Unified treatment of complete orthonormal sets for wave functions, and Slater orbitals of particles with arbitrary spin in coordinate, momentum and four-dimensional spaces

The new analytical relations of complete orthonormal sets for the tensor wave functions and the tensor Slater orbitals of particles with arbitrary spin in coordinate, momentum and four-dimensional spaces are derived using the properties of tensor spherical harmonics and complete orthonormal scalar basis sets of ψα-exponential type orbitals, ?α-momentum space orbitals and zα-hyperspherical harmonics introduced by the author for particles with spin s=0, where the . All of the tensor wave functions obtained are complete without the inclusion of the continuum and, therefore, their group of transformations is the four-dimensional rotation group O(4). The analytical formulas in coordinate space are also derived for the overlap integrals over tensor Slater orbitals with the same screening constant. We notice that the new idea presented in this work is the combination of tensor spherical harmonics of rank s with complete orthonormal scalar sets for radial parts of ψα-, ?α- and zα-orbitals, where .     

Interpretation of the Experimental Electron Momentum Spectra of 5e1/2 and 5e3/2 Orbitals of CF3I with Relativistic Calculations

With our newly developed method, we calculate the spin-orbit splitting states 5e1/2 and 5e3/2 of the CF3I molecule incorporating the relativistic effects. Our theoretical results agree excellently with the recent experimental observations. The present study shows that relativistic effects can evidently change the electron momentum distributions of molecular orbitals when a medium Z element is included, such as iodine.     

Relativistic core binding energies of selected atoms: Comparison with experiment and other calculations

Single-configuration Dirac—Fock binding energies are reported for K 2p, Rb 3p, Cs 3d, Mg 1s, Zn 2p, and Cd 3d as well as for the corresponding levels in the cations of these elements and the neighboring rare gases. The results are compared with similar Dirac—Slater and Hartree—Fock calculations, and with experiment. It is found that the calculated binding energies are generally in very good agreement with experiment and are superior to the Dirac—Slater and Hartree—Fock results. The relativistic contribution is shown to be significant even for lighter atoms.     

Relativistic effects on atomic and molecular properties of the heaviest elements

Interaction of superheavy element 112 and its homolog Hg with inert and gold surfaces was studied on the basis of atomic and molecular fully-relativistic (4-component) DFT electronic structure calculations. Performance of additional non-relativistic calculations allowed one to demonstrate the role and magnitude of relativistic effects on adsorption energies and bond distances of the studied systems. For example, on quartz, element 112 will be stronger adsorbed than Hg by about 5 kJ/mol (or at 5 degrees higher temperatures) due to the stronger van der Waals interaction. This is caused by the relativistically contracted smallest atomic radius of element 112. Non-relativistically, the trend would be opposite. On surface of gold, element 112 will be about 20 kJ/mol weaker adsorbed than Hg (i.e., it will be deposited at about 100 degrees lower temperatures than Hg). Such a decrease in ΔH_ads comes at the account of the weaker interaction of the relativistically stabilized 7s_1/2(112) orbital with valence orbitals of gold. Still, the relatively large adsorption energy of element 112 is indicative that it is a transition metal forming intermetallic compounds with Au and other metals due to the involvement of the relativistically destabilized 6d orbitals. The influence of relativistic effects on the adsorption energy depends, however, on the adsorption position.     

Theoretical Study of Relativistic Retardation Effects： the Abnormal Fine Structure of OⅡ

Using multi-configuration Dirac-Fock and relativistic configuration interaction methods with high-order corrections, we report our precise calculation results of the fine-structure energy levels of the ground-state configuration of OⅡ（1s^22s^22p^3）. Our calculated fine-structure splittings of ^2D3/2,5/2 and ^2p1/2,3/2 are abnormal We elucidate that the transverse （Breit） interaction, i.e. relativistic retardation effect, plays an important role for the abnormal fine-structure splittings. Our calculation results are in good agreement with experimental measurements.     

Electron density and Fisher information of Dirac-Fock atoms

Numerical calculations on the gradient and Laplacian forms of the position space Fisher information measure are reported using the 1-normalised Dirac-Fock densities (shape function), σ(r), for atoms H-Lr. It is shown that the difference in effective electrostatic potentials, corresponding to the gradient and the Laplacian form of Fishers' information, is completely defined by the shape function (the density per particle) at the nucleus, σ(r=0). The influence of relativistic effects on the Fisher information is recovered for the first time.     

