Link to PDF version of Power Point Slides for Lecture 6, Part 1

Link to PDF version of Power Point Slides for Lecture 6, Part 2

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 it is easy to show that plane wave calculations are not feasible for elements heavier than Li, because too many Fourier components are required for a realistic calculation. Plane waves are useful only only if there is a way to treat core electrons.

1. The Schrodinger equation for a periodic crystal
2. Calculations in a Plane Wave Basis
1. Simple expressions for Bloch states, Schr. Eq.
2. Self-consistent Kohn-Sham calculation .
1. From output eigenvectors - find new density and potential
2. Example of clever algorithm -- efficient in real space
3. Fast Fourier Transform (FFT) – introduction of grid in real space
3. Repeat to self-consistency
3. Tests for convergence -- MUST be done for calculations to be correct!
1. Convergence in number of plane waves
2. Convergence in number of iterations
3. Convergence in number of points sampled in the BZ
Accurate treatment of core electrons
4. How to use planes for surface calculations
1. "Supercells"
2. Must use many plane waves - but still efficient
5. Examples

1. Why eliminate core electrons?
1. Must do something to have a feasible number of plane waves
2. Eliminating the core electrons with pseudopotentials
1. "Ab Initio" Norm-Conserving Pseudopotentials
1. Key point is to do calculation on an atom - easy in spherical coordinates
2. Define pseudopotential for valence states by finding smooth functions that have the same form outside the core and same eigenvalues as the original atomic valence states
3. Elegant proof that if the normalization of the wavefunction inside the core region, the the resulting pseudopotential is valid to linear order for a range of energies
2. Great Advantages of Plane waves with pseudopotentials
   1. Great advantage - Plane wave programs that use pseudopotentials are very simple and very fast - FFTs Can readily calculate forces == the method used for essentially all "ab initio" molecular dynamics. The method easiest to use for response functions, phonon frequencies, etc.
3. Augmented methods
   
1. APW methods solve for both core and valence states by using spherical waves inside a sphere and plane waves outside
   1. Most accurate - most difficult
   2. Matching is non-linear problem - much easier by linearizing - similar to pseudopotentials that are correct to linear order for changes in energy
   3. Most useful when both localized and delocalized valence states are important - like transition metals
   4. Great disadvantage - cannot use FFTs directly since functions are defined differently inside and outside the spheres
2. PAW methods add localized core functions to plane waves that extend throughout crystal
   1. Great advantage - Plane wave programs that are like pseudopotentials - very simple and very fast - FFTs
   2. Unlike pseudopotentials the core states are kept and the method is more useful when both localized and delocalized valence states are important - like transition metals
4. Comparison of results
   
5. Examples of results for solids
   
6. Conclusions