MAE715: Atomistic Modeling of Materials
Cornell University, Spring 2008
Meeting room and time: Mondays and Wednesdays (Rhodes 178) 4:30-5:45 pm
Lecture notes and literature material
- Introduction to the course: Course syllabus, requirements,
format of homeworks and project, textbooks, examples of atomistic simulations. Introduction to the fundamentals of Quantum Mechanics:
Departure from classical mechanics, eigenstates, time-independent Schrödinger equation,
free particle, particle in a box.
- More on the fundamentals of Quantum Mechanics:
Time-dependent Schrödinger equation,
harmonic oscillator, superposition state, quantum measurements and the collapse of the
wavefunction, Hamiltonian, the momentum operator, uncertainty principle, expectation values, etc.
- Angular momentum and the Hydrogen atom:
Angular momentum quantization, the Hydrogen atom, atomic orbitals, multi-electron configurations in other elements.
- Operators and approximation methods in
Quantum Mechanics:
Hermitian operators, commutation, general form of
uncertainty principle, spin dependence, the variational principle.
- Electron states in a molecule:
An application of the variational principle in diatomic molecules, the homo- and hetero-nuclear diatomic molecule,
bond order, bond energy, electronegativity.
- Energy Models: empirical potential methods, many-body potentials,
pair potentials, Lennard-Jones, Born-Mayer, Buckingham, Morse, fiting potentials, failures and successes of
potentials (metals, organic materials, oxides, etc.)
- Empirical potential methods, supercells, relaxation,
numerical methodology, convergence,
potentials for different materials classes, embedded atom methods (EAM), applications to metals, potentials for Si, applications to
Si-surface reconstruction, etc.
- Applications of empirical potential methods to organic
& biological materials, polymers, and oxides.
- Perturbation methods:
Time-independent and time-dependent perturbation theory.
- Adiabatic / Born-Oppenheimer approximation: Introduction to the
many interacting electron problem, the Schrödinger equation for the electrons, Hartree and Rydberg atomic units,
single- and two-particle operators, the total energy.
- Introduction to Hartree-Fock methods: The Hartree approximation,
Slatter
determinant, exchange, mean field approximation.
- Derivation of the Hartree-Fock equations.
- Hartree-Fock methods continued: Koopmans' theorem, the
Hartree-Fock-Roothaan method,
applications (successes and failures), restricted, unrestricted, extended, and projected Hartree-Fock methods.
- Selection of basis sets: Slater (atomic) orbitals, Gaussians, plane
waves, augmented plane waves, APW, ASW, muffin-tin-orbitals (LMTO).
- Semi-empirical methods: The Hückel and the extended Hückel method, the
Pariser-Parr-Pople (PPP) method, zero-differential overal (ZDO) methods, complete neglect of differential
overal (CNDO) methods, intermediate neglect of differential overlap (INDO and MINDO), etc.
- Introducing static correlation effects: Möller-Plesset (MP) perturbation
theory, configuration interaction (CI), multiple configuration methos (MC-SCF). A review of the notation for
creation and annihilation operators and wavefunctions for excited configurations is given here.
- Essentials of the free electron gas, Fermi energy, density of states,
Fermi-Dirac distribution, chemical potential, electronic heat capacity. Energy bands and general band structure, periodic potential and Bloch's theorem,
band gaps, crystal momentum, band structure representation.
- Introduction to density functional theory: The Thomas-Fermi approach,
local density approximation (LDA), the Hohenberg-Kohn theorems, the independent electron approximation, the
Kohn-Sham equations, exchange correlation.
- The total energy pseudopotential method, implementation
of DFT in periodic solids, basis selection (plane waves and Gaussians).
A review of
the total energy pseudopotential method is provided on this important article.
- Integration in k-space with some applications are discussed here.
A complete set of notes by K. Burke of the ABC of DFT is available. Additional
presentation material of DFT calculations (P. Kratzer),
the plane wave pseudopotential method (E. Pehlke),
and
pseudopotential construction (M. Fuchs) are given here.
- Notes on
spin polarization, performance of LDA/GGA on structure
prediction and energetics, local spin density approximation (LSDA), spin polarization density, error cancellation
in DFT.
- Non-zero temperature DFT calculations: T-dependence of
structure, phase diagrams, thermal expansion, elastic and other properties.
- Phonon calculations: Frozen phonon approximation,density functional
perturbation theory, thermodynamic properties of solids by phonons.
- Modeling direct dynamics vs. statistical approach: time limitations of
molecular dynamics, introducing a statistical mechanics approach, microscopic states, probability, partition
function, entropy, electronic excitations and electronic entropy.
- Introduction to Molecular Dynamics. A comprehensive web resource on classical MD can be found
on Furio Ercolessi's Molecular Dynamics primer.
- More on Molecular Dynamics: thermalization, initial condition
selection, integration, ergodicity, energy, property, equation of state
and structure calculations in liquids, thermostats, temperature and pressure
setting in MD, time and space correlations, phase transitions.
- Computing macroscopic properties from micro-fluctuations using MD: Green-Kubo techniques. Lagrangian and Hamiltonian approaches to
MD dynamics with application to Nosé-Hoover thermostat. Introduction to the basics of First-Principles Molecular Dynamics.
- More on ab Initio MD: Global potential energy surfaces,
Ehrenfest Molecular Dynamics, Born-Oppenheimer Molecular Dynamics and Car-Parrinello Molecular Dynamics.
