Physics 250-06 “Advanced Electronic Structure” Linear Methods in Band Theory Contents:

1. Physics 250-06 “Advanced Electronic Structure” • Linear Methods in Band Theory • Contents: • KKR-ASA method and canonical energy bands • 2. Envelope Functions • 3. LAPW and LMTO Methods

2. Non linear equations of APW Method Non linear equations of KKR Method How to get rid off the energy dependence coming from the orbitals?

3. KKR partial waves Basic idea of KKR method is to construct a partial wave Consider its Bloch sum And demand tail-cancellation:

4. Non-linear KKR Equations where potential parameters function is and where KKR structure constants are

5. Energy linearizaton Andersen proposed to split energy dependence coming From inside the spheres and from interstitials. Since interstitial region is small, Andersen proposed to fix this energy kappa to some value (originally to zero) Energy Bands E k2=E-V0 Average kinetic energy of electron in the interstitial region MT-zero V0

6. Partial waves of fixed energy tails Consider as before Bloch sum and demand tail-cancellation:

7. KKR equations become where potential parameters function is and where the fixed energy structure constants are To minimize the error of fixing the energy, Andersen proposed to enlarge MT spheres to atomic Spheres. This method has the name KKR-ASA.

8. Canonical band structures (Andersen , 1973) At the absence of hybridzation, a remarkable consequence of KKR ASA equations is canonical energy bands: For a given l block, one can diagonalize the structure constants and obtain (2l+1) non-linear equations whose solutions give rise to band structures E(kj), so called canonical band structures.

9. Canonical band structures It is remarkable how easy we can understand the s,p,d energy bands of various materials, including its canonical dispersion prescribed by the lattice, as well as their position and the bandwidth prescribed by the potential parameters.

10. Center of the Band The key property is to understand the behavior of logarithmic derivative as a function of energy Because if (condition of center of the band) This would mean that a given nl band appears always when as a non-linear equation for Enl

11. Logarithmic Derivatives Behavior of Logarithmic Derivative Consider s-wave: 1s has no nodes, 2s has 1 node,… or Nodes=n-l-1. From the point of view of node appears When which means that log. derivative diverges! So logartihmic derivatives behave as tan(E), they diverge each time a new node of radial wave function appears.

12. Logarithmic Derivative as a Function of Energy E1 E2 r E3 S Energy Window for 3s states E3 Energy Window for 2s states E2 E1 Energy Window for 1s states

13. Logarithmic Derivative as a Function of Energy Centers of the nl band E1 E E2 r E3 New node of wave function appears! 4s 1s 3s 2s

14. Linearized Solutions If Dl(E) can be expanded in Teilor series around some energy En, we obtain potential function in a linearized form which solves the band structure problem Cl gives the center of the l-band, wl gives its width while denominator 1-gS gives additional distortion of the band

15. Canonical d-band for fcc material wl Cl

16. Comparison with bands of Cu

17. Canonical f-band for fcc material wl Cl

18. Comparison with bands for Ce

19. Energy Linearization (Andersen, 1973) General idea to get rid of E-dependence: use Teilor series and get LINEAR MUFFIN-TIN ORBITALS (LMTOs) Before doing that, consider one more useful construction: envelope function. In fact, concept of envelope functions is very general. By choosing appropriate envelope functions, such as plane waves, Gaussians, spherical waves (Hankel functions) we will generate various electronic structure methods (APW, LAPW, LCGO, LCMTO, LMTO, etc.)

20. Construction of augmented plane waves

21. Construction of Augmented Spherical Wave Linear combinations of local orbitals should be considered. However, it looks bad since Bessel does not fall off sufficiently fast! Consider instead:

22. Envelope Functions • Algorithm, in terms of which we came up with the • augmented spherical wave (MUFFIN-TIN ORBITAL) • construction: • Step 1. Take a Hankel function • Step 2. Augment it inside the sphere • by linear combination: • Step 3. Construct a Bloch sum

23. Envelope Functions • Why take Hankel function as an envelope? • Step 1. Take ANY function • which has one center expansion • in terms of • Step 2. Augment it inside the sphere • by linear combination: • Step 3. Construct a Bloch sum

24. Envelope Functions Envelope functions can be Gaussians or Slater-type orbitals. They can be plane waves which generates augmented plane wave method (APW)

25. Linearization over Energy General idea to get rid of E-dependence: use Teilor series and get read off the energy dependence. Introduction of phi-dot function gives us an idea that we can always generate smooth basis functions by augmenting inside every sphere a linear combinations of phi’s and phi-dot’s The resulting basis functions do not solve Schroedinger equation exactly but we got read of the energy dependence! The basis functions can be used in the variational principle.

26. Linear Augmented plane wave (LAPW) method Augmented plane waves: become smoothlinear augmented plane waves:

27. Linear Muffin-Tin Orbitals Consider local orbitals. Energy-dependent muffin-tin orbital defined in all space: becomes energy-independent provided we also fix to some number (say 0)

28. Linear Muffin-Tin Orbitals Bloch sum should be constructed and one center expansion used: Final augmentation of tails gives us LMTO:

29. Linear Muffin-Tin Orbitals In more compact notations, LMTO is given by where we introduced radial functions which match smoothly to Hankel and Bessel functions.

30. Linear Muffin-Tin Orbitals Another way of constructing LMTO. Consider envelope function as Inside every sphere perform smooth augmentation which gives again LMTO construction.

31. Linear Muffin-Tin Orbitals We could do the same trick for a single Hankel function Inside every sphere perform one- Center expansion and augmentation Bloch summation is trivial.

32. Linear Muffin-Tin Orbitals LMTO definition (k dependence is highlighted): which should be used as a basis in expanding Variational principle gives us matrix eigenvalue problem.

33. Linear Muffin-Tin Orbitals Accuracy and Atomic Sphere Approximation: LMTO is accurate to first order with respect to (E-En) within MT spheres. LMTO is accurate to zero order (k2 is fixed) in the interstitials. Atomic sphere approximation can be used: Blow up MT-spheres until total volume occupied by spheres is equal to cell volume. Take matrix elements only over the spheres. ASA is accurate method which eliminates interstitial region and increases the accuracy. Works well for close packed structures, for open structures needs empty spheres.