg, Wiley (2000).
Electronic Structure Calculations for Solids and Molecules , Jorge Kohanoff, Cambridge University
Press (2006).
Electronic Structure – Basic Theory and Practical Methods, R. Martin, Cambridge University Press
(2004).

• Introduction – the course in a nutshell (Chap. 1)
• Theory – From Quantum Mechanics to Density Functional Theory (Chap. 1)
• Density Functional Theory (DFT) – nuts & bolts (Chap. 3)
–
Reciprocal space total energy formalism
– Approximations – k-point sampling, pseudopotentials, exchange-correlation
• Simple molecules & solids
– Structure (Chap. 2)
–
Geometry optimization (Chap. 3)
– Vibrations (Chap. 5)
–
Electronic structure (Chap. 8)
• Surface science
–
Periodic boundary conditions, relaxation, reconstruction, surface energy (Chap. 4)
– Chemisorption and reaction on surfaces (Chap. 4, 5, 6)
• Chemical processes & transition state theory (Chap. 6)
• Non-zero temperatures – Thermodynamics
– Phase diagrams (Chap. 7)
–
Thermal properties
– specific heat, thermal expansion
• Response functions – elastic, dielectric, piezoelectric constants
• Beyond standard DFT
–
Ab initio molecular dynamics (Chap. 9)
– DFT + X, DFT + U, GW (Chap. 10)
• Summary, outlook, accuracy of DFT (Chap. 10)

2. Overview of theory: Total energy methods and density functional theory (DFT)
3. Predictions of known properties using DFT (i.e., validation)
4. Practical value of DFT calculations: Insights, and design of new materials (i.e., success stories)

•
Anytime the future has to be predicted or forecasted, simulation is used, generally based on well understood scientific notions/principles
• Anytime experimental analysis is too expensive or too impractical, simulation becomes necessary
• Simulations complement experiments and could provide insights
•
Examples from everyday contemporary experience
– Weather modeling involves solution of Newton’s equations of motion and fluid dynamics
– Astrophysical predictions (eclipses, comets) involve solution of Newton’s equations of motion and/or general relativity
– Other notable examples: economic (stock market) modeling, drug design, mechanical properties (auto industry), electromagnetic simulations (microelectronics industry)
•
Challenges:
– Models by themselves may not be representative of the real situation
– Practical treatment of model (or numerical solution) is time intensive
– Sometimes the physical principles (or theory) involved are not well known
– Unknown extraneous factors, e.g., stock market
– Major numerical problems – non-linear systems, e,g, chaotic pendulum, weather
•
Fortunately, in Computational Materials Science (CMS), we need to worry mainly about the first two challenges, and the others are listed in decreasing order of importance

•
Theory & experiment go hand in hand.
– A set of results may come out of experiment, but one needs a theory to put it all in a “framework of understanding”. A theory cannot be formulated in the absence of experimental data
– Goal of science: construct theory based on available experimental data, make predictions using theory outside the regime of experimental input, and modify theory if predictions are not satisfactory
•
A model is a representation of physical reality, along with a set of assumed equations that govern that reality
• Simulation is the process of using the model using numerical techniques
• CMS involves theory, modeling and simulation, with the terminologies generally used interchangeably!
Experiments
(Observations of Reality)
Validation
Theory
(Understanding of Reality)
Models of Reality
(in terms of atoms, electrons, etc.)
Simulations

• Let us consider an example: tensile testing
– In the elastic region, we know that there is a linear relationship between stress and strain, which is at the heart of elasticity theory; merely having experimental data on stress versus strain for a few materials does not constitute true understanding; the experimental data together with the realization that stress = constant*strain constitutes understanding
– Now, one can do two types of computations: (1) use the constant obtained from experimental data to look at complicated geometries (FEM, used widely in the auto industry), or (2) we can use a more fundamental theory to determine the constant from first principles; the second approach results in a even more fundamental understanding of the origin of the constant, namely, in terms of atomic level bond stretching
• CMS sometimes complements experimental studies, and sometimes provides insights, and increasingly is being used to design materials

Time
Years
Hours
Minutes
Seconds
Microseconds
Nanoseconds
Picoseconds
Femtoseconds
Quantum
Mechanics
Molecular
Mechanics
Mesoscale
Modeling
(Semi-classical)
Finite element
Analysis
(Continuum/classical)
Engineering
Design
Å m m nm mm m Distance e.g., density functional theory (DFT)
• If more than one “box” is involved in a computation  multi-scale modeling

• Central themes
– Our system is represented as a collection of atoms, or a collection of electrons and ions
– We can determine the total potential energy of this collection of particles
– Equilibrium (stable, unstable and metastable) situations correspond to features (minima, maxima, saddlepoint) in the total potential energy function
• Analogous approaches in other types of computations
– Electromagnetic simulations involve minimization of electromagnetic energy density
– Mechanical simulations involve minimization of strain energy
• Let us for a moment assume that we do have a prescription for computing the total energy of a group of atoms, given their spatial positions …
– What kind of properties can we compute? And how?

