Three-electron Atomic Systems Research Articles

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 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.

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.

Compact and accurate variational wave functions of three-electron atomic systems constructed from semi-exponential radial basis functions

The semi-exponential basis set of radial functions (A.M. Frolov, Physics Letters A {\bf 374}, 2361 (2010)) is used for variational computations of bound states in three-electron atomic systems. It appears that semi-exponential basis set has a substantially greater potential for accurate variational computations of bound states in three-electron atomic systems than it was originally anticipated. In particular, the 40-term Larson's wave function improved with the use of semi-exponential radial basis functions now produces the total energy \linebreak -7.47805413551 $a.u.$ for the ground $1^2S-$state in the ${}^{\infty}$Li atom (only one spin function $\chi_1 = \alpha \beta \alpha - \beta \alpha \alpha$ was used in these calculations). This variational energy is very close to the exact ground state energy of the ${}^{\infty}$Li atom and it substantially lower than the total energy obtained with the original Larson's 40-term wave function (-7.477944869 $a.u.$).

Semi-exponential basis for highly accurate computations of three-electron atomic systems

An advanced basis set of semi-exponential radial functions is proposed and applied to highly accurate computations of three-electron atomic systems. This set is found to be significantly more effective in highly accurate, bound state computations of three-electron atomic systems, than currently used set of Hylleraas radial functions.

Compact variational wave functions for bound states in three-electron atomic systems

The variational procedure to construct compact and accurate wave functions for three-electron atoms and ions is developed. The procedure is based on the use of six-dimensional Gaussoids written in the relative four-body coordinates r 12, r 13, r 23, r 14, r 24, and r 34. The nonlinear parameters in each basis function have been carefully optimized. Using these variational wave functions, we have determined the energies and other bound state properties for the ground 12 S-states in a number of three-electron atoms and ions. The three-electron atomic systems considered in this work include the neutral Li atom and nine positively charged lithiumlike ions: Be+, B2+, C3+, ..., Na8+, and Mg9+. Our variational wave functions are used to determine the hyperfine structure splitting and field shifts for some lithium-like ions. The explicit formulas of the Q −1 expansion are derived for the total energies of these three-electron systems.

Double autoionization of hollow-atom states

A time-dependent close-coupling method for three-electron atomic systems is formulated to calculate the double autoionization of hollow-atom states. Initial excited states are obtained by relaxation of the Schr\"odinger equation in imaginary time, while autoionization rates are obtained by propagation in real time. A 12-coupled-channels nonperturbative calculation on a three-dimensional radial lattice yields a double-autoionization rate for the $\mathrm{Li}(2{s}^{2}2p)\ensuremath{\rightarrow}{\mathrm{Li}}^{2+}(1s)+2{e}^{\ensuremath{-}}$ transition that that is somewhat smaller than earlier many-body perturbation theory calculations and in reasonable agreement with rates extracted from resonance profiles found in more recent $\ensuremath{\gamma}+\mathrm{Li}$ experiments.

Density functional studies on the hollow resonances in the Li-isoelectronic sequence (Z = 4–10)

Following from our work on triply excited hollow resonances in three-electron atomic systems, a density functional theory (DFT)-based formalism is employed to investigate similar resonances in the Li-isoelectronic series (Z = 4–10). A combination of the work-function-based local nonvariational exchange potential and the popular gradient plus Laplacian-included Lee–Yang–Parr correlation energy functional is used. The generalized pseudospectral method provides nonuniform and optimal spatial discretization of the radial Kohn–Sham equation. First, all the eight n = 2 intrashell states of B2+, N4+ and F6+ are presented, which are relatively less studied in the literature compared to the remaining four members. Then calculations are performed for the eight 2l2l′nl″ (3 ⩽ n ⩽ 6) hollow resonance series, namely 2s2ns 2Se, 2s2np 2Po, 2s2nd 2De, 2s2pns 4Po, 2s2pnp 4De, 2p2ns 4Pe, 2p2np 4Do and 2p2ns 2De, of all the seven positive ions. Next, as an illustration, higher resonance positions of the 2s2ns 2Se series are calculated for all the ions with a maximum of n = 25. The calculated excitation energies are in excellent agreement with the available literature data (for the n = 2 intrashell states the deviation is within 0.125% and excepting only one case, the same for the resonance series is well below 0.5%). With an increase in Z, the deviations tend to decrease. Radial densities are also presented for some of the selected states. The only result available in the literature for the lower resonances (corresponding to a maximum of n = 17) have been reported very recently. The n > 16 (17 for F6+) resonances are examined here for the first time. This gives a promising viable and general DFT scheme for the accurate calculation of these and other hollow resonances in many-electron atoms.

