MAE715: Atomistic Modeling of Materials

Cornell University, Spring 2009


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.


  2. 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.


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


  4. Operators in Quantum Mechanics: Hermitian operators, commutation, general form of uncertainty principle, spin dependence. Perturbation methods: Time-independent and time-dependent perturbation theory.


  5. 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.


  6. 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.


  7. Introduction to Hartree-Fock methods: The Hartree approximation, Slatter determinant, exchange, mean field approximation. Derivation of the Hartree-Fock equations.


  8. 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.


  9. 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.


  10. 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.


  11. 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.


  12. 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.


  13. 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.


  14. 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.


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


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


  17. Empirical potential methods: pair potentials, Lennard-Jones, Born-Mayer, Buckingham, Morse, fiting potentials, failures and successes of potentials (metals, organic materials, oxides, etc.)


  18. 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.


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


  20. 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.


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


  22. 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.


  23. 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.


  24. 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.


  25. 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.


  26. Advanced Monte Carlo methods and free energy computation. Papers that are referred in these notes can be downloaded from this directory.


  27. 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.


  28. Multiple time scales and coarse graining methods.
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Homework

  1. 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

  2. Homework 2 pdf -- Using the variational principle to compute ground and excited states, LCAO approximations in hetero-nuclear diatomic molecules.

    Homework 2 solution pdf

  3. Homework 3 pdf -- Introduction to Hartree-Fock methods, calculating the He ground state energy

    Homework 3 solution pdf

  4. 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

  5. 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

  6. 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

  7. 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

  8. Homework 8 pdf -- Computing pseudo-potentials using Quantum Espresso.

    Homework 8 solution pdf

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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:

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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