Dirac-Fock Approximation Research Articles

RELATIVISTIC 2S1/2 (L1) ATOMIC SUBSHELL RADIATIONLESS TRANSITION PROBABILITIES FOR Yb AND Hg

Radiationless transition rates to L1 vacancy states have been calculated ab initio in the Dirac–Fock approximation. The calculations include quantum-electrodynamic corrections. Results in the jj coupling scheme for all possible L1 transitions are tabulated for elements Yb and Hg.

Relativistic 2s1/2 (L1) atomic subshelldecay rates and fluorescence yields for Yb and Hg

De-excitation characteristics of 2s1/2 atomic hole states are poorly known. Measurements are hampered by the fact that the K-L1 radiative decay is dipole-forbidden in the non-relativistic limit, precluding standard coincidence experiments. Calculations call for state-of-the-art approaches. We report results of radiative and radiationless transition rates to L1 vacancy states carried out by two independent methods, i.e., a Dirac-Fock approximation and a second-order relativistic many-body approach. Calculations were performed for the elements Yb and Hg. Comparisons are made with measurements and with other theoretical results. Disagreement between experimental fluorescence yields and between experiment and theory persist even with the present, more advanced, methods.

Ab initio calculation of the enhancement of the electric dipole moment of an electron in the YbF molecule

The ground state of the molecule is modelled in the unrestricted Dirac-Fock approximation. Molecular orbitals are taken to be linear combinations of kinetically balanced four-component spherical Gaussian spinors. The nuclei are assumed to be spherical distributions of charge, the radial dependences of which are taken to be Gaussian. Uncontracted basis sets with 27s, 27p, 12d and 8f exponents on the Yb centre and 15s and 10p exponents on the F centre yield an equilibrium bond length of , a dissociation energy of and a dipole polarizability of ; the experimental values of these quantities are, respectively, , or , and . The matrix element of the operator , the interaction between an electronic electric dipole moment and the molecular electric field, is estimated to be atomic units within this approximation.

Atomic background in x-ray absorption spectra of fifth-period elements: Evidence for double-electron excitation edges.

A systematic x-ray absorption study to detect atomic background features at the K edges of fifth-period elements is reported. Samples of ${\mathrm{MoC}}_{\mathit{x}}$ (Z=42) and pure Rh (Z=45), Pd (Z=46), Ag (Z=47), In (Z=49), and Sn (Z=50) are measured at high temperature, mostly in the liquid phase, to reduce the intensity of the structural extended x-ray absorption fine structure (EXAFS) contribution. Main anomalies, showing a regular trend as a function of Z, are identified in the region around 100 eV and in the range 200--800 eV above an edge. They are assigned to the opening of absorption channels creating double-hole configurations such as [1s4p], [1s,3d], and [1s,3p]. Differences of self-consistent calculations in the Dirac-Fock approximation have been performed and the dominant transitions unambiguously identified. The results are relevant for the improvement of the EXAFS analysis at these edges.

Radiative Corrections in the Many-Particle and Dirac-Fock Approximations: Two Branches from a Common Origin

Starting from a common origin, the Bethe-Salpeter equation for an N-electron atom, expressions are derived for the energy through O(α 5mc 2) in the many-particle approach on the one hand and the Green′s function approach, leading to the Dirac-Fock equation in zeroth order on the other hand. For the sake of specificity, the case of He- and Li-like atoms is considered. The principal thrust of this, at first glance familiar, analysis is to emphasize the need for a careful treatment of the α 5mc 2 energy contributions, in order to avoid overlaps and double counting of contributions. Numerical examples are provided to illustrate the problem. The variation in the final results between correct and incorrect expressions is small, but serves to emphasize the need for more precise future calculations and more accurate experiments.

Solution of the one-electron Dirac equation for the heavy diatomic quasi-molecule NiPb 109+ by the finite element method

A two-dimensional, fully numerical approach to the four-component first-order Dirac equation using the finite element method is employed for diatomic systems. Using the Dirac-Fock approximation with only 2601 grid points we achieve for the heavy quasi-molecule NiPb 109+ at R = 0.002 au a relative accuracy better than 10 −8 for orbital energies (nuclear repulsion energies ignored).

Energy levels and transition amplitudes for alkali-metal atoms in the Brueckner approximation.