Highly accurate evaluation of atomic three-electron integrals of lowest orders.

Calculations of three-electron atomic systems in Hylleraas coordinates require integrals involving all the interparticle distances r(ij), which have usually been evaluated by introducing series expansions. For integrals with the smallest powers of r(ij) these expansions do not converge at a satisfactory rate, leading some investigators to introduce convergence-acceleration procedures. This paper recommends the alternative of evaluating these integrals in closed form and presents stable explicit formulas for so doing. Some of the formulas are more compact versions of those in the literature; others have not been previously reported. It is also shown that finite-difference methods can be used with advantage to obtain additional low-order integrals. Sample integral values have been provided for test purposes.

Hyperspherical angular adiabatic separation for three-electron atomic systems

The hypothesis of treating three-electron systems as a two-electron core plus a bound electron in the hyperspherical adiabatic approach introduces a second adiabatic separation into hyperangular equations. Differently from the main radial-angular hyperspherical adiabatic separation, the resulting couplings are necessary to guarantee permutational and rotational invariance of the Hamiltonian. Thus, any kind of approximation, disregarding such couplings, represents a loss of symmetry. This paper explores the consequences of such approximations in the potential curve calculations for the lithium atom, showing that this symmetry breaking is quite smooth and can be recovered within a good precision adding few couplings to the system of angular differential equations.

Calculations of magnetic moments for three-electron atomic systems.

The first fully correlated calculations of the magnetic moment in lithium are presented. Relative to the free-electron value, the Zeeman gJ factor for the ground state lithium gJ/g(e)-1 is calculated to a computational accuracy of 200 parts in 10(9), including relativistic and radiative corrections of orders alpha2, alpha2m/M, and alpha3. The isotope shifts in gJ are predicted precisely for various isotopes. The extensions to the first excited S state of lithium and the ground state of Be+ are made.

Auger decay of multiply excited atomic systems

By studying the branching ratios in the Auger decay of many triply excited three-electron atomic systems, we try to discover the physical reason behind the relative magnitude of the partial widths. It appears that Auger decay is most likely to happen when the two interacting electrons are in close proximity and the overlap of their charge distribution is significant in coordinate space. It is also pointed out that when an ${\mathrm{nl}}_{1}{\mathrm{nl}}_{2}{n}^{\ensuremath{'}}{l}_{3}{(n}^{\ensuremath{'}}gng~2)$ state decays into an ${n}^{\ensuremath{''}}{l}_{1}{n}^{\ensuremath{'}}{l}_{3}(ng{n}^{\ensuremath{''}})+\ensuremath{\epsilon}{l}_{2}$ channel, the ${n}^{\ensuremath{'}}{l}_{3}$ orbital of the initial state is more diffused than that of the two-electron target. Therefore, some ${n}^{\ensuremath{''}}{l}_{1}{\mathrm{ml}}_{3}(mg{n}^{\ensuremath{'}})+\ensuremath{\epsilon}{l}_{2}$ channels will also contribute significantly to the Auger width of this state. The argument presented in this work is also applied to two- and four-electron systems. Explicit examples are presented.

