Korringa–Kohn–Rostoker approximation

Electronic structure methods
Valence bond theory
Generalized valence bond Modern valence bond Resonance
Molecular orbital theory
Hartree–Fock method Semi-empirical quantum chemistry methods Møller–Plesset perturbation theory Configuration interaction Coupled cluster Multi-configurational self-consistent field Quantum chemistry composite methods Quantum Monte Carlo Linear combination of atomic orbitals
Electronic band structure
Nearly free electron model Tight binding Muffin-tin approximation Density functional theory k·p perturbation theory Empty lattice approximation
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The Korringa–Kohn–Rostoker approximation or KKR method^[1]^[2]^[3] is used to calculate the electronic band structure of solids. It is a Green's function method which matches the different types of one-electron wave functions and their derivatives which are used in a muffin-tin approximation.

Introduction

In solid state physics the properties of electrons and potentials are determined by the spherical symmetry of the interior of the atoms and the point group symmetry of the crystal lattice. The wave function of an electron and the potential in the KKR method are both composed of two parts. A linear combination of spherical harmonics, which is multiplied by a radial wave function, is used for the wave function on the inside of the atoms where the potential is the potential of the atom. Linear combinations of plane waves are used for the wave function in the regions between the atoms where the potential is constant.

Mathematical formulation

The Schrödinger equation can be expressed like

[\nabla^2 + E]\psi(\bold{r}) = V(\bold{r})\psi(\bold{r})

where $V(\bold{r})$ is the potential of the solid and $\psi(\bold{r})$ is the wave function of the electron that has to be calculated.

The unperturbed Green's function is defined as the solution of

[\nabla^2 + E]G(\bold{r},\bold{r}') = \delta(\bold{r}-\bold{r}')

A plane wave can be expanded as

e^{i\bold{k}\cdot \bold{r}} = \sum_l (2 l + 1) i^l j_l(kr)P_l(\cos\theta)

where $j_l(kr)$ are spherical Bessel functions and $P_l(\cos\theta)$ are Legendre polynomials.

References

↑ J. Korringa (1947). "On the calculation of the energy of a Bloch wave in a metal". Physica XIII (6–7): 392–400. Bibcode:1947Phy....13..392K. doi:10.1016/0031-8914(47)90013-x.
↑ W. Kohn, N. Rostoker (1954). "Solution of the Schrödinger Equation in Periodic Lattices with an Application to Metallic Lithium". Phys.Rev. 94 (5): 1111–1120. Bibcode:1954PhRv...94.1111K. doi:10.1103/physrev.94.1111.
↑ W. Jones, N. H. March (1973). Theoretical Solid State Physics. Wiley and Sons - Dover Publications. ISBN 0-486-65015-4.