Three-electron Systems Research Articles

Spectra of the Energy Operator of Three-Electron Systems in the Impurity Hubbard Model. Second Doublet State

Spectra of the Energy Operator of Three-Electron Systems in the Impurity Hubbard Model. Second Doublet State

Variational Calculation of Lithium-Like Ions from B+2 to N+4 Using β-Type Roothaan–Hartree–Fock Wavefunction

Within the KaKB, KaLa, and KBLa shells in the position space, the properties of a series of three-electron systems, for instance, B+2, C+3, and N+4 ions, have been studied. This required the partitioning of the two-particle space-spin density and was explicit for the Hartree–Fock description which have been proposed by considering a basis set based on single-zeta B-type orbitals (BTOs). The one- and two-body radial electronic densities R(r1), R(r1, r2), moments ⟨rn1⟩, X-ray form factor F(s), nucleus density R(0), nuclear magnetic shielding constant qd, and the diamagnetic susceptibility бs in the position space are reported. Our results are realized via the Mathematica program and compared with previous theoretical values in the literature.

Electron-impact double ionization of He (1s2s 3 S)

A time-dependent close-coupling method for three-electron atomic systems is used to calculate the electron-impact double ionization of the He (1s2s 3 S) excited state. Partial cross sections are calculated for total angular momenta from . An exponential fitting expression is used to extend the calculated partial cross sections to . The total cross sections at three different incident energies are compared with recent crossed-beams measurements and are found to lie within the experimental error bars. The magnitude of the double ionization cross section is found to be similar to that of the double ionization cross section from the ground state of He.

The Structure of Essential Spectra and Discrete Spectrum of Three-Electron Systems in the Impurity Hubbard Model—Quartet State

We consider a three-electron system in the Impurity Hubbard model with a coupling between nearest-neighbors. Our research aim consists of studying the structure of essential spectrum and discrete spectra of the energy operator of three-electron systems in the impurity Hubbard model in the quartet state of the system in a v-dimensional lattice. We have reduced the study of the spectrum of the three-electron quartet state operator in the impurity Hubbard model to the study of the spectrum of a simpler operator. We proved the essential spectra of the three-electron systems in the Impurity Hubbard model in the quartet state is the union of no more than six segments, and the discrete spectrum of the system is consists of no more than four eigenvalues.

Observation of strong two-electron–one-photon transitions in few-electron ions

We resonantly excite the $K$ series of O$^{5+}$ and O$^{6+}$ up to principal quantum number $n=11$ with monochromatic x rays, producing $K$-shell holes, and observe their relaxation by soft-x-ray emission. Some photoabsorption resonances of O$^{5+}$ reveal strong two-electron--one-photon (TEOP) transitions. We find that for the $[(1s\,2s)_1\,5p_{3/2}]_{3/2;1/2}$ states, TEOP relaxation is by far stronger than the radiative decay and competes with the usually much faster Auger decay path. This enhanced TEOP decay arises from a strong correlation with the near-degenerate upper states $[(1s\,2p_{3/2})_1\,4s]_{3/2;1/2}$ of a Li-like satellite blend of the He-like $K\alpha$ transition. Even in three-electron systems, TEOP transitions can play a dominant role, and the present results should guide further research on the ubiquitous and abundant many-electron ions where electronic energy degeneracies are far more common and configuration mixing is stronger.

Revised and extended analysis of doubly ionized bismuth: Bi III

The third spectrum of bismuth (Bi III) with its ground configuration 5d106s26p has been investigated in the wavelength region 278–2000 Å using spectrograms recorded on a 3-m normal incidence vacuum spectrograph. The observed spectrum was interpreted using semi-empirical calculations with Cowan's code in the framework of the quasi-Relativistic Hartree-Fock method. Analyses made by previous investigators on one and three-electron systems were checked thoroughly, while the tentative identification of the 2S1/2 level of the 5d106s6p2 configuration is confirmed. For the first time we had investigated the 6p3, 6s6p6d and 6s6p7s odd configurations based on the identification of 99 spectral lines. In total, 67 energy levels are now established in the 5d10[6s2np (n = 6–8) + 6s2nf (n = 5–7) + 6s2nh (n = 6–7)], 6s6p2 and 5d96s26p2 configurations, among which 23 energy levels were newly located. The least squares parametric fitting calculations were performed to interpret the final results of the analysis. Optimization of the observed energy levels was carried out with the computer program LOPT. The ionization energy was found to be 206,242.0 ± 1.4 cm−1 (25.5708±0.0002 eV).

