Three-electron Atoms Research Articles

General model and toolkit for the ionization of three or more electrons in strongly driven atoms using an effective Coulomb potential for the interaction between bound electrons

We formulate a three-dimensional semi-classical model to address triple and double ionization in three-electron atoms driven by intense infrared laser pulses. During time propagation, our model fully accounts for the Coulomb singularities, the magnetic field of the laser pulse and for the motion of the nucleus at the same time as for the motion of the three electrons. The framework we develop is general and can account for multi-electron ionization in strongly-driven atoms with more than three electrons. To avoid unphysical autoionization arising in classical models of three or more electrons, we replace the Coulomb potential between pairs of bound electrons with effective Coulomb potentials. The Coulomb forces between electrons that are not both bound are fully accounted for. We develop a set of criteria to determine when electrons become bound during time propagation. We compare ionization spectra obtained with the model developed here and with the Heisenberg model that includes a potential term restricting an electron from closely approaching the core. Such spectra include the sum of the electron momenta along the direction of the laser field as well as the correlated electron momenta. We also compare these results with experimental ones.

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.

On the hyperfine structures of the ground state(s) in the 6Li and 7Li atoms

Hyperfine structure of the ground $2^{2}S-$states of the three-electron atoms and ions is investigated. By using our recent numerical values for the doublet electron density at the atomic nucleus we determine the hyperfine structure of the ground (doublet) $2^{2}S-$state(s) in the ${}^{6}$Li and ${}^{7}$Li atoms. Our predicted values (228.2058 $MHz$ and 803.5581 $MHz$, respectivly) agree well with the experimental values 228.20528(8) $MHz$ (${}^{6}$Li) and 803.50404(48) $MHz$ (${}^{7}$Li (R.G. Schlecht and D.W. McColm, Phys. Rev. \textbf{142}, 11 (1966))). The hyperfine structures of a number of lithium isotopes with short life-times, including ${}^{8}$Li, ${}^{9}$Li and ${}^{11}$Li atoms are also predicted. The same method is used to obtain the hyperfine structures of the three-electron ${}^{7}$Be$^{+}$ and ${}^{9}$Be$^{+}$ ions in their ground $2^{2}S-$states. Finally, we conclude that our approach can be generalized to describe the hyperfine structure in the triplet $n{}^{3}S-$states of the four-electron atoms and ions.

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.

Two-photon triple ionization of Li

A time-dependent close-coupling method is used to calculate the two-photon triple ionization of Li. Our method combines approaches previously used for two-photon double ionization of two-electron atoms with approaches that have been used for single-photon triple ionization of three-electron atoms. At a photon energy of 115.0 eV the three outgoing electrons share an energy of 31.4 eV. We explore the energy and angle differential cross sections of the three outgoing electrons and compare and contrast these distributions with those arising from single-photon triple ionization of Li.

The Coulomb, exchange, and correlation components of the electron-electron repulsion in harmonium atoms.

Highly accurate Coulomb, exchange, and correlation components of the electron-electron repulsion energies of the three-electron harmonium atoms in the (2)P- and (4)P+ states are obtained for 19 values of the confinement strength ω ranging from 10(-3) to 10(3). The computed data are consistent with their ω → 0 and ω → ∞ asymptotics that are given by closed-form algebraic expressions. Robust approximants that accurately reproduce the actual values of the energy components while strictly conforming to these limits are constructed, opening an avenue to stringent tests capable of predicting the performance of electronic structure methods for systems with varying extents of the dynamical and nondynamical electron correlation. The values of the correlation components, paired with the computed 1-matrices are expected to be particularly useful in the context of benchmarking of approximate density matrix functionals.

Semiempirical Quantum Chemistry Model for the Lanthanides: RM1 (Recife Model 1) Parameters for Dysprosium, Holmium and Erbium