• Only one degree of freedom  R
AB
• If we new E(R
AB
), then we can determine potential energy surface (PES)
Energy
Curvature = m w
2 m = m
A m
B
/(m
A
+m
B
)
Note: slope = -force
Bond energy
R
0
R
AB
Thus, IF E(R
AB
) is known, then we can trivially determine equilibrium bond length, bond energy and vibrational frequency!

• Consider the reaction: A-B + C  A + B-C
• Two degrees of freedom in 1-dimensional A-B-C system  R
AB
, R
BC
• If we new E(R
AB
, R
BC
), then we can determine potential energy surface (PES)
AB + C atoms
E
E A + BC
Activation barrier
Enthalpy
Along dotdashed line (“reaction coordinate”)
Transition state (saddle point)

• Only one degree of freedom  lattice parameter a, or Volume V (= a 3 )
• If we new E(V), then we can determine potential energy surface (PES)
Energy
Curvature = B/V
0
B = bulk modulus
Note: slope = stress
Cohesive energy
V
0
V
Thus, IF E(V) is known [equation of state], then we can trivially determine equilibrium lattice parameter, cohesive energy and bulk modulus!

1. Introductory comments
3. Predictions of known properties using DFT (i.e., validation)
4. Practical value of DFT calculations: Insights, and design of new materials (i.e., success stories)

• Until ~ 1950s, no reliable prescription was available to practically compute the energy
– [breakthrough: quantum mechanics, 1920s; Wigner & Seitz, 1930s; more later]
• Hence energy as a function of geometry was “parameterized” using experimental data then, and even to-date!
– Lennard-Jones, Morse, etc. (physicists), embedded-atom method, etc.
(materials scientists), force fields (chemists)
– Referred to as empirical or semi-empirical methods (as experimental data was used entirely or partially)
• Today, reliable parameter-free methods are available to compute energy, which come with a (rapidly diminishing) price tag of large computational time
– Density functional theory (DFT), and higher level quantum mechanics based methods

• Suppose that our system contains M atoms, and that atoms interact “pairwise”
E

1
2
 i i , j
 j
V ( r ij
)
[summation over i and j run over the number of atoms M]
• Lennard-Jones: 
• Morse:
V ( r ij
)

A r ij
6

B r ij
12
V ( r ij
)  V
0
 e
 2 d ( r ij
 r
0
)  2 e
 d ( r ij
 r
0
)

• By fitting A and B (Lennard-Jones), or V
0
, d and r
0
(Morse) to experimental data so that equilibrium bulk properties are
• We could make the scheme more sophisticated by defining E in terms of 3- or many-atom interactions (e.g., embedded atom method) or angular (e.g., Stillinger-Weber)


• Building blocks are N electrons and M nuclei, rather than M atoms
• The N-electron, M-nuclei Schrodinger (eigenvalue) equation:

( r
1
, r
2
,..., r
N
, R
1
, R
2
,..., R
M
)

E

( r
1
, r
2
,..., r
N
, R
1
, R
2
,..., R
M
)
The N-electron, M-nuclei wave function
The total energy that we seek
The N-electron, M-nuclei Hamiltonian
  
M 
I

1
2
2 M
I

I
2 
N  i

1
2 m
2
 i
2 
1
2
M 
I

1
M 
J

I
Z
I
R
I
Z e
J

R
J
2

1
2
N  i

1
N  j

I r i e
2
 r j

M  N 
I

1 i

1 e
2
R
Z
I
I
 r i
Nuclear kinetic energy
Electronic kinetic energy
Nuclear-nuclear repulsion
Electron-electron repulsion
Electron-nuclear attraction
• The problem is completely parameter-free , but formidable! Why?
– Cannot be solved analytically when N+M > 3 (really?!?)
– Too many variables (for a 100 atom Pt cluster, the wave function is a function of 23,000 variables!!!)