We use the B-spline approximation to solve the Brueckner equation. The Brueckner equation is obtained from Dyson equation for the one-particle propagator by cutting the kernel at second order in the electron-electron interaction. Thus the Brueckner approximation is a straightforward extension of the frozen-core Dirac-Fock approximation to include second-order correlations in a self-consistent way. We also give the explicit expression for the gauge-invariant transition in the Brueckner approximation. The numerical results of energies and transition amplitudes for alkali-metal atoms in the Brueckner approximation differ from experimental values by percentages ranging from 0.1% to 2%, increasing from Li to Cs. These results are comparable to existing accurate theoretical values.

Fully numerical relativistic calculations for diatomic molecules using the finite-element method.

A two-dimensional, fully numerical approach to the four-component Dirac equation utilizing the finite-element method for diatomic systems is presented. The convergence properties of the calculations, which are affected by the singularities of the relativistic wave functions, are studied and improved. In the Dirac-Fock approximation for ${\mathrm{H}}_{2}$, an absolute accuracy of about ${10}^{\mathrm{\ensuremath{-}}10}$ a.u. for the total energy and the 1(${\mathit{j}}_{\mathit{z}}$=1/2) orbital energy is achieved with only 2116 grid points, where ${\mathit{j}}_{\mathit{z}}$ is the component of the total angular momentum along the internuclear axis. Relativistic total energies, orbital energies, and their relativistic corrections were calculated in the Dirac-Fock-Slater approximation for some small diatomic systems such as LiH, ${\mathrm{Li}}_{2}$, BH, and ${\mathrm{C}}_{2}$. The accuracies available previously have been improved by several orders of magnitude, yet without presently exceeding 2601 grid points.

Dipole oscillator strengths for the alkalilike ions

We calculate the oscillator strengths of some transitions for the alkalilike ions in the Dirac–Fock approximation. The oscillator strengths calculated have included a nonlocal term derived previously by Feldman and Fulton from quantum electrodynamics. The numerical values of the length and velocity forms are shown to be equal. Most results of the present calculation are in good agreement with the available experimental data. Thus the formalism, with no parameters to be adjusted in contrast to the semiempirical methods, offers a simple and effective way to obtain qualitatively satisfactory oscillator strengths for large alkalilike ions.

Leading radiative corrections for atoms in the Dirac-Fock approximation

Previous work by the first two authors, in which analytical expressions were obtained for the low virtual photon frequency (Bethe log and constant log) contributions to leading radiative corrections in the Dirac-Fock (DF) approximation of many-electron atoms, has been extended to all leading radiative corrections, including those contributed by high virtual photon frequencies. The photon cutoff frequency cancels out when the low and high frequency contributions are combined. The resulting analytic expressions have a simple structure. They can be divided into contributions from core and valence electrons and photon exchange between valence and core electrons. The results obtained correspond in some detail to those obtained many years ago in leading order for positronium, Helium and Hydrogen (including leading recoil effects in radiative corrections for the latter). Leading next order αZ corrections are also obtained, so that extension of the results to isoelectronic sequences of atoms with moderate values of Z can also be considered.

Transition amplitudes in the Dirac–Fock approximation

Transition amplitudes for several iso-electronic sequences are computed on the frozen-core Dirac–Fock approximation using the B-spline method. The length and velocity forms of the oscillator strengths are shown to be equal numerically if the nonlocal effective vertex is used. These gauge-invariant values, when compared with other theoretical results aiming for the exact values of the oscillate strengths, show satisfactory quality. We also find a peculiar feature for some transitions.

Leading radiative corrections for atoms in the Dirac-Fock approximation: G. Feldman, T. Fulton, and J. Ingham. Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218

Leading radiative corrections for atoms in the Dirac-Fock approximation: G. Feldman, T. Fulton, and J. Ingham. Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218

Relativistic self-consistent calculations for small diatomic molecules by the finite element method

A two-dimensional, fully numerical approach to the four-component first-order Dirac equation using the finite element method is employed for diatomic systems. Using the Dirac—Fock approximation with only 2116 grid points we achieve for H 2 an absolute accuracy of about 10 −10 au for the ground-state total energy. For the many-electron systems Li 2 and BH, we obtain a similar accuracy within the Dirac—Fock—Slater approximation which allows us to determine the relativistic contribution to the total as well as orbital energies very precisely.

