speaking-notes.org



Up
Introduction
KKR introduction

  master-thesis speaking notes for this slide
    
      4 KKR-GF
    
  
  KKR SCF cycle
    
      
dft20 lecture 20 KKR
        
          p. 18
            
              1) potential V -> solutions R, H
                
                  from 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 iffMD KKRimp tutorial: t = V + V G0 t = ∫_V ∑_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
    
      blugelDensityFunctionalTheory2006 - 6 The Green function method of Korringa, Kohn and Rostoker
    
    
      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.
    
  
  Observables and electron density
    
      from lit-rev - kkr.org
        
          […] charge density \(n(\bm{r})\) can be directly expressed by an energy integral
            over the imaginary part of the Green function
        
      
      from msc2a_theory
        
          The integral sums over all occupied states up to the Fermi energy \(E_F\) at
            zero absolute temperature
        
      
      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.