( r
1
, r
2
,..., r
N
, R
1
, R
2
,..., R
M
)

E

( r
1
, r
2
,..., r
N
, R
1
, R
2
,..., R
M
)
Density Functional Theory (DFT)
[W. Kohn, Chemistry Nobel Prize, 1999]
The “average” potential seen by electron i



 2
2 m

2

V eff
( r

)

 i
( r )
  i
 i
( r ) r
( r )
 occ 
 i
( r )
2 i
1-electron wave function
(function of 3 variables!)
1-electron energy
(band structure energy) r
(r) , or from
 i and
 i
( i labels electrons)
• Still parameter-free , but has a few acceptable approximations
• DFT is versatile: in principle, it can be used to study any atom, molecule, liquid, or solid (metals, semiconductors, insulators, polymers, etc.), at any level of dimensionality (0-d, 1-d, 2-d and 3-d)

1. Introductory comments
2. Overview of theory: Total energy methods and density functional theory (DFT)
4. Practical value of DFT calculations: Insights, and design of new materials (i.e., success stories)

[Yin and Cohen, PRB 26, 5668 (1982)]
Slope is transition pressure
• The correct equilibrium phase (diamond cubic) is predicted
• The lattice parameter of the equilibrium phase, and the pressure for the diamond cubic  beta-tin phase transition (common tangent) are also predicted to a good level of accuracy

• Structural details predicted typically to within 1% of experiments for a wide variety of solids and molecules
• Results from various sources [collected in
Ramprasad, Shi and
Tang, in Physics and
Chemistry of
Nanodielectrics,
Dielectric Polymer
Nanocomposites
(Springer)]

– Polarization = dipole moment per unit cell volume
– … inadequate … depends on how unit cell is defined
… unless we are in the
“Clausius-Mossotti” limit …
Each unit cell will give a different net dipole
Dipole per unit cell well defined
Resolution provided by Resta and King-Smith & Vanderbilt, in terms of phases of the wavefunctions (Berry’s phase)

Experiments
Boron Nitride
Tin
Kern et al, PRB 59, 8551 (1999) Pavone et al, PRB 57, 10421 (1998)

DFT

1. Introductory comments
2. Overview of theory: Total energy methods and density functional theory (DFT)
3. Predictions of known properties using DFT (i.e., validation)

• Extreme geophysical pressures may be difficult to create in the lab, but can be simulated easily

Liquid
Solid

• Can a material contract when heated?

• When heated to high temperatures in ultra high vacuum the surface atoms of the Si (111) surface rearrange to form the 7x7 reconstructed surface

Bi makes Cu-Cu bonds softer (hence, brittleness NOT due to electronic effects)
Grain boundary decohesion due to larger size of Bi atoms

• High activity of transition metals in oxidation catalysis is due to the presence of surface oxides under catalytic conditions

• Systems that can be represented in terms of up to a few 100 atoms per repeating unit cell are okay
– Geometric details to within 1% of experiments
– Other properties to within 5% of experiments
• Challenges
– Investigations requiring a large number of atoms
– Systems in which periodicity is absent
– Band gaps and excited state energies
– Non-zero and high temperatures
– Highly “correlated-electron” systems

1. Introductory comments
2. Overview of theory: Total energy methods and density functional theory (DFT)
3. Predictions of known properties using DFT (i.e., validation)
4. Practical value of DFT calculations: Insights, and design of new materials (i.e., success stories)

• Introduction – the course in a nutshell (Chap. 1)
• Theory – From Quantum Mechanics to Density Functional Theory (Chap. 1)
• Density Functional Theory (DFT) – nuts & bolts (Chap. 3)
–
Reciprocal space total energy formalism
– Approximations – k-point sampling, pseudopotentials, exchange-correlation
• Simple molecules & solids
– Structure (Chap. 2)
–
Geometry optimization (Chap. 3)
– Vibrations (Chap. 5)
–
Electronic structure (Chap. 8)
• Surface science
–
Periodic boundary conditions, relaxation, reconstruction, surface energy (Chap. 4)
– Chemisorption and reaction on surfaces (Chap. 4, 5, 6)
• Chemical processes & transition state theory (Chap. 6)
• Non-zero temperatures – Thermodynamics
– Phase diagrams (Chap. 7)
–
Thermal properties
– specific heat, thermal expansion
• Response functions – elastic, dielectric, piezoelectric constants
• Beyond standard DFT
–
Ab initio molecular dynamics (Chap. 9)
– DFT + X, DFT + U, GW (Chap. 10)
• Summary, outlook, accuracy of DFT (Chap. 10)