The Ground State (1s<sup>2</sup>2s)<sup>2</sup>S and the Low-lying Excited (1s<sup>2</sup><i>n</i>s)<sup>2</sup>S States Energy Calculations of Li-Like Ions Using Special Forms of the Hylleraas-Type Wave Functions

In this paper, we have applied the complex rotation method to the calculations energies of the ground state (1s22s) 2S and the low-lying excited (1s2ns)2S states with n = 3, 4 and 5; of lithium and its isoelectronic series. For the ground state, the calculations were made up to Z = 20 and for the low-lying excited states up to Z = 10. These energies calculations were made using new special forms of Hylleraas-type wave functions designed by combining incomplete radial hydrogenic wave functions and Hylleraas-type wave functions. Using a numerical calculation program, the values of the resonance energies are calculated. Our results are compared with the results of ab-initio calculations using Hylleraas type wave functions and with semi-empirical results by Screening Constant by Unit Nuclear Charge (SCUNC) formalism. Analysis of the present results is achieved by calculating the ratio and the difference between our values and those of other authors. The results obtained are in good agreement with those of the theoretical methods available in the literature. This agreement shows the adequacy of our wave function with small bases to satisfactorily describe the ground state and the low-lying excited states of the three-electron atomic systems.

Spectra of the Energy Operator of Three-Electron Systems in the Impurity Hubbard Model.Second Doublet State

We consider the three-electron systems in the impurity Hubbard model and investigated the spectra of the system in the second doublet state in the ν-dimensional lattice Zν .

Double ionization of a three-electron atom: Spin correlation effects

We study the effects of spin degrees of freedom and wave function symmetries on double ionization in three-electron systems. Each electron is assigned one spatial degree of freedom. The resulting three-dimensional Schr\"odinger equation is integrated numerically using grid-based Fourier transforms. We reveal three-electron effects on the double ionization yield by comparing signals for different ionization channels. We explain our findings by the existence of fundamental differences between three-electronic and truly two-electronic spin-resolved ionization schemes. We find, for instance, that double ionization from a three-electron system is dominated by electrons that have the opposite spin.

Spin-free wave functions for small molecules

In an earlier paper, we described a simple version of spin-free calculations of the energy of two three-electron systems, HeH and H3. Here, we extend our treatment to LiH+ and to two four-electron systems, LiH and BeH+. The necessary extensions are not trivial, but we are still able to obtain energies in atomic units consistent to at least four figures using very simple approximate wave functions. The wave functions used are given in the text, and are based on two-centre molecular orbitals consisting of exponential functions with varied exponents. Our procedures may be viewed as a convenient fusion of the classic valence-bond and molecular-orbital approaches to molecular structure.

Microcalorimeters for X-Ray Spectroscopy of Highly Charged Ions at Storage Rings

X-ray spectroscopy of highly charged heavy ions is an important tool for the investigation of many topics in atomic physics. Such highly charged ions, in particular hydrogen-like uranium, are investigated at heavy ion storage rings, where high charge states can be produced in large quantities, stored for long times and cooled to low momentum spread of the ion beam. One prominent example is the determination of the 1s Lamb Shift in hydrogen-like heavy ions, which has been investigated at the Experimental Storage Ring (ESR) at the GSI Helmholtz Centre for Heavy Ion Research. Due to the large electron binding energies, the energies of the corresponding photon transitions are located in the X-ray regime. To determine the transition energies with high accuracy, highly resolving X-ray spectrometers are needed. One concept of such spectrometers is the concept of microcalorimeters, which, in contrast to semiconductor detectors, uses the detection of heat rather than charge to detect energy. Such detectors have been developed and successfully applied in experiments at the ESR. For experiments at the Facility for Antiproton and Ion Research (FAIR), the Stored Particles and Atoms Collaboration (SPARC) pursues the development of new microcalorimeter concepts and larger detector arrays. Next to fundamental investigations on quantum electrodynamics such as the 1s Lamb Shift or electron–electron interactions in two- and three-electron systems, X-ray spectroscopy may be extended towards nuclear physics investigations like the determination of nuclear charge radii.

Simple spin-free calculations for small molecules

A simple treatment of spin-free quantum theory is applied to three-electron systems, constructing the matrix representations and the corresponding projection operators. The theory is used to treat two example molecules HeH using a model where a helium atom interacts first with a proton and then an electron; and linear H3 where a basis of three 1s functions is used to calculate a set of allowed states. For both systems the energies compare well to earlier, more extensive, calculations.

Three-electron atoms and ions in a magnetic field

The energies and physical properties of three-electron systems are studied using an explicitly correlated Gaussian basis optimized by the stochastic variational method. The stability of the system as a function of the nuclear charge is analyzed. The role of the Coulomb and magnetic interactions in shaping the structure of these systems is discussed. The accuracy of the energies is substantially improved for high magnetic fields.

Quantum electrodynamicsmα6andmα7lnαcorrections to the fine splitting in Li andBe+

We derive quantum electrodynamics corrections to the fine structure in three-electron atomic systems at $m{\ensuremath{\alpha}}^{6}$ and $m{\ensuremath{\alpha}}^{7}ln\ensuremath{\alpha}$ orders and present their numerical evaluations for the Li atom and ${\mathrm{Be}}^{+}$ ion.