Propagation regimes of intense circularly polarized laser beam in magnetoactive plasma

This paper presents an analytical and numerical investigation of an intense circularly polarized wave propagating along the static magnetic field parallel to oscillating magnetic field in magnetoactive plasma. In the relativistic regime such a magnetic field is created by pulse itself. The authors have studied different regimes of propagation with relativistic electron mass effect for magnetized plasma. An appropriate expression for dielectric tensor in relativistic magnetoactive plasma has been evaluated under paraxial theory. Two modes of propagation as extraordinary and ordinary exist; because of the relativistic effect, ultra-strong magnetic fields are generated which significantly influence the propagation of laser beam in plasma. The nature of propagation is characterized through the critical-divider curves in the normalized beam width with power plane For given values of normalized density (ω_p/ω) and magnetic field (ω_c/ω) the regions are namely steady divergence (SD), oscillatory divergence (OD) and self-focusing (SF). Numerical computations are performed for typical parameters of relativistic laser-plasma interaction: magnetic field B = 10-100 MG; intensity I = 10¹⁶ to 10²⁰ W/cm²; laser frequency ω = 1.1 × 10¹⁵ s⁻¹; cyclotron frequency ω_c = 1.7 × 10¹³ s⁻¹; electron density n_e = 2.18 × 10²⁰ cm⁻³. From the calculations, we confirm that a circularly polarized wave can propagate in different regimes for both the modes, and explicitly indicating enhancement in wave propagation, beam focusing/self-guiding and penetration of E-mode in presence of magnetic field.     

Relativistic effects on LEED intensities from Au(111)

Comparison of relativistically and nonrelativistically calculated intensity versus energy profiles in low energy electron diffraction (LEED) from the (111) surface of Au (Z = 79) reveals that relativistic corrections are quite significant. They can however, be obtained in very good approximation by quasirelativistic calculations, in which spin-averaged relativistic phase shifts are used as input for the nonrelativistic multiple scattering formalism. Further, relativistic effects on intensities are found to be comparable to differences arising from different approximations to the exchange part of the ion core potential.     

Alternative evaluation of statistical indicators in atoms: The non-relativistic and relativistic cases

In this work, the calculation of a statistical measure of complexity and the Fisher-Shannon information is performed for all the atoms in the periodic table. Non-relativistic and relativistic cases are considered. We follow the method suggested in [C.P. Panos, N.S. Nikolaidis, K.Ch. Chatzisavvas, C.C. Tsouros, arXiv:0812.3963v1] that uses the fractional occupation probabilities of electrons in atomic orbitals, instead of the continuous electronic wave functions. For the order of shell filling in the relativistic case, we take into account the effect due to electronic spin-orbit interaction. The increasing of both magnitudes, the statistical complexity and the Fisher-Shannon information, with the atomic number Z is observed. The shell structure and the irregular shell filling is well displayed by the Fisher-Shannon information in the relativistic case.     

Relativistic calculation of spectra of 2-2 transitions in O- and F-like atomic ions

We have studied theoretically the structure of 1s²2s^n₁2p^n₂ levels in O- and F-like ions with z = 25 ÷ 40. Relativistic perturbation theory is used for the total interelectronic interaction. Allowance is made for Coulomb and relativistic interelectronic interactions. The first order perturbation theory for retardation of the interaction has been calculated fully. The complete calculation has also been made for the nonrelativistic correction in the second order and the Hartree-Fock part of the nonrelativistic correction in the third order. The Hartree-Fock relativistic correction in the second order has been taken into account fully for F-like ions and partially for O-like ions. Corrections to higher orders for z ? 28 have been found empirically and extrapolated to the region z = 29 ÷ 40. For energies of 2-2 transitions, calculation errors for z ? 28 do not exceed 900 cm^-1 and do not increase with z. Typical errors obtained with conventional, more cumbersome calculations are 5000 cm^-1 at z = 30.     

多电子原子能量的相对论修正

以Breit-Pauli哈密顿的球张量形式为基础,借助不可约张量理论,建立了计算多电子原子能量的相对论修正的一种解析理论形式,导出了多电子原子相对论修正项(包括相对论质量修正项、单体和双体达尔文修正项、自旋-自旋接触相互作用项和轨道-轨道相互作用项)在斯莱特表象中的矩阵元的解析表达式,完成了所有角向积分和自旋求和计算.利用所建立的理论,对类锂体系(1s)2(2p)2P态能量的相对论修正进行了具体计算.     

Conformal symmetry of relativistic and nonrelativistic systems and AdS/CFT correspondence

The nonlinear realization of conformal so(2,d) symmetry for relativistic systems and the dynamical conformal so(2,1) symmetry of nonrelativistic systems are investigated in the context of AdS/CFT correspondence. We show that the massless particle in d-dimensional Minkowski space can be treated as the system confined to the border of the AdS_d+1 of infinite radius, while various nonrelativistic systems may be canonically related to a relativistic (massless, massive, or tachyon) particle on the AdS₂ × S^d−1. The list of nonrelativistic systems “unified” by such a correspondence comprises the conformal mechanics model, the planar charge-vortex and three-dimensional charge-monopole systems, the particle in a planar gravitational field of a point massive source, and the conformal model associated with the charged particle propagating near the horizon of the extreme Reissner-Nordström black hole.     

