Relativistic Theory for Energy-Band Calculation
05 Jul 1966-Journal of the Physical Society of Japan (The Physical Society of Japan)-Vol. 21, Iss: 7, pp 1273-1281
TL;DR: In this article, a relativistic generalization of the Green's function method for the energy-band calculation is presented, where the wave function within the atomic spheres is expanded in terms of four-component spherical waves.
Abstract: A relativistic generalization of the Green's function method for the energy-band calculation is presented. The wave function within the atomic spheres is expanded in terms of four-component spherical waves. The resulting expression which gives the relationship between E and k is very similar to the nonrelativistic one. Matrix elements between the spherical waves can be easily computed provided structure constants used in the nonrelativistic calculations are available.
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TL;DR: The KKR-GF method as mentioned in this paper represents the electronic structure of a system directly and efficiently in terms of its single-particle Green's function (GF), which is in contrast to its original version and many other traditional wave-function-based all-electron band structure methods dealing with periodically ordered solids.
Abstract: The modern version of the KKR (Korringa–Kohn–Rostoker) method represents the electronic structure of a system directly and efficiently in terms of its single-particle Green's function (GF). This is in contrast to its original version and many other traditional wave-function-based all-electron band structure methods dealing with periodically ordered solids. Direct access to the GF results in several appealing features. In addition, a wide applicability of the method is achieved by employing multiple scattering theory. The basic ideas behind the resulting KKR-GF method are outlined and the different techniques to deal with the underlying multiple scattering problem are reviewed. Furthermore, various applications of the KKR-GF method are reviewed in some detail to demonstrate the remarkable flexibility of the approach. Special attention is devoted to the numerous developments of the KKR-GF method, that have been contributed in recent years by a number of work groups, in particular in the following fields: embedding schemes for atoms, clusters and surfaces, magnetic response functions and anisotropy, electronic and spin-dependent transport, dynamical mean field theory, various kinds of spectroscopies, as well as first-principles determination of model parameters.
758 citations
TL;DR: In this article, a summary of the relativistic calculations on multielectron or multicenter problems is provided, including relativism effects on the chemical properties of the periodic system of elements.
Abstract: Publisher Summary This chapter provides a summary of the relativistic calculations on multielectron or multicenter problems. The Dirac–Fock Hamiltonian and the main quantum electrodynamical (QED) corrections are discussed and the atomic and bandstructure calculations are reviewed. Then the construction of relativistic molecular orbitals and the solvable one-electron molecular and solid-state models are described. The simplest possible system for studying relativistic effects in chemical bonding is H2+. Several variational linear combinations of atomic orbitals (LCAO)-type solutions of the Dirac equation for H2+ show that the relativistic effects decrease the electronic energy by about –7 ╳ 10–6 a.u. The Dirac-Fock and Dirac-Slater molecular calculations, the relativistic semiempirical methods, and the perturbation treatments of relativistic effects are also described. In relativistic treatments of several spectroscopic properties, the entire formulation must be changed if relativistic wavefunctions are used. Some of its examples are considered. The chapter also presents a preliminary account of the relativistic effects on the chemical properties of the periodic system of elements.
422 citations
TL;DR: In this article, a new method was presented to calculate binding energies and eigenfunctions for molecules, using the Dirac-Slater Hamiltonian, for a series of molecules, including dihydrides H2X (X=O, S, Se, Te, Te), diatomic indium halides InX(X=F, Cl, Br, I), and metal chlorides XCl (X =B, Al, Ga, In, Tl).
Abstract: A new method is presented to calculate binding energies and eigenfunctions for molecules, using the Dirac–Slater Hamiltonian. A numerical basis set of four component wavefunctions is obtained from atom‐like Dirac–Slater wavefunctions. A discrete variational method (DVM) has been applied to generate the binding energies and eigenfunctions for the molecule. Results are given for a series of molecules, including dihydrides H2X (X=O, S, Se, Te), diatomic indium halides InX (X=F, Cl, Br, I), and metal chlorides XCl (X=B, Al, Ga, In, Tl). Comparison is made with results from nonrelativistic calculations using the DVM with numerical Hartree–Fock–Slater‐type wavefunctions and with other types of nonrelativistic calculations. In particular, relativistic level shifts and spin–orbit splitting have been analyzed. The theoretical ionization energies are compared with experimental results. Generally a very good agreement is obtained between the experimental and theoretical binding energies for the valence levels, calcu...
244 citations
01 Jan 1999
TL;DR: In this article, the properties of the underlying Dirac equation, set up within the framework of density functional theory (DFT), are discussed together with the Breit-interaction and Brooks' orbital polarization mechanism.
