MAE715: Atomistic Modeling of Materials
Cornell University, Spring 2009
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:
Eigenstates, time-independent Schrödinger equation,
free particle, particle in a box. Time-dependent Schrödinger equation,
harmonic oscillator, superposition state, Hamiltonian, the momentum operator, uncertainty principle & expectation values.
- Angular momentum and the Hydrogen atom:
Angular momentum quantization, the Hydrogen atom, atomic orbitals, multi-electron configurations in other elements.
- Operators in
Quantum Mechanics:
Hermitian operators, commutation, general form of
uncertainty principle, spin dependence. Perturbation methods:
Time-independent and time-dependent perturbation theory.
- Approximation methods
in Quantum Mechanics:
The variational principle. LCAO (Linear Combination
of Atomic Orbitals) approach. An application of the variational principle in diatomic molecules,
the homo- and hetero-nuclear diatomic molecule,
bond order, bond energy, electronegativity.
- 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.
- The Hartree-Fock-Roothaan method, Koopmans' theorem,
restricted, unrestricted, extended, and projected Hartree-Fock methods,
applications. Chapter 4 from J.M. Thissen's computational physics book provides an
excellent supplementary reading
with emphasis on the implementation of HF 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.
- 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, Perdew-Zunger parametrization.
- 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 here.
- 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.
- Empirical potential methods: pair potentials,
Lennard-Jones, Born-Mayer, Buckingham, Morse, fiting potentials, failures and successes of
potentials (metals, organic materials, oxides, etc.)
- Implementation of empirical potential methods: supercells, relaxation,
numerical methodology, convergence,
potentials for different materials classes, effective medium theories, embedded atom methods (EAM),
applications to metals, potentials for Si, Si-surface reconstruction, etc.
- Applications of empirical potential methods to organic
& biological materials, polymers, and oxides.
- 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). Also see lecture notes on intro to ab initio MD by Jurg Hutter.
- Statistical
mechanics and Monte Carlo: Sampling techniques, Metropolis algorithm, implementation of MC, scaling and sampling errors,
Ising spin model, 1st and 2nd order phase transitions, applications to polymers and bio-molecules. 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. 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.
- Advanced Monte Carlo methods and free energy computation. 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.
- Multiple time scales and coarse graining methods.
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Homework
- Homework 1 pdf -- Review of QM, The Hydrogen atom, radial and angular parts of atomic orbitals, Fermi Holes and Fermi Heaps.
Homework 1 solution pdf
- Homework 2 pdf -- Using the variational principle to compute ground and excited states, LCAO approximations
in hetero-nuclear diatomic molecules.
Homework 2 solution pdf
- Homework 3 pdf -- Introduction to Hartree-Fock methods, calculating the He ground state energy
Homework 3 solution pdf
- Homework 4 pdf -- Implementation of the Hartree-Fock-Roothaan method using a Gaussian basis with
application to the Hydrogen molecule, tight-binding approximations (Slater-Koster
method) for carbon systems
with applications to
fullerenes and carbon nanotubes. Data and programs for this HW can be downloaded from here.
Homework 4 solution pdf
- Homework 5 pdf -- Implementation of the LDA approximation of the density functional theory (DFT)
for the Helium atom using the
Perdew-Zunger parametrization of the correlation energy.
Homework 5 solution pdf
- Homework 6 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. The
discussion of this homework set from the recitation is available here.
Homework 6 solution pdf
- Homework 7 pdf -- Computing pseudo-potentials, Quantum Espresso calculations for
structure and property predictions, Band diagrams for Si. A number of needed input files
are contained on this compressed directory.
Homework 7 solution pdf
- Homework 8 pdf -- Computing pseudo-potentials using Quantum Espresso.
Homework 8 solution pdf
- Homework 9 pdf -- Phonon calculations using the linear response method
and the small displacement method of Dario Alfe. A useful presentation
of this HW is given here and guidelines on compiling Quantum Espresso and Phon in Windows
are also provided.
Homework 9 solution pdf
- Homework 10 pdf -- Empirical potential methods using GULP. The gulp1.3 manual can be downloaded here. A nice introduction to Gulp can be found in this
comprehensive paper. The executables can be
obtained directly from GULP. The EAM potential libraries are given here.
The input files needed for Problem 1 are given in this compressed directory.
The
discussion of this HW in class is also available.
Useful background material that can be consulted for this homework is given below:
Homework 10 solution pdf
- Homework 11 pdf -- Molecular Dynamics Simulation of a Lenard Jones Liquid using the MDLab software developed by D. Spangberg. The software
can also be downloaded here with a documentation.
Homework 11 solution pdf
- Homework 12 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.
Homework 12 solution pdf
- Homework 13 pdf -- Monte Carlo simulation for alloys: Pattern formation using
Glauber dynamics and investigating phase transitions using the Potts and Ising models. The MC software needed for the first
problem
can be downloaded here and for the second here. You will need a visualization software such as VMD for the
first problem.
Homework 13 solution pdf
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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, zabaras@cornell.edu
Teaching Assistant: Adam Shai, ass42@cornell.edu
Computer Laboratory: Tuesdays, 4:30-5:45 pm, Swanson Lab, Rhodes Hall 163, held by
Adam Shai.
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 inlcudes weekly 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 as much as possible but an earlier exposure to
quantum mechanics, physical chemistry, solid state physics and programming 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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