Hyperspherical analysis of doubly and triply excited states of Li

A computational method for calculating the hyperspherical adiabatic potential curves for three-electron atomic systems is presented. This method allows us to obtain accurate potential curves for any symmetries more efficiently. The potential curves for the $\mathrm{Li}{(}^{2}{S}^{e})$ symmetry are analyzed. For the ground state, the energy calculated using the single channel adiabatic approximation is in good agreement with experiment. For doubly excited states, in the region of small and medium hyperradius the potential curves are similar to those for the doubly excited states of two-electron atoms and these curves can be classified using the same set of $K$, $T$, and $A$ quantum numbers. For triply excited states, the potential curves are used to show the different Rydberg series that converge to the doubly excited states of ${\mathrm{Li}}^{+}$. We also illustrate the rotor structure in the energies of triply excited states of Li.

Density functional calculations on triply excited states of lithium isoelectronic sequence

Triply excited states of many-electron atomic systems are characterized by the presence of strong electron correlation, closeness to more than one threshold, and degeneracy with many continua; therefore, they offer unusual challenges to theoretical methodologies. In the present article, we computed with reasonable accuracy all the n=2 intrashell triply excited states (2s22p 2P; 2s2p2 2D, 4P, 2P, 2S; and 2p3 2D, 2P, 4S) of three-electron atomic systems (Z=2, 3, 4, 6, 8, 10) by using a density functional formalism developed recently in our laboratory, based on the nonvariational Harbola–Sahni exchange potential in conjunction with a parametrized local Wigner and Lee–Yang–Parr correlation potentials. Nonrelativistic energies and densities are obtained by solving a Kohn–Sham-type differential equation. The calculated results are compared with available experimental and other theoretical data. The 2p3(4S)1s2p2(4P) transition wavelength for the isoelectronic series is also computed. The overall good agreement of our results with the literature data indicates the reliability of the present density functional methodology for excited states of many-electron systems. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 317–332, 1997

Hyperspherical approach to three-electron atomic systems.

The hyperspherical coordinates approach to three-electron atomic systems is further refined by taking into account the symmetry of the electrons analytically. Spins are treated explicitly for both doublet and quartet cases to reduce the three-electron Schr\odinger equation to a set of coupled two-dimensional partial differential equations in a compact, symmetric form ready for numerical calculations. It is shown that the resulting equations can be accurately solved using B splines. By adopting the adiabatic approximation, the hyperspherical potential curves for singly, doubly, and triply excited states for the $^{2}$${\mathit{P}}^{\mathit{o}}$ symmetry of Li are obtained. \textcopyright{} 1996 The American Physical Society.

Analysis of hyperspherical channels of three-electron atomic systems.

Analysis of hyperspherical channels of three-electron atomic systems.

Hyperspherical approach to three-electron atomic systems

A computational procedure for carrying out calculations on three-electron atomic systems in hyperspherical coordinates is presented. By adopting an adiabatic approximation, the solution of the Schrodinger equation at a fixed hyperradius is reduced to the solution of a set of coupled partial differential equations involving the two hyperangles. The derivation of such a set of coupled equations is explicitly illustrated.

Excited states of three-electron atomic systems with two vacancies in the K shell

Excited states of three-electron atomic systems with two vacancies in the K shell

Autoionising states of three-electron atomic systems

The excitation energies and the probabilities for the radiative and radiationless decay of the autoionising states belonging to the 2s22p, 2s2p2 and 2p3 configurations have been calculated for three-electron atomic systems on the basis of perturbation theory with respect to inter-electron interaction in a broad range of values of atomic number Z. Using the results of the diagonalisation calculations of the autoionising states of two-electron systems, the authors have obtained estimates of corrections to the autoionisation widths calculated by perturbation theory. The calculations of the radiative transition probabilities for Fe and Ni ions are in good agreement with the 1976 results of Nussbaumer.