Abstract: Relativistic effects, in particular the spin-orbit coupling, give rise for magnetic systems to a great number of interesting and technologically important phenomena. The formal and technical aspects of corresponding fully relativistic theoretical investigations are reviewed. The properties of the underlying Dirac equation, set up within the framework of density functional theory (DFT) are discussed together with the Breit-interaction and Brooks’ orbital polarization mechanism. As an example for a corresponding band structure method, the Korringa-Kohn-Rostoker (KKR) Green’s function method is adopted. In particular, some technical aspects specific to this technique are discussed. The numerous applications that will be presented are primarily meant to demonstrate the many different facets of relativistic - this means in general - of spin-orbit induced effiects in magnetic solids. In addition, these also demonstrate the tremendous flexibility of band structure schemes based on the Green’s function formalism.
168 citations
TL;DR: An overview of band structure calculations on the fourth and fifth group transition metal monocarbides, mononitrides, and monoxides, published since the review article by Calais as mentioned in this paper, is given here.
Abstract: An overview is given here of band structure calculations on the fourth and fifth group transition metal monocarbides, mononitrides, and monoxides, published since the review article by Calais [J.-L. Calais, Adv. Phys. 26, 847 (1977)]. Furthermore, the relations of three categories of experimental properties, which allow insight into the electronic structure of the above mentioned compounds, and the results of band structure calculations are discussed. Theoretical predictions are compared with experimental findings. The considered experimental properties are valence band photoemission spectra, valence band x-ray emission spectra, and optical properties.
153 citations
References
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TL;DR: In this article, the problem of solving the Schrodinger equation in a periodic lattice is studied from the point of view of the variation-iteration method, which leads to a very compact scheme if the potential $V(r)$ is spherically symmetrical within the inscribed spheres of the atomic polyhedra and constant in the space between them.
Abstract: The problem of solving the Schr\"odinger equation in a periodic lattice is studied from the point of view of the variation-iteration method. This approach leads to a very compact scheme if the potential $V(r)$ is spherically symmetrical within the inscribed spheres of the atomic polyhedra and constant in the space between them. The band structure of the lattice is then determined by (1) geometrical structure constants, characteristic of the type of lattice and (2) the logarithmic derivatives, at the surface of the inscribed sphere, of the $s, p, d, \dots{}$ functions corresponding to $V(r)$. By far the greater part of the labor is involved in the calculation of (1), which needs to be done only once for each type of lattice; (2) can be obtained by numerical integration or directly from the atomic spectra. Although derived from a different point of view, this scheme turns out to be essentially equivalent to one proposed by Korringa on the basis of the theory of lattice interferences. The present paper also contains an application to the conduction band of metallic lithium.
1,251 citations
TL;DR: In this article, the mathematical basis of calculations of energy bands in periodic lattices using the Green's function method is presented and the usefulness of the method's usefulness discussed, and the original formulation by Kohn and Rostoker is modified to achieve more efficient and accurate evaluation of "structure constants" using symmetry considerations and the full Ewald summation procedure.
Abstract: The mathematical basis of calculations of energy bands in periodic lattices using the Green's function method is presented and the method's usefulness discussed. The original formulation of the method by Kohn and Rostoker is modified to achieve more efficient and accurate evaluation of "structure constants" using symmetry considerations and the full Ewald summation procedure. Formulas are derived giving the wave function both inside and outside the sphere inscribed in the unit cell. The method is demonstrated with the 3-dimensional Mathieu potential. Convergence is found to be very rapid both in this test case and in practical calculations on metals, and accurate energies and wave functions can be obtained without elaborate calculation even at points of low symmetry within the Brillouin zone.
233 citations
130 citations
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TL;DR: In this article, the relativistic terms of the Dirac Hamiltonian were obtained by two successive applications of the Foldy-Wouthuysen transformation to the nonrelativistic Hamiltonian, and the resulting Hamiltonian matrix was diagonalized to give the energy levels for PbTe.
Abstract: The augmented-plane-wave method for calculating the band structure of a solid has been extended to include the relativistic terms of the two-component Hamiltonian which is obtained by two successive applications of the Foldy-Wouthuysen transformation to the Dirac Hamiltonian. Basis functions for the secular equation which are basis partners for the irreducible representations of the double group are constructed from the eigenfunctions of the nonrelativistic Hamiltonian. Expressions for the relativistic matrix elements between these basis functions are found and numerically evaluated, and the resulting Hamiltonian matrix is diagonalized to give the energy levels for PbTe. The calculated bands, with only slight modification, appear to be consistent with available experimental information.
97 citations
TL;DR: In this article, a method for studying the band structure of complex crystals (i.e., crystals having more than one atom per unit cell) is developed, which is a generalization of one proposed independently and arrived at by different approaches by Korringa and Kohn and Rostoker.
Abstract: A method for studying the band structure of "complex" crystals (i.e., crystals having more than one atom per unit cell) is developed. This method is a generalization of one proposed independently and arrived at by different approaches by Korringa and Kohn and Rostoker for the study of the band structure of "simple" crystals. The general approach leads to a promising method when the crystalline potential can reasonably be approximated by a potential which is spherically symmetric within nonoverlapping spheres about the lattice sites and is constant elsewhere. Important virtues of the method are its expected accuracy and the fact that the largest part of the labor involved is in the computation of certain "structure constants" which are applicable to all crystals with the same crystallographic structure.
74 citations