Born Reciprocity and the Granularity of Spacetime

The Schrödinger-Robertson inequality for relativistic position and momentum operators X ^μ, P _ν, μ, ν = 0, 1, 2, 3, is interpreted in terms of Born reciprocity and ‘non-commutative’ relativistic position-momentum space geometry. For states which saturate the Schrödinger-Robertson inequality, a typology of semiclassical limits is pointed out, characterised by the orbit structure within its unitary irreducible representations, of the full invariance group of Born reciprocity, the so-called ‘quaplectic’ group U(3, 1) #x2297;_s H(3, 1) (the semi-direct product of the unitary relativistic dynamical symmetry U(3, 1) with the Weyl-Heisenberg group H(3, 1)). The example of the ‘scalar’ case, namely the relativistic oscillator, and associated multimode squeezed states, is treated in detail. In this case, it is suggested that the semiclassical limit corresponds to the separate emergence of spacetime and matter, in the form of the stress-energy tensor, and the quadrupole tensor, which are in general reciprocally equivalent.     

Measurement of the tensor analyzing power for relativistic-deuteron fragmentation as a means for studying the deuteron structure within light-front dynamics

New data on the vector (A_y) and tensor (A_yy) analyzing powers for the reaction ⁹Be (d, p)X at a primary deuteron momentum of 5 GeV/c for a proton emission angle of 178 mrad are obtained by using the synchrophasotron of the Joint Institute for Nuclear Research (JINR, Dubna). The experimental data on A _yy are analyzed within the approach based on light-front dynamics, the relativistic wave function obtained by Karmanov and his colleagues being used for the deuteron. It is shown that, in contrast to what one has from calculations with standard nonrelativistic deuteron wave functions, all relevant data can be explained in this approximation without resort to additional degrees of freedom.     

Relativistic calculation of atomic X-ray transition rates

The fully relativistic transition probabilities in atomic X-ray emission are calculated for a wide range of selected elements of the periodic table, fromZ=21 toZ=93. The treatment includes the effect of the retardation of the radiation field. The atomic model used is the self-consistent relativistic Hartree-Fock, with the Slater approximation for the exchange term. The formulas for EL and ML multipole transition rates are derived, and transition rates are calculated for all transitions to theK-shell,L-subshells, andM-subshells. The contributions of all multipoles, as determined by the angular momentum and parity selection rules, are included in the present work. The calculated values of (Kα ₂/Kα ₁), (Kβ ₁/Kα ₁), (Kβ ₃/Kβ ₁) and (Kβ ₂ Kβ ₁) transition rates are in good agreement with the available experimental data. Rates for the forbidden (M1) 2s _1/2→1s _1/2 transitions are also presented.     

Heavy Quark Potential and Quarkonium Spectrum in Nonperturbative pNRQCD

The charmonium and bottomonium mass spectra are investigated systematically in potential nonrelativistic QCD with the heavy quark potential computed by lattice QCD simulations nonperturbatively. The potential consists of a static potential and relativistic corrections classified in powers of the inverse of heavy quark mass m, and the effects of the O(1/m) correction, the O(1/m ²) spin–orbit and spin–tensor corrections on the mass spectra are examined systematically. The pattern of the mass spectra is found to be in fairly good agreement with the experimental data, in which the O(1/m) correction gives an important contribution.     

The study of the 4s4p configuration of the Zn isoelectronic sequence using the relativistic jj-coupling approach

The 4s4p configuration of Zn is analyzed using the Relativistic jj-coupling approach. The experimentally determined relativistic Slater integrals are compared with the results of numerical codes, both quasi- and fully-relativistic ones. In this work, they are estimated, semi-empirically, the two J = 1 levels up Z = 70 and the ¹P₁ level up Z = 92 by judicious interpolation and extrapolation of energies. The comparison with extensive relativistic configuration-interaction calculations indicates that differences between both approaches are of the order of measurement accuracies.     

Conformal and poincaré tensor calculi in N=1 supergravity

We present the superconformal tensor calculus for N=1 supergravity in a complete form; irreducible multiplets, their multiplication and embedding formulae and invariant action formulae. It is further clarified in detail how the various versions of N=1 Poincaré supergravity (i.e. with different sets of auxiliary fields) are reproduced from the unique superconformal theory. The tensor calculi for all the known versions of Poincaré supergravity are derived explicitly.     

Electronic Curves Crossing in Methyl Iodide by Spin--Orbit Ab Initio Calculation

An ab initio investigation of electronic curve crossing in a methyl iodide molecule is carried out using spin-orbit multiconfigurational quasidegenerate perturbation theory. The one-dimensional rigid potential curves and optimized effective curves of low-lying states, including spin-orbit coupling and relativistic effects, are calculated. The spin-orbit electronic curve crossing between ^3Qo＋ and ^1Q1, and the shadow minimum in potential energy curve of ^3Qo＋ at large internuclear distance are found in both sets of the curves according to the present calcu- lations. The crossing position is in the range of Re-1 = 0.2370 3：0.0001 nm. Comparisons with other reports are presented.     

Studying Li and He− in the 1s2s2p configuration

Expressions are derived for the nonrelativistic energy of the 1s2s2p configuration. In both the possible electron momentum binding systems (1s2s)2p and 1s(2s2p) values are determined for the parameters of the generalized hydrogenlike analytical radial orbitals for a lithium atom and a negative helium ion. The Li and He^– energy spectra in the 1s2s2p configuration are compared with the theoretical calculations of other authors and with experimental data.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 70–74, June, 1977.