Link to HTML version of Power Point Slides for Lecture 12

Link to WEIN2k - LAPW code. This code is available only upon request to the developers. It is probably the best developed code that uses the augmentation methods

See the overview of methods on pages 233-235 of the text, especially the last part on atomic sphere methods. The key idea of these methods is to divide space into two types of regions: 1) spheres around each atom and 2) the interstitial region between the spheres. The Kohn-Sham equations are solved in each region and the conditions for a solution in all space are cast as a set of boundary matching conditions. There are various ways to do this which has lead to the APW, KKR and MTO methods. A key idea is "linearization" which greatly simplifies the methods and has lead to the LAPW and LMTO methods.

For discussion see the text, Chapters 16-17.
Nitty-gritty details are given in the monograph by D. J. Singh, "Planewaves, Pseudopotentials, and the APW Method", Kluwer Academic Publishers, Boston, 1994.

These methods provide many possibilities for projects - to go beyond the limited discussion possible in class

1. The muffin-tin approximation for the potential
• Good starting point for essentially all materials - very realistic in close packed systems, e.g., transition metals, . . .
• Can be generalized - NOT an essential approximation
2. The APW method (Slater)
• Solve equation in each sphere in spherical coordinates
• Solve equation in interstitial using plane waves
• Solution for eigenstate with wavevector k: find the eigenvalue for which the two solutions match at the boundary
• Leads to non-linear APW equations
3. The KKR method
• Solve equation in each sphere in spherical coordinates
• Define the coupling between spheres i and j for a state at energy E in terms of a Green's function G_ij(E), which can be found analytically.
• Solution for eigenstate with wavevector k: find the eigenvalue for which the Green's function equations are satisfied
• Leads to the non-linear KKR equations
4. The MTO method
• Recast KKR in terms of solutions for isolated sphere
• A "muffin tin orbital" (MTO) is a spherical solution inside sphere matched to a Bessel or Hankel function outside the sphere
• Defines an atom-centered basis - like an energy dependent local orbital adapted to the actual problem
5. Advantages
• Can have a small basis: the "best of both worlds"
• Physically interpretable (like local orbitals)
• Straightforward to carry to convergence (like plane waves)
6. Disadvantages
• Generalized Eigenvalue equation
• Non-linear equations
• Methods are much more complicated (but straightforward) for the "full potential" beyond the muffin-tin approximation
7. Linearization
• Elegant idea - expand in terms of the solution in the sphere at a fixed energy E₀ and a second solution the is the derivative of the wavefunction with respect to energy
• Final energy can be accurate to third order - (E_actual - E₀)³ using clever ideas that are examples of the "2n+1" theorem
• Leads to the LAPW and LMTO methods
8. Examples of results for molecules and solids