Complexes of dysprosium, holmium, and erbium find many applications as single-molecule magnets, as contrast agents for magnetic resonance imaging, as anti-cancer agents, in optical telecommunications, etc. Therefore, the development of tools that can be proven helpful to complex design is presently an active area of research. In this article, we advance a major improvement to the semiempirical description of lanthanide complexes: the Recife Model 1, RM1, model for the lanthanides, parameterized for the trications of Dy, Ho, and Er. By representing such lanthanide in the RM1 calculation as a three-electron atom with a set of 5 d, 6 s, and 6 p semiempirical orbitals, the accuracy of the previous sparkle models, mainly concentrated on lanthanide-oxygen and lanthanide-nitrogen distances, is extended to other types of bonds in the trication complexes’ coordination polyhedra, such as lanthanide-carbon, lanthanide-chlorine, etc. This is even more important as, for example, lanthanide-carbon atom distances in the coordination polyhedra of the complexes comprise about 30% of all distances for all complexes of Dy, Ho, and Er considered. Our results indicate that the average unsigned mean error for the lanthanide-carbon distances dropped from an average of 0.30 Å, for the sparkle models, to 0.04 Å for the RM1 model for the lanthanides; for a total of 509 such distances for the set of all Dy, Ho, and Er complexes considered. A similar behavior took place for the other distances as well, such as lanthanide-chlorine, lanthanide-bromine, lanthanide, phosphorus and lanthanide-sulfur. Thus, the RM1 model for the lanthanides, being advanced in this article, broadens the range of application of semiempirical models to lanthanide complexes by including comprehensively many other types of bonds not adequately described by the previous models.

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.

Nuclear(n;t)reaction in the three-electron6Li atom

The nuclear $(n;t)$ reaction of the three-electron ${}^{6}$Li atom with thermal or slow neutrons is considered. An effective method has been developed for determining the probabilities of the formation of various atoms and ions in different bound states. We discuss a number of fundamental questions directly related to numerical computations of the final-state atomic probabilities. A few appropriate variational expansions for atomic wave functions of the incident lithium atom and final helium atom and/or tritium negatively charged ion are discussed. It appears that the final ${}^{4}$He atom arising during the nuclear $(n,{}^{6}$Li; ${}^{4}$He$,t)$ reaction in the three-electron Li atom can also be created in its triplet states. The formation of the quasistable three-electron ${e}_{3}^{\ensuremath{-}}$ during the nuclear $(n;t)$ reaction at the Li atom is briefly discussed. Bremsstrahlung emitted by atomic electrons accelerated by the rapidly moving nuclear fragments from this reaction is analyzed. The frequency spectrum of the emitted radiation is investigated.

Testing quantum electrodynamics in the lowest singlet states of the beryllium atom

High-precision results are reported for the energy levels of the $2\phantom{\rule{0.16em}{0ex}}{}^{1}S$ and $2\phantom{\rule{0.16em}{0ex}}{}^{1}P$ states of the beryllium atom. Calculations are performed using fully correlated Gaussian basis sets and taking into account the relativistic, quantum electrodynamics (QED), and finite nuclear mass effects. Theoretical predictions for the ionization potential of the beryllium ground state $75\phantom{\rule{0.16em}{0ex}}192.699(7)\phantom{\rule{0.28em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ and the $2\phantom{\rule{0.16em}{0ex}}{}^{1}P\ensuremath{\rightarrow}2\phantom{\rule{0.16em}{0ex}}{}^{1}S$ transition energy $42\phantom{\rule{0.16em}{0ex}}565.441(11)\phantom{\rule{0.28em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ are compared to known but less accurate experimental values. The accuracy of the four-electron computations approaches that achieved for the three-electron atoms, which enables determination of the nuclear charge radii and precision tests of QED.

The three-electron harmonium atom: The lowest-energy doublet and quadruplet states

Calculations of sub-μhartree accuracy employing explicitly correlated Gaussian lobe functions produce comprehensive data on the energy E(ω), its components, and the one-electron properties of the two lowest-energy states of the three-electron harmonium atom. The energy computations at 19 values of the confinement strength ω ranging from 0.001 to 1000.0, used in conjunction with a recently proposed robust interpolation scheme, yield explicit approximants capable of estimating E(ω) and the potential energy of the harmonic confinement within a few tenths of μhartree for any ω ≥ 0.001, the respective errors for the kinetic energy and the potential energy of the electron-electron repulsion not exceeding 2 μhartrees. Thanks to the correct ω → 0 asymptotics incorporated into the approximants, comparable accuracy is expected for values of ω smaller than 0.001. Occupation numbers of the dominant natural spinorbitals and two different measures of electron correlation are also computed.

Note: The weak-correlation limit of the three-electron harmonium atom

Asymptotic energy expressions for the weak-correlation limits of the two lowest energy states of the three-electron harmonium atom are obtained in closed forms. When combined with the known results for the strong-correlation limit, these expressions, which are correct through the second order of perturbation theory, yield robust Padé approximants that allow accurate estimation of energies in question for all magnitudes of the confinement strength.

Benchmark Full Configuration Interaction Calculations on the Lowest-Energy (2)P and (4)P States of the Three-Electron Harmonium Atom.

