MAE715: Atomistic Modeling of Materials

Cornell University, Spring 2008


Meeting room and time: Mondays and Wednesdays (Rhodes 178) 4:30-5:45 pm

Professor Nicholas Zabaras



Lecture notes and literature material

  1. 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.


  2. 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.


  3. Angular momentum and the Hydrogen atom: Angular momentum quantization, the Hydrogen atom, atomic orbitals, multi-electron configurations in other elements.


  4. Operators and approximation methods in Quantum Mechanics: Hermitian operators, commutation, general form of uncertainty principle, spin dependence, the variational principle.


  5. 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.


  6. 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.)


  7. 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.


  8. Applications of empirical potential methods to organic & biological materials, polymers, and oxides.


  9. Perturbation methods: Time-independent and time-dependent perturbation theory.


  10. 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.


  11. Introduction to Hartree-Fock methods: The Hartree approximation, Slatter determinant, exchange, mean field approximation.


  12. Derivation of the Hartree-Fock equations.


  13. Hartree-Fock methods continued: Koopmans' theorem, the Hartree-Fock-Roothaan method, applications (successes and failures), restricted, unrestricted, extended, and projected Hartree-Fock methods.


  14. Selection of basis sets: Slater (atomic) orbitals, Gaussians, plane waves, augmented plane waves, APW, ASW, muffin-tin-orbitals (LMTO).


  15. 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.


  16. 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.


  17. 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.


  18. 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.


  19. 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.


  20. 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.


  21. 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.


  22. Non-zero temperature DFT calculations: T-dependence of structure, phase diagrams, thermal expansion, elastic and other properties.


  23. Phonon calculations: Frozen phonon approximation,density functional perturbation theory, thermodynamic properties of solids by phonons.


  24. 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.


  25. Introduction to Molecular Dynamics. A comprehensive web resource on classical MD can be found on Furio Ercolessi's Molecular Dynamics primer.


  26. 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.


  27. 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.


  28. 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).


  29. 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.


  30. 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.


  31. Free energy computation and coarse graining methods. Papers that are referred in these notes can be downloaded from this directory.


  32. 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.


  33. 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.


  34. The quasi-continuum method and its variants, kinetic Monte Carlo (KMC), other coarse graining (space and time) methods.


  35. Large scale electronic structure calculations: maximally localized Wannier functions, linear scaling methods, alloy semiconductors, alloy disorder using perturbation theory and cluster expansins.


  36. Course closing: a number of case studies of recent interest.
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Homework

  1. 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:

  2. 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.

  3. 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.

  4. 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:

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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