Bound state spectra and properties of the doublet states in three-electron atomic systems

The bound state spectra of the doublet states in three-electron atomic systems are investigated. By using different variational expansions we determine various bound state properties in these systems. Such properties include the electron–nucleus and electron–electron delta-functions and cusp values. The general structure of the bound state spectra in several three-electron atomic systems (Li, Be+, C3+ and F6+) is investigated with the use of the Hylleraas-Configuration Interaction and the Configuration Interaction wave functions. The advantage of our Configuration Interaction based procedure is that it provides high numerical accuracy for all rotationally excited states, including the bound states with L⩾7.

Spectral properties of three-electron systems in the Hubbard model

We investigate spectral properties of a three-electron system in the Hubbard model framework. We prove that the essential spectrum of the system in a quartet state consists of a single segment and the three-electron bound state is absent. We show that the essential spectrum of the system in doublet states is the union of at most three segments. We also prove that three-electron bound states exist in doublet states.

Two-dimensional pseudospectral Hartree-Fock method for low-Zatoms in intense magnetic fields

The energy levels of the first few low-lying states of helium and lithium atoms in intense magnetic fields up to $\approx 10^8-10^9$~T are calculated in this study. A pseudospectral method is employed for the computational procedure. The methodology involves computing the eigenvalues and eigenvectors of the generalized two-dimensional Hartree-Fock partial differential equations for these two- and three-electron systems in a self-consistent manner. The method exploits the natural symmetries of the problem without assumptions of any basis functions for expressing the wave functions of the electrons or the commonly employed adiabatic approximation. It is seen that the results obtained here for a few of the most tightly bound states of each of the atoms, helium and lithium, are in good agreement with findings elsewhere. In this regard, we report new data for two new states of lithium that have not been studied thus far in the literature. It is also seen that the pseudospectral method employed here is considerably more economical, from a computational point of view, than previously employed methods such as a finite-element based approach. The key enabling advantage of the method described here is the short computational times which are on the order of seconds for obtaining accurate results for heliumlike systems.

Relativistic configuration interaction calculations for some two- and three-electron systems with screened hydrogenic spin orbitals

An approach is proposed for the generation of virtual spin orbitals required to construct a configuration state function basis set for relativistic configuration interaction calculations. Screened hydrogenic spin orbitals, corresponding to noninteger nuclear numbers determined separately for each orbital employing the so-called condensed-space concept are proposed and used here. To test the method calculations are performed of ionization and transition energies of two- and three-electron systems such as He, He-like Ne and He-like U, Li, Li-like Ne and Li-like U, including Breit interaction and quantum electrodynamic corrections.

A Computational Study of RXHn X–H Bond Dissociation Enthalpies

Bond dissociation enthalpies can exhibit dramatic variations resulting from substituent effects. These variations result from changes in electronic structure that accompany bond dissociation. We have studied bond dissociation enthalpies (BDEs) at the W1BD level of theory for a series of RX-H compounds where X = CH2, NH, O, PH, and S. The substituents, R, included H, H3C, H2N, HO, F, H2P, HS, and Cl. The experimentally available BDEs were reproduced in a very satisfactory fashion. Most of the substituent effects could be related to a conjugative interaction in the two-center three-electron systems in the radicals formed by H abstraction. Comparisons of isoelectronic species demonstrate the important roles that variations in electronegativity and hybridization can play in determining BDEs. The OH and SH BDEs were linearly related, and the same was found with N-H and P-H BDEs. The relative effects of H2P and HS as substituents, in contrast to H2N and HO, could be related to the need to promote a 3s electron to 3p before H2P can participate in π-bonding. We must also include electronegativity differences to account for the observed transfer of a sulfur lone pair to an oxygen but the absence of the reverse process.

Hylleraas-configuration-interaction analysis of the low-lying states in the three-electron Li atom and Be+ion

The total energies of twenty eight bound S-, P-, D-, F-, G-, H-, and I-states in the three-electron systems Li atom and Be+ ion, respectively, are determined with the use of the Configuration Interaction (CI) with Slater orbitals and L-S eigenfunctions, and the Hylleraas-configuration-interaction (Hy-CI) methods. We discuss the construction and selection of the configurations in the wave functions, optimization of the orbital exponents and advanced computational techniques. Finally, we have developed a very effective procedure which allows one to determine the energies of the excited states in three-electron atoms and ions to high accuracy by using compact wave functions. For the ground and low lying excited states our best accuracy with the Hy-CI method was approximately 1.10-6 a.u. and 1.10-4 a.u. for other excited states. Analogous accuracy of the CI method is substantially lower approximately 1.10-3 a.u. Many of the rotationally excited (bound) states in the three-electron Li atom and Be+ ion have never been evaluated to such an accuracy.