Full configuration interaction calculations carried out in conjunction with careful optimization of basis sets and judicious extrapolation schemes for 12 values of the confinement strength ω ranging from 0.1 to 1000.0 provide benchmark energies for the (2)P ground state and the (4)P first excited state of the three-electron harmonium atom, allowing for numerical verification of the recently obtained second-order energy coefficients and confirming the few available results of Monte Carlo studies. The final energy values, obtained by correcting the extrapolated data for residual errors in the low-order energy coefficients, possess accuracy of ca. 20 μHartree for the doublet state and ca. 10 μHartree for the quartet one, making them suitable for calibration and testing of approximate electron correlation methods of quantum chemistry. The energy limits for individual angular momenta ranging from 1 to 4 are also available, facilitating comparisons with results of calculations involving finite basis sets. An example of application involving the BLYP and B3LYP functionals is provided.

Complete Feshbach-type calculations of energy positions and widths of autoionizing states in Li-like atoms

Applications of the Feshbach formalism to systems of more than two active electrons are very scarce due to practical limitations in the construction of the projection operators $\mathcal{P}$ and $\mathcal{Q}$ that are inherent to the theory. As a consequence, most previous applications rely on the use of approximate quasiprojection operators, whose theoretical justification is not yet clear. In this work, an implementation of the Feshbach formalism for three-electron atoms is presented that includes all the ingredients of the original formalism. Energy positions and autoionization widths of the lowest ${}^{2}{S}^{e}$, ${}^{2}{P}^{o}$, and ${}^{2}{D}^{e}$ autoionizing states of Li and Ne${}^{7+}$ have been evaluated. The results show that the use of quasiprojection operators is justified for the evaluation of resonant positions. However, for the ${}^{2}{S}^{e}$ states of Li, the use of quasiprojection operators can lead to errors in the autoionization widths of the order of $100%$.

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.

Properties and hyperfine structure of the beryllium-muonic atoms

The bound-state properties and hyperfine structure splitting in the ground -states of the three-electron beryllium-muonic atoms 9Be4+μ−e−3 and 10Be4+μ−e−3 are determined numerically with the use of highly accurate variational five-body wavefunctions. All computations of the beryllium-muonic atoms are performed with the use of the two independent electron spin functions.

Wigner molecules: The strong-correlation limit of the three-electron harmonium

At the strong-correlation limit, electronic states of the three-electron harmonium atom are described by asymptotically exact wave functions given by products of distinct Slater determinants and a common Gaussian factor that involves interelectron distances and the center-of-mass position. The Slater determinants specify the angular dependence and the permutational symmetry of the wave functions. As the confinement strength becomes infinitesimally small, the states of different spin multiplicities become degenerate, their limiting energy reflecting harmonic vibrations of the electrons about their equilibrium positions. The corresponding electron densities are given by products of angular factors and a Gaussian function centered at the radius proportional to the interelectron distance at equilibrium. Thanks to the availability of both the energy and the electron density, the strong-correlation limit of the three-electron harmonium is well suited for testing of density functionals.

Energies and oscillator strengths of lithium in a strong magnetic field

Accurate energy levels, transition wavelengths, and dipole oscillator strengths involving the 1s(2)2s(0), 1s(2)2p(-1), and 1s(2)3d(-2), states of lithium in strong magnetic field up to 1.27 x 10(6) T are calculated using the modified full core plus correlation method. The contributions to the energy levels from electron correlation is found to be smaller than 0.06 a.u. in the present calculations. Taking evidence from a detailed convergence study into account, the results for the energy levels are expected to be accurate to within 10(-5) a.u. The wavelength for the 1s(2)2p(-1)-1s(2)3d(-2) transition shows a stationary line characteristic, useful for the understanding of observed spectra from white dwarf stars. The absorption oscillator strengths in the length and velocity gauges given in this work are, to the best of our knowledge, the first values for a three-electron atom in a strong magnetic field.

Ground-state multiplicities of atoms and positive ions

A procedure is developed to establish the ground-state multiplicities for atoms with any number of electrons. The procedure is applied to two- and three-electron systems. The results are that all neutral and positive two- and three-electron atoms have singlet (S=0) and doublet (S=1/2) ground states, respectively.

Wave functions for two- and three-electron atoms and isoelectronic ions

We have developed simple wave functions for two- and three-electron atoms and ions, which have the correct structure when one of the electrons is far away, or when two of the particles are close to each other. These essentially parameter-free wave functions allow us to deduce fairly accurate values for the energies, , for multipolar polarizabilities of two-electron atoms and ions, and for the coefficients of the asymptotic density.