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

It is pointed out that the Lippmann-Schwinger integral equation, as it is usually written, does not necessarily have a unique solution when applied to the motion of three or more bodies. It follows that a certain amount of caution must be exercised in using the Lippmann-Schwinger equation as the basis for an approximation procedure.

Application of formal scattering theory to many-body problems

In the synchro-cyclotron (or frequency-modulated cyclotron) the higher energies available are obtained at the expense of a decrease in the ion current compared with that available from the conventional cyclotron. This decrease results from the fact that during only a small fraction of the frequency-modulation cycle is it possible for ions to be captured into phase stable orbits that do not return to the center during the first phase oscillation. By solving the phase equation, it is possible to obtain a general expression for this fraction, which is defined as the capture efficiency. At a constant dee voltage and varying rate of frequency modulation, the capture efficiency has a maximum at an equilibrium phase angle of 30\ifmmode^\circ\else\textdegree\fi{} (corresponding to an energy gain per turn equal to half the maximum available). For larger equilibrium phase angles the efficiency decreases as a result of the smaller range of phase stability, while for smaller phase angles it decreases as a result of return of particles to the center. The maximum efficiency is proportional to the square root of the dee voltage or alternatively to the square root of the rate of frequency modulation, and depends on the charge and mass of the ions only through the ratio of charge to mass. Comparisons of the theoretical expectations with available experimental data show satisfactory agreement. Capture efficiencies for present designs of synchro-cyclotrons are of the order of 0.1 to 2 percent.

Theory of the Synchro-Cyclotron

It is shown that for nuclei in which the ground state wave function is symmetric in the space coordinates of all the nucleons, such as ${\mathrm{H}}^{2}$, ${\mathrm{H}}^{3}$, ${\mathrm{He}}^{3}$, ${\mathrm{He}}^{4}$, the integrated bremsstrahlung-weighted cross section for electric dipole absorption is simply related to the mean-square radius of the nucleus, independent of the existence of correlations between the motions of the nucleons. This relationship was first derived by Levinger and Bethe on the assumption of absence of correlation, and was found to be at variance with known experimental results in heavy nuclei. Experimentally, the relationship is well verified by data on ${\mathrm{H}}^{2}$ and ${\mathrm{He}}^{4}$. The relationship of the integrated bremsstrahlung-weighted cross section to certain nuclear parameters in the case of ${\mathrm{Li}}^{6}$ is briefly discussed.

Photodisintegration of the Lightest Nuclei

The phenomenon of vacuum polarization can be studied apart from other higher-order electrodynamic effects through its modification of the electrostatic interaction of heavy charged particles. In particular the principal deviations of the $2p\ensuremath{-}1s$ level separations from the Bohr formula in light mu-mesonic atoms ($Z\ensuremath{\le}6$) are expected to arise from vacuum polarization effects rather than relativistic effects (ordinary hydrogenic fine structure) or finite nuclear size. The vacuum polarization contribution to the electrostatic interaction of two protons requires slight changes in the usual analysis of proton-proton scattering data to obtain information about nuclear forces. The effect of vacuum polarization on the electrostatic energies of nuclei is also briefly discussed.

/pdf/some-physical-consequences-of-vacuum-polarization-2mleidg4od.pdf

Some physical consequences of vacuum polarization

It is demonstrated that any scattering amplitude can formally be expressed as a sum of terms each corresponding to the exchange of a particular isotopic spin without requiring the concept of crossing. Employing only the optical theorem and invariance under isotopic spin rotations the following theorem is proved: The contribution to the imaginary part of the forward scattering ampiitude arising from the exchange of zero isotopic spin cannot be arbitrarily small compared to contributions from the exchange of any other isotopic spin or spins. From the same argument follows a second theorem: If the total cross section for two particles is independent of their isotopic spin state, then only the exchange of isotopic spin zero contributes to the imaginary part of the forward scattering amplitude. In combination with the hypothesis that high-energy scattering is dominated by the exchange of a sinkle Regge pole, it follows that the Pomeranchuk-pole must have isotopic spin zero. The relation to the Pomeranchuk-Okun' rule is discussed. (auth)

Leslie L. Foldy

Papers

Application of formal scattering theory to many-body problems

Theory of the Synchro-Cyclotron

Photodisintegration of the Lightest Nuclei

Some physical consequences of vacuum polarization

Isotopic spin of exchanged systems