- from [[https://iffmd.fz-juelich.de/jAngJ9qIQgSwPObGOSU1dw#][iffMD KKRimp tutorial]]: this is the single-site problem
- from dft20 p.27: secular equation: local solution of TISE in each cell
with basis RL YL
- from [[https://iffmd.fz-juelich.de/jAngJ9qIQgSwPObGOSU1dw#][iffMD KKRimp tutorial]]: t = V + V G0 t = \int_V \sum_L J V R
- 2) Algebraic Dyson equation -> structural GF
- the ADE IS the SGF
- the SGF contains all possible scattering paths btw any two cells
- Sol found by Fourier transform (k-space), matrix inv, back-transform
(otherwise infinite sum)
- from msc2a. for KKRimp, one gets the impurity region block GII from
impurity SGF inversion in real space and discarding all blocks GRI,
GIR, GRR. Host G0 enters as a boundary condition but does not change.
- 3) GF = SiSca + Musca(structural GF)
- From lit-rev - kkr.org
- [[file:~/src/iffgit.fz-juelich.de/phd-project-wasmer/learn/literature-review/notes/topics/kkr.org::#h-5A792112-F416-4096-8067-E61A5BC02A38][blugelDensityFunctionalTheory2006 - 6 The Green function method of Korringa, Kohn and Rostoker]]
#+begin_quote
In order to solve the Schrödinger equation, the scattering properties of each
scattering center (atom) are determined in a first step and described by a
scattering matrix, while the multiple-scattering by all atoms in the lattice
is determined in a second step by demanding that the incident wave at each
center is the sum of the outgoing waves from all other centers. In this way, a
separation between the potential and geometric properties is achieved.
A further significant development of the KKR scheme came when it was
reformulated as a KKR Green function method [75, 76]. By separating the
single-site scattering problem from the multiple-scattering effects, the
method is able to produce the crystal Green function efficiently by relating
it to the Green function of free space via the Dyson equation. In a second
step the crystal Green function can be used as a reference in order to
calculate the Green function of an impurity in the crystal [77], again via a
Dyson equation. This way of solving the impurity problem is extremely
efficient, avoiding the construction of huge supercells which are needed in
wavefunction methods.
#+end_quote
- Observables and electron density
- from lit-rev - kkr.org
#+begin_quote
[...] charge density \(n(\bm{r})\) can be directly expressed by an energy integral
over the imaginary part of the Green function
#+end_quote
- from msc2a_theory
#+begin_quote
The integral sums over all occupied states up to the Fermi energy \(E_F\) at
zero absolute temperature
#+end_quote
- expensive energy integrals are calculated efficiently via contour integration
(less E points)
- Some KKR applications besides impurity embeddings
- surfaces, layered systems, transport and spectroscopic properties,
linear-scaling DFT with accurate long-range interactions (KKRnano),
disordered systems (CPA), conventional superconductivity (BdG-DFT), etc.
