Lecture 9: Plane Wave Calculations in Crystals: Self-Consistent DFT
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Link to HTML version of Power Point Slides for Lecture 9

Link to ABINIT - Open Source Kohn-Sham DFT plane wave code with excellent tutorial and many features.

The full solution of the Kohn-Sham Equations requires a self-consistent solution for the density and the total energy.  What quantities should be correct in a Kohn-Sham calculation?  How are self-consistent calculations done in a crystal?  Typical results?

There are good references for plane wave pseudopotential calculations:
The textbook, Chapter 13.
Monograph by D. J. Singh, "Planewaves, Pseudopotentials, and the APW Method", Kluwer Academic Publishers, Boston, 1994.
Review by W. E. Pickett, "Pseudopotential Methods in Condensed Matter Applications", Computer Physics Reports 9, 115 (1989).
Review by M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias and J. D. Joannopoulos, "Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients", Rev. Mod. Phys.64, 1045, (1992).

  1. The Kohn-Sham Equations for Solids
    1. Solution for given Kohn-Sham Hamiltonian same as in previous notes
    2. Solve for each k independently
    3. Ground State given by occupying lowest states
      1. Insulators - fixed number of bands needed
      2. Metals require finding Fermi energy after all k-points are calculated
    4. All total quantities given by sum over filled states in the Brillouin Zone
  2. Self-consistent calculation
    1. Output gives the density for any input potential
    2. But in DFT the potential depends upon the density
    3. Must iterate to get self-consistent potential and density
    4. Flow-Chart
  3. Integration over Brillouin Zone
    1. Easiest is regular grid - "Monkhorst -Pack grid"
    2. Remarkably efficient for insulators
  4. Output results
    1. Energy, Force, Stress, .....
    2. Eigenvalues (bands), Bloch functions, .....
  5. Examples of results for solids: Calculations using norm-conserving pseudopotentials
    1. Total Energy and charge density
    2. Equilibrium Volume of simple crystals,  Carbon, silicon, ...
    3. Phase transitions -  silicon, carbon under pressure
    4. Hellmann-Feynman Theorem and forces
    5. Static Linear Response:  Phonons
    6. Surfaces, Interfaces, ...
  6. Open Source code: ABINIT available at http://www.abinit.org/
  7. Conclusions