Excellent
review material on Ab initio molecular dynamics can be found on this
article of Dominik Marx (a shortned version is also available).
- Statistical mechanics and Monte Carlo: Sampling techniques, Metropolis algorithm,
Ising spin model, applications to other two state systems, 1st and 2nd order phase transitions. The original article of
Metropolis et al. is given here. A more detailed presentation of MC and statistical mechanics with
computational details is
also available.
- Implementation of MC: More on MC moves, scaling and sampling errors, applications to polymers and biomolecules.
An overview of
MC is given in an article by D. Frenkel. For applications to polymers, see articles by
Baschnagel et al. and K. Kremer.
- Free energy computation and coarse graining methods. Papers
that are referred in these notes can be downloaded from this directory.
- Ab initio thermodynamics and structure prediction.
Computation of phase diagrams is discussed here: Part A and Part B. Papers
that are referred in these notes can be downloaded from this directory.
- Multibody expansions:
Review of Ab initio structure prediction, cluster and multibody expansions,
weighted multibody expansions, ab initio based phase diagram prediction for alloys, review of
ATAT.
- The quasi-continuum method and its variants, kinetic Monte Carlo (KMC), other
coarse graining (space and time) methods.
- Large scale electronic structure calculations:
maximally localized Wannier functions, linear scaling methods, alloy semiconductors, alloy disorder using
perturbation theory and cluster expansins.
- Course closing: a number of case studies of recent interest.
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Homework
- Homework 1 pdf -- Empirical potential methods. The executables can be
obtained directly from GULP or can be directly extracted here for Windows or Linux compilers. The EAM potential libraries are given here so you dont need to search the Gulp directories.
The input files needed for Problem 1 are given in this compressed directory.
The gulp1.3 manual can be downloaded here. A nice introduction to Gulp can be found in this
comprehensive paper.
The
discussion of this HW in class is also available. Your questions and answers related to this homework will be
given here (some student questions from last year are already attached).
Useful background material that can be consulted for this homework is given below:
- Homework 2 pdf -- DFT calculations using Quantum Espresso (the pwscf3.2 executables, users guide and manual can be downloaded here): k-point integration, energy cutoffs, error cancellation in DFT,
computing equilibrium lattice constants and material properties. A number of typical examples and the corresponding input files
using PWSCF are contained on this file. In class
discussion of this homework set will become available here. Your questions and answers related to this homework will be
given here. For those having problems to compile
PWSCF on the CTC nodes, we are attaching an executable (ready to use) here.
- Homework 3 pdf -- More DFT calculations using Quantum Espresso: band diagram calculation, structure prediction, phonon calculations. A
number of needed input files
are contained on this compressed directory. A number of executables (e.g. Ph.x) are
required that are attached
here in case you have difficulties to compile the original files.
- Homework 4 pdf -- Molecular Dynamics Simulations using LAMMPS: mechanical and thermophysical property calculation,
grain boundaries, and phase transformations. A
number of needed input files
are contained on this compressed directory. A copy of our presentation on how to use
LAMMPS is given here.
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Basic course info
Credit: 4 Units.
Lectures: Mondays and Wednesdays 4:30 -- 5:50 pm, Rhodes 178.
Professor: Nicholas
Zabaras, 101 Frank H. T. Rhodes Hall, (607) 255-9104, zabaras@cornell.edu
Office hours: Fridays 4:00 -- 5:30 p.m. (or by appointment)
References: The course lectures will become available on the course web site. For in depth study,
a list of articles from the current literature will also be provided to enhance the material of the lectures. There is
no required text for this course.
Some important books that can be used for general reading in the subject areas of the course include the following:
Web resources on atomistic modeling and DFT:
References on quantum mechanics:
References on electronic structure of materials:
References on electronic structure calculations:
References on modern computational chemistry:
References on molecular modeling:
Homework: The class inlcude 4-5 homeworks that emphasize atomistic
computer simulations using commercial and/or academic software in order
to model, understand, and predict the properties of engineering materials.
Final project: A project is required emphasizing physical or computational aspects of
atomistic modeling of materials. Students are encouraged to investigate topics not covered in class.
A written report and oral presentation are required (reports and presentations to be posted on the course web site).
Grading: Homeworks 70% (all with computer assignments) and a final project 30%.
Prerequisites: The material covered is self-contained but an earlier exposure to an
undergraduate quantum mechanics, physical chemistry or modern physics course is desirable.
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Course description
Catalog description: The course is intended for engineering, physics and chemistry graduate
students with interests in the simulation of materials at the atomic
scale using academic and commercial software. Emphasis
is given to models of interatomic forces from
Lennard-Jones models to self-consistent all-electron solution
of the quantum mechanical problem. Specific topics include:
energy models, density functional theory and the total-energy
pseudopotential method, Monte Carlo and
molecular dynamics simulations,
free energy and phase transitions,
fluctuations and transport properties,
first-principles MD,
Ab-initio thermodynamics
and structure prediction, coarse-graining methods and mesoscale models.
The course includes advanced applications of materials to nanotechnology.
Course objectives:
This course provides graduate students with a single source introduction to
all aspects of atomistic modeling of materials including direct experience with simulations of
classical energy models, electronic structure (ab initio) approaches, Monte Carlo
sampling techniques and molecular dynamics as applied to diverse materials problems.
Intended audience: Graduate Students in Engineering, Physics and Chemistry with interests in
materials, nanotechnologies and modeling at the atomic/electronic scale.
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