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* [[file:../README.org][Up]]
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* Introduction
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* KKR introduction
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- master-thesis speaking notes for this slide
  - [[file:~/src/iffgit.fz-juelich.de/phd-project-wasmer/publishing/master-thesis/notes/speaking/MScThesis_Presentation_Speaking_Notes.org::#h-cb86fd2d-4acd-4934-a5dc-d54797fbc4b6][4 KKR-GF]]
- KKR SCF cycle
  - [[file:~/Desktop/Studium/Kurse_RWTH/DensityFunctionalTheory/20W/Lecture 17-20 Electronic Struct___eudopotential, PAW, FLAPW, KKR/Lecture20_KKRGF-Method.pdf][dft20 lecture 20 KKR]]
    - p. 18
      - 1) potential V -> solutions R, H
        - 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.