Collision strengths for electron impact excitation with Ca XV: a relativistic approach

Collision strengths for electron impact excitation between the lowest 20 levels among the 1s22s22p2, 2s2p3 and 2p4 configurations of Ca XV have been computed for collision energies of 15, 30, 45 and 60 Ryd in a Dirac-Fock approximation. Comparisons have been made with the R-matrix and distorted-wave collision strengths. Differences are discussed and the accuracy is assessed.

Electron-impact ionization of highly charged ions in a level-to-level multiconfiguration Dirac-Fock approximation.

The electron-impact ionization cross section for highly charged ions is obtained in a level-to-level multiconfiguration approximation. The target atomic ion is described by a Dirac-Fock approximation while the interaction between electrons includes static, magnetic, and retardation effects. Level-to-level cross-section results for several highly charged ions are presented to illustrate the computational method.

Log terms for leading radiative corrections in atoms in the Dirac-Fock approximation

Analytical expressions are obtained for the log terms arising from the low photon frequency part of the radiative corrections in the Dirac-Fock (DF) approximation of many-electron atoms. The atoms considered are iso-electronic sequences of non-degenerate electronic cores, plus or minus one electron. The nuclear charge is taken to be small enough that the Lamb shift is still a small correction to the DF energies. If the unshielded nuclear charge is a point charge, the principal effect of the shielding by the electrons in the DF approximation is to replace the absolute square of the hydrogenic wave function at the origin of coordinates by the corresponding expression for the DF wave function. A preliminary numerical test of this result leads to better agreement between theory and experiment for the Li iso-eletronic sequence than has hitherto been the case.

Electron-impact ionization of highly charged ions in lowest-order QED theory.

The electron-impact ionization cross section for highly charged ions is calculated as a triple partial-wave expansion of scattering amplitudes obtained from lowest-order QED theory. The interaction between electrons includes static, magnetic, and retardation effects. The target atomic ion is described by a Dirac-Fock approximation and the radial partial waves are computed in relativistic distorted-wave potentials. The QED cross-section results for ${\mathrm{U}}^{91+}$ and ${\mathrm{U}}^{90+}$ are compared with nonrelativistic and relativistic theories as well as a recent heavy-ion channeling experiment.

Numerical study of the role of shielding and exchange effects and of correlation and relativistic corrections in the Dirac-Fock approximation.

The purpose of this work is to provide a quantitative underpinning to the qualitative understanding of the various physical effects which play a role in the spectra of many-electron atoms. Shielding and exchange contributions are studied, as functions of the principal quantum number n, the orbital angular momentum l, and the nuclear charge Z, for neutral alkali-metal atoms in the Dirac-Fock (DF) approximation. Fine-structure effects and valence-state correlation corrections to the DF approximation are also considered for these atoms. Breit interaction and correlation corrections are discussed for the core states of noble gases. Previous numerical computations of these various contributions to atomic energy levels by others are extended to provide a larger data set to serve as a basis for the study of estimates and trends.

Ground state correlation energy of the Be-sequence forZ=4?20 in MCDF approximation

The ground state (J=0) electronic correlation energy of the 4-electron Be-sequence is calculated in the Multi-Configuration Dirac-Fock approximation forZ=4–20. The 4 electrons were distributed over the configurations arising from the 1s, 2s, 2p, 3s, 3p and 3d orbitals. Theoretical values obtained here are in good agreement with experimental correlation energies.

Radiative corrections for many-electron atoms in perturbation theory

The techniques of Erickson and Yennie for treating the Lamb shift are generalized to obtain finite expressions for additional radiative corrections. These terms were previously formally generated by us in their unrenormalized form in perturbation theory, and are of significance for the case of many-electron atoms. For the sake of brevity, only log corrections (terms proportional to ln( Zα) 2 and the analogues of the Bethe-log term) are considered. Explicit finite expressions, suitable for immediate numerical evaluation, are obtained. The effect of these corrections is first to replace the unshielded nucleus by one that is shielded by the core electrons. A secondary effect accounts for the change of the energy of a given orbital state due to the radiative effects on the core electrons during their interaction with that orbital state. Because of the simple physical interpretation of the perturbation theory results, the corresponding radiative correction contributions to the Dirac-Fock approximation can be obtained in the simplest approximation (neglect of Bethe-log terms) and can be conjectured in the general case.