The Korringa–Kohn–Rostoker (KKR) method is used to calculate the electronic band structure of periodic solids. In the derivation of the method using multiple scattering theory by Jan Korringa and the derivation based on the Kohn and Rostoker variational method, the muffin-tin approximation was used. Later calculations are done with full potentials having no shape restrictions.
Introduction All solids in their ideal state are single crystals with the atoms arranged on a periodic lattice. In condensed matter physics, the properties of such solids are explained on the basis of their electronic structure. This requires the solution of a complicated many-electron problem, but the density functional theory of Walter Kohn makes it possible to reduce it to the solution of a Schrödinger equation with a one-electron periodic potential. The problem is further simplified with the use of group theory and in particular Bloch's theorem, which leads to the result that the energy eigenvalues depend on the crystal momentum k {\displaystyle {\bf {k}}} and are divided into bands. Band theory is used to calculate the eigenvalues and wave functions. As compared with other band structure methods, the Korringa-Kohn-Rostoker (KKR) band structure method has the advantage of dealing with small matrices due to the fast convergence of scattering operators in angular momentum space, and disordered systems where it allows to carry out with relative ease the ensemble configuration averages. The KKR method does have a few "bills" to pay, e.g., (1) the calculation of KKR structure constants, the empty lattice propagators, must be carried out by the Ewald's sums for each energy and k-point, and (2) the KKR functions have a pole structure on the real energy axis, which requires a much larger number of k points for the Brillouin Zone (BZ) integration as compared with other band theory methods. The KKR method has been implemented in several codes for electronic structure and spectroscopy calculations, such as MuST, AkaiKKR, sprKKR, FEFF, GNXAS and JuKKR.
Mathematical formulation The KKR band theory equations for space-filling non-spherical potentials are derived in books and in the article on multiple scattering theory. The central quantity of the KKR method is the (energy-dependent) Green's function, G ( E ) {\displaystyle G(E)} for the Kohn–Sham equations, defined as
G ( E ) = ( E + i ϵ − H K S ) − 1 , {\displaystyle G(E)=(E+i\epsilon -H_{\mathrm {KS} })^{-1},}
where E {\displaystyle E} is the energy, i {\displaystyle i} is the imaginary unit, ϵ {\displaystyle \epsilon } is a small positive constant (taken eventually towards the limiting value of zero), and H K S {\displaystyle H_{\mathrm {KS} }} is the Kohn–Sham Hamiltonian. For an infinite periodic system, with a complete set of eigenfunctions for H K S {\displaystyle H_{\mathrm {KS} }} , denoted | ψ i ⟩ {\displaystyle |\psi _{i}\rangle } and having associated eigenvalues ϵ i {\displaystyle \epsilon _{i}} , this Green's function can be given in a spectral representation,
G ( E ) = ∑ i | ψ i ⟩ ⟨ ψ i | E − ϵ i , {\displaystyle G(E)=\sum _{i}{\frac {|\psi _{i}\rangle \langle \psi _{i}|}{E-\epsilon _{i}}},}
though a major advantage of the KKR method is that the use of multiple scattering theory facilitates recovery of Green's functions for which there is not a convenient spectral representation, such as disordered systems. In real space, the Green's function is represented as
G ( r , r ′ , E ) = ⟨ r | G ( E ) | r ′ ⟩ . {\displaystyle G(\mathbf {r} ,\mathbf {r} ',E)=\langle \mathbf {r} |G(E)|\mathbf {r} '\rangle .}
From this real-space representation of the Green's function, the electron density, n ( r ) {\displaystyle n(\mathbf {r} )} , and the electronic density of states, ρ ( E ) {\displaystyle \rho (E)} , are obtained simply via
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