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. is used to construct Hamiltonian and overlap
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=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:
Structure constants
Potential Parameters

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