Reading Material: Koonin p. 1 - p. 20, Example 1.
Homework Set #1a
Assigned: Due 1/18
Reading Material: Koonin p. 25 - p. 36
Th - 1/18 Solving the 1-body Schrodinger Eq. Boundary Value and Eigenvalue Problems
Reading Material: Koonin Ch. 3, p 55-72, Example 3.
Homework Set #1b
Assigned: Due 2/2
Homework Set #1a due
Reading Material: Koonin Ch. 3. Project 3
Th - 1/25 Hartree-Fock Theory and Spherical Atoms
Functionals in Quantum Mechanics; Hohenberg-Kohn Theorem; Kohn-Sham Hartree-like methods; Comparison with Hartree-Fock approaches. (Continue discussion of project 3, numerical calculations of the Hartree-Fock atom)
Reading Material: Notes and selected references
Homework Set #2 Assigned: Due 2/20 NOTE CHANGE
Homework Set #1b due on 2/2
Changing the programs to replace the non-local Fock exchange by an exchange correlation functional - example of local approximation to exchange only - Wigner interpolation formula for correlation - Example of general approach of iterative solution of self-consistent equations by Greens function methods
Th - 2/8 Matrix Mechanics: Bases for Quantum Mechanics -- Hilbert Spaces
Representing the wavefunction in a basis - completeness - examples of bases - plane waves - Gaussians - completeness - Transformation of bases - representing the operators as matrices - finite matrix operations on the computer - multiplication - inversion - diagonalization - computational algorithms - "complexity" of the computations Libraries of computational algorithms
CLASS MEETS IN ME343 WORKSTATION LAB
Lecture and Demonstration by Keith Glassford -
Introduction to GAUSSIAN - methods available
include RHF, UHF, Various density functionals -
Example of H2O molecule, H2 molecule
Th - 2/15 Quantum States in a periodic potential: Crystals
Reciprocal space, the Bloch theorem, the plane wave basis
See the Constructing programs for computation
in crystals
Reiteration of equations to solve for plane wave basis
Schrodinger Eq. as a matrix equation in reciprocal lattice vectors G,G'
Constructing the Reciprocal Space
Fortran 90 progam styles
Fortran 90 programs for reciporcal lattice
Homework 3a - with examples and help files for Fortran 90
Constructing the Reciprocal Space
Setting up the Matrices
Assign Homework Set #3a
Homework Set #2 due
Th - 2/22 Constructing programs for computation in crystals - Continued
Reiteration of matrix equations for Schr. Eq. in plane wave basis
Setting up the Hamiltonian matrices
Setting up arrays using Fortran 90
Diagonalization routine from Lapack - on EWS workstations
Information on Lapack routines on network
Discussion of Projects
First ideas due today. Project outlines really due Mar. 7.
Setting up and Diagonalizing the Hamiltonian matrices
Bands calculated by varying the k vector in the first Brillouin Zone
Examples of Results: 1d Matheiu Eq., Nearly free electron metals,
Insulators like He
Why Pseudopotentials?
Th - 2/29 Constructing programs for computation in crystals - Continued
The Brillouin Zone for 3d crystals
Rationale for the screening form Vtot(G)=Vion(G)/Epsilon(G)
Examples: He - insulator at ordinary volumes - metal under great compression
The diamond structure - Ge
Homework Set 2 solution now
in homework directory
Assign Homework Set #3b
Further discussion of bands in H, He, and Ge
General forms of iterative methods - example is our atomic program
Time dependent Schrodinger Eq. - parabolic Diff. Eq., See Koonin, Ch. 7 and EXample 7
Iterative methods to find the lowest states, See Koonin, Ch. 7.4
Car-Parrinello and related methods - very brief discussion
Th - 3/7 Examples of time-dependent calculations, work on programs
Further discussion of time dependent Schrodinger Eq. and interative methods
Demonstrations:
Tim Bergfeld: Making movies on the WWW of the time dependence
using Koonin's Example 7
Nick Rigakis: Mathematica program for the time dependent
Schrodinger Eq. - movies and Fourier transforms
Th - 3/14 Spring Break - No Class
Th - 3/21 APS Meeting - No Class - Work on Projects
Self consistent Hartree-Fock and Density function methods
Solution of 1 d equations by numerical differentiation: atoms
Solution by matrix diagonalization: examples: Gaussian for molecules, plane waves for crystals
Solution of time dependence and solution of large matrix equations by iterations
Iterative methods for eigenequations: Car-Parrinello Molecular dynamics methods
Th - 3/28 Many-Body problems - Introduction
Continuation of Car-Parrinello methods
Introduction to many-body quantum problems
Interacting particles - Schrodinger Eq. in many dimensions
Example Hydrogen molecule - Monte Carlo sampling
Model problems: Ising, Heisenberg, Hubbard
Size of basis grows exponentially - as N!
Lanczos Methods - iterative method to find a few lowest states of
large eigensystems
Lecture notes will be passed out in class.
The same notes are in a
postscript file, lecture notes on QMC
Also material from Koonin, Chapt. VIII, will be used.
Th - 4/4 Variational Monte Carlo - continued (Ceperley)
See lecture notes described for the previous lecture and material from Koonin, Chapt. VIII.
Note: Monte Carlo lectures by Ceperley will continue April 18 and 23
Homework 4 assigned for a variational MOnte Carlo calculations. A The Lanczos Method - General properties - and discussion of homework assignment 4 for Variational Monte Carlo
Algortithms for random sampling
Specific discussion for Homework 4
The Lanczos Method
Generates a basis in which the hamiltonian is tri-diagonal
Operation of H on trial functions in independent particle systems
Sprectra and measurable response functions
Discussion of solution of homework 3b
Structure of the program
Algorithms and codes for the components of the programs
Examples of many-body problems, huge hilbert spaces
Spin systems, the Ising model
The 1d Ising model in a transverse field - maps onto model
of a quantum tunneling ferroelectric - and simple model for
lattice gauge theories (see Kogut review)
Solutions by Lanczos iterations
Continued discussion of Lanczos type calculations and the Ising model
Further discussion of the band calculations of homework 3, if needed
Th - 4/18 Quantum Monte Carlo Methods - Diffusion Monte Carlo - Lectures by D. M. Ceperley
Lectures on Diffusion Monte Carlo (Called "path integral Monte Carlo"
by Koonin - see section VIII.3)
Th - 4/25 Review and/or Presentation of student projects
Final Exam: Wednesday, May 8, 8:00 - 11:00 AM