In this paper, potential‐dependent transformations are used to transform the four‐component Dirac Hamiltonian to effective two‐component regular Hamiltonians. To zeroth order, the expansions give second order differential equations (just like the Schrodinger equation), which already contain the most important relativistic effects, including spin–orbit coupling. One of the zero order Hamiltonians is identical to the one obtained earlier by Chang, Pelissier, and Durand [Phys. Scr. 34, 394 (1986)]. Self‐consistent all‐electron and frozen‐core calculations are performed as well as first order perturbation calculations for the case of the uranium atom using these Hamiltonians. They give very accurate results, especially for the one‐electron energies and densities of the valence orbitals.

Relativistic regular two‐component Hamiltonians

Ever since its discovery, the Berry phase has permeated through all branches of physics. Over the last three decades, it was gradually realized that the Berry phase of the electronic wave function can have a profound effect on material properties and is responsible for a spectrum of phenomena, such as ferroelectricity, orbital magnetism, various (quantum/anomalous/spin) Hall effects, and quantum charge pumping. This progress is summarized in a pedagogical manner in this review. We start with a brief summary of necessary background, followed by a detailed discussion of the Berry phase effect in a variety of solid state applications. A common thread of the review is the semiclassical formulation of electron dynamics, which is a versatile tool in the study of electron dynamics in the presence of electromagnetic fields and more general perturbations. Finally, we demonstrate a re-quantization method that converts a semiclassical theory to an effective quantum theory. It is clear that the Berry phase should be added as a basic ingredient to our understanding of basic material properties.

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Berry phase effects on electronic properties

Quantum electrodynamical corrections to the fine structure of helium

https://th.fhi-berlin.mpg.de/site/uploads/Publications/CPC-180-2175-2009.pdf

Ab Initio Molecular Simulations with Numeric Atom-Centered Orbitals

We introduce density functional theory and review recent progress in its application to transition metal chemistry. Topics covered include local, meta, hybrid, hybrid meta, and range-separated functionals, band theory, software, validation tests, and applications to spin states, magnetic exchange coupling, spectra, structure, reactivity, and catalysis, including molecules, clusters, nanoparticles, surfaces, and solids.

Density functional theory for transition metals and transition metal chemistry

By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and $c$. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.

On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit

By combining two irreducible representations of the proper inhomogeneous Lorentz group, certain irreducible unitary representations of the complete Lorentz group including space and time inversion are obtained, together with a Schr\"odinger equation whose solutions constitute the representation space for these representations. The representations thus define a "canonical" form for covariant particle theories, in which not only the wave equations but the manner in which the wave functions transform under Lorentz transformations is prescribed. It is shown that by a suitable choice of representation, the Dirac, Klein-Gordon, and Proca equations can all be reduced to this canonical form. It is further shown that in the representation space provided, several possibilities exist for the identification of the transformations to be associated with space inversion, time inversion, and charge conjugation, thus suggesting the existence of several distinct relativistic theories for particles of any given spin. Conjectures are made as to the physical significance of these different possibilities when the equations are second-quantized. It is shown that each of the conventional theories employs only one of the available possibilities for these transformations, the choices being different for integral and half-integral spin theories.

Synthesis of Covariant Particle Equations

The possibility of covariantly describing a system of a fixed number of particles interacting directly is explored by attempting a direct "integration" of ihe commutation relations for the inhomogeneous Lorentz group under restrictions appropriate to the term "system of a fixed number of particles." By direct interaction is meant the fact that interaction between the particles is expressed directly in terms of coordinates, momenta, and spins for the particles rather than through the agency of a mediating field. The integration is carried out in considerable generality with the assumption that the infinitesimal generators of the group have expansions in inverse powers of the square of the velocity of light. The result coincides with that obtained earlier by Bakamjian and Thomas, but the method employed yields greater insight into the generality of the result, as well as into how further conditions beyond covariance, such as the property which is here called "separability of the interaction," can be incorporated in the result. The relationship of the result to the complete reducibility of a representation of the inhomogeneous Lorentz group is pointed out. Possible generalizations and applications of the procedures here employed are discussed. (auth)

Relativistic particle systems with interaction

The ground state energy and elementary excitations of a charged gas of bosons at high densities are examined by use of the method developed by Bogoliubov for boson gases. It is conjectured, but not herein established, that this method yields exact results in the high-density limit analogous to those obtained by Gell-Mann and Brueckner, and by Sawada, in the corresponding case of a charged fermion gas. The ground state energy is essentially correlation energy, and is therefore negative, and its magnitude varies as the one-fourth power of the density at high densities. The elementary excitations have for low momenta the energy appropriate to plasma waves, and for high momenta the energy appropriate to single-particle excitation. There is therefore an energy gap, suggesting that the gas is both a superfluid and a superconductor at low temperatures. At low densities the behavior of a charged gas is independent of statistics; hence, such a gap must disappear as the system is expanded.

Charged Boson Gas

A framework for describing those electromagnetic properties of Dirac (spin-\textonehalf{}) particles which determine their behavior when moving with low momentum through weak, slowly varying, external electromagnetic fields is developed by finding the most general interaction terms which may be added to the Dirac equation for the particle subject to appropriate conditions. The interaction terms found form an infinite series involving arbitrarily high derivatives of the electromagnetic potentials evaluated at the position of the particle. The series of coefficients of these terms then characterize the properties to be described and can be interpreted as a series of moments of the charge and current distribution associated with the Dirac particle. The first coefficient represents the charge of the particle, the second its anomalous magnetic moment. The third is a measure of the spatial extent of the particle's charge distribution, and the corresponding term describes a direct interaction of the particle and the charge distribution responsible for the external electromagnetic field. Higher terms in the series describe direct interactions of the particle with various derivatives of the external charge and current distribution. The correct physical interpretation of the terms is examined by transforming to the Foldy-Wouthuysen (nonrelativistic) representation of the Dirac equation. The consistency of the framework developed with field theoretical results is discussed. Limitations of the characterization derived here and the possibilities of broadening the assumptions on which it is based are examined. The results are applied in the succeeding paper to the interpretation of the electromagnetic properties of nucleons with particular reference to the electron-neutron interaction.

Leslie L. Foldy

Papers

On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit

Synthesis of Covariant Particle Equations

Relativistic particle systems with interaction

Charged Boson Gas

The Electromagnetic Properties of Dirac Particles