Physics 250-06 “Advanced Electronic Structure”

1. Physics 250-06 “Advanced Electronic Structure” • Solution of Electronic Structure for Muffin-Tin Potential • Contents: • 1. Augmented Plane Wave Method (APW) of Slater • 2. Green Function Method of Korringi, Kohn, and Rostoker (KKR) • 3. Tail Cancellation Condition in KKR Method (Andersen) a. J. M. Ziman,Principles of the Theory of Solids (Chapter 3) b. J. M. Ziman, the Calculation of Bloch functions, Solid State Phys. 26, 1 (1971).

2. Solving Schrodinger’s equation for solids Solution of differential equation is required Properties of the potential

3. LCAO method is great since it gives as Tight-Binding Description, however: Problems with LCAO Method: Atomic wave functions are tailored to atomic potential (not to self-consistent potential). Atomic wave functions have numerical tails which are difficult to handle with.

4. Muffin-tin Construction: Space is partitioned into non-overlapping spheres and interstitial region. potential is assumed to be spherically symmetric inside the spheres, and constant in the interstitials. Muffin-tin potential Muffin-tin sphere

5. With muffin tin potential solutions are known: Radial Schroedinger equation inside the sphere Helmholtz equation outside the sphere: spherical waves Muffin-tin sphere Or plane waves

6. Hence two methods have been invented to solve • the electronic structure problem: • Augmented plane wave method (APW) of Slater using plane • wave representation for wave functions in the • Interstitial region • Green function method of Korringi, Kohn and Rostocker • (KKR) which uses spherical waves in the interestitial • region. • You can also say it is augmented spherical wave (ASW) method • but historically this name ASW appeared much later.

7. APW Method of Slater Trying to solve variationally: in the interstitial region. inside the spheres

8. Construction of augmented plane waves

9. Resulting APW is used to construct Hamiltonian and overlap matrices and lead to generalized eigenvalue problem: Since APWs are not smooth additional surface Intergrals should be added in the energy functional Leading to variational solutions. Discussed in J. M. Ziman, the Calculation of Bloch functions, Solid State Phys. 26,1. (1971)

10. Matrix elements of H and O inside the spheres =E =1

11. Matrix Elements in the interstitial

12. Integrals between Bessel functions

13. Major difficulty of APW approach: implicit energy dependence Therefore, to find the roots, the determinant should be evaluated as a function of E on some energy grid and see at which E it goes to zero: In practice, this determinant is very strongly oscillating which can lead to missing roots!

14. Alternative view on APWs: kink cancellation Each APW by construction is continuous but not smooth. for r>S for r<S Request that linear combination of APWs becomes smooth: This occurs for selected set of energies only! Discussed in J. M. Ziman, the Calculation of Bloch functions, Solid State Phys. 26,1. (1971)

15. Green Function (KKR) Method First rewrite Schroedinger equation to intergral form Introduce free electron Green function (of Helmgoltz equation) We obtain:

16. Green Function (KKR) Method Second, using Bloch property of wave functions rewrite the integral over crystal to the integral over a single cell

17. Use expansion theorems: Summation over lattice is trivial are called structure constants

18. Consider now the solutions in the form of linear combinations of radial Schroedinger’s equation with a set of unknown coefficients: Using expansion theorems and many other tricks we finally obtain the conditions of consistency:

19. Consequences of KKR equations: Spectrum is obtained from highly non-linear eigenvalue problem Information about the crystal structure and potential is split: Potential Parameters Structure constants

20. Reformulation of KKR method as tail-cancellation condition (Andersen, 1972) Radial Schroedinger equation inside the sphere Helmholtz equation outside the sphere: spherical waves Muffin-tin sphere

21. Solution of Helmholtz equation outside the sphere where coefficients provide smooth matching with

22. Our construction is thus Linear combinations of local orbitals should be considered. However, it looks bad since Bessel does not fall off sufficiently fast! Consider instead:

23. Using expansion theorem, the Bloch sum is trivial: where structure constants are: Convenient notations which aquire spherical harmonics inside spherical functions:

24. A single L-partial wave is not a solution: However, a linear combination can be a solution Tail cancellation is needed which occurs at selected

25. is a good basis, basis of MUFFIN-TIN ORBITALS (MTOs), which solves Schroedinger equation for MT potential exactly! For general (or full) potential it can be used with variational principle