Lecture 8: Plane Wave Calculations in Crystals: Bloch Theorem, Nearly Free Electrons, Empirical Peudopotentials
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No Power Point Slides for Lecture 8 - See Ch. 12 in text and manual for TBPW code linked below

Link to TBPW - simple pedagogical plane wave code.

The big picture: Fourier Transforms always work in periodic systems. Direct proof of the Bloch theorem; alternative way to understand the Brillouin Zone, etc.
But are plane waves useful?  How many Fourier components are required for a realistic calculation?
Easy to show that plane wave calculations are out of the question for heavy elements.
Plane waves are widely used only because of pseudopotentials!; MANY aspects become much easier, e.g.,  Hellmann-Feynman Forces, Car-Parrinello type simulations, ....

  1. The Schrodinger equation for a periodic crystal
    1. General Expressions for Eqs. in a Plane Wave Basis
      1. Express periodic potential in Fourier components: Veff(G)
      2. Express wave function in basis of plane waves
      3. Equations lead directly to Bloch theorem, definition of Brillouin Zone, etc.
    2. Form of equations if V(r) is a sum of spherical potentials around each atom
      1. Form factors and structure factors
  2. The nearly free electron approximation
    1. Free electrons
    2. Effect of a weak lattice potential
    3. Remarkable fact that this seems to describe bands in many crystals
  3. Empirical Pseudopotentials
    1. Approximate the potential due to the electrons as spherical atomic-like
    2. Phillips-Kleinman - 1959  - can describe bands by a few parameters
    3.  More recent accurate work - Chelikowsky and Cohen Book
  4. Computer program for plane wave calculation
    1. "TBPW" F90 code available at http://www.mcc.uiuc.edu/software/ under "instructional software"
    2. Flow Chart
    3. Identify parts of program common to all band structure calculations
      1. Crystal structure information
      2. Diagonalization routines
    4. Identify parts of program specific to plane waves
      1. Form of hamiltonian: essential ingredient is the set of Fourier components of the potential
      2. Empirical Pseudopotential: specified by a parameterized analytic form, or by values of the Fourier components of at the reciprocal lattice vectors
    5. Example input file
  5. Examples of Results in crystals - Si, GaAs
  6. Beyond the local approximation for the potential - needed for accurate calculations
  7. Conclusions