Three-electron Atoms Research Articles

Comprehensive analysis of electron correlations in three-electron atoms

We study the electron correlations in singly, doubly, and triply excited states of a three-electron atom. While electron correlation in general is weak for singly excited states, correlation plays major roles in determining the characteristics of doubly and triply excited states. Using the adiabatic approximation in hyperspherical coordinates, we show that the distinction between singly, doubly, and triply excited states is determined by the radial correlations, while finer distinctions within doubly or triply excited states lie in the angular correlations. Partial projections of the body-fixed frame wave functions are used to demonstrate the characteristic nodal surfaces which provide clues to the energy ordering of the states. We show that doubly excited states of a three-electron atom exhibit correlations that are similar to the doubly excited states of a two-electron atom. For the triply excited states, we show that the motion of the three electrons resemble approximately that of a symmetric top.

Description of correlated densities for few-electron atoms by simple functional forms

Simple analytical functional forms for the electron density of two- and . three-electron atoms which reproduce fairly the correlated exact values are presented. . The procedure is based on the fitting of an auxiliary fr function which has adequate properties for this purpose and can be extended to more complex atoms. Q 1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 443)454, 1999

Ground and excited states of few-electron systems in spherical quantum dots

Energy spectra of two- and three-electron systems confined in semiconductor quantum dots, i.e., artificial helium and lithium atoms, are studied by the variational method under the assumption of the spherically symmetric confinement potential of finite depth. It is shown that the electron pairs and triples can form bound states if the quantum ‘capacity’, V0R2, of the quantum dot, is sufficiently large (V0 is the potential-well depth and R is the quantum-dot radius). The conditions of binding have been determined for the ground and excited states. The binding energy and dipole transition energy have been calculated for several QDs. It is found that the dipole transition energy for the one-, two-, and three-electron artificial atoms is nearly independent of the number of electrons.

Nodal structures of intrashell states of three-valence-electron atoms

The nodal structures of intrashell states of a three-valence-electron atom are investigated. We expand the wave function in the body frame of the three electrons and show that the effect of symmetry such as rotation, space inversion, and permutation is to impose nodal surfaces on the components of the wave function. The equilibrium configuration is deduced to be a coplanar equilateral triangle with the core at the center and the three electrons at the vertices. Three basic modes of internal oscillations have been identified and the relative energy levels of the atom are interpreted in terms of the degrees of excitations of these basic modes. Calculations are carried out using a model of three electrons moving on the surface of a sphere. Results from this r-frozen model, as well as the limited data in the literature, are used to illustrate conclusions drawn only from the symmetry consideration alone. We thus claim that the relative energy levels of intrashell states of a three-electron atom is determined less by the detailed dynamics and interactions, but more by the inherent quantum symmetry. {copyright} {ital 1997} {ital The American Physical Society}

Analysis of Triply Excited States of Atoms in Hyperspherical Coordinates

A new method for obtaining accurate adiabatic potential curves of a three-electron atom in hyperspherical coordinates has been developed. These potential curves are used to study singly, doubly, and triply excited states all at the same time. The group of potential curves that supports triply excited states of atoms is analyzed. For the Li atom, these curves are used to interpret and identify the triply excited states observed recently in the measurements using synchrotron radiation. For the ${\mathrm{H}}^{2\ensuremath{-}}$ ion, all the potential curves are repulsive, which clearly implies that there are no resonances of any type for this system.

Precision radio-frequency measurements of the high-L Rydberg states of lithium.

The n=10 G-to-H and H-to-I intervals in lithium atoms have been measured to an accuracy of approximately 10 parts per million. The experiment utilizes a beam of Rydberg lithium atoms produced via a resonant charge-exchange technique and the measurements were performed using precision rf spectroscopy coupled with a laser-excitation and Stark-ionization detection technique. The results are compared with recently published polarization-model calculations of the fine structure of lithium and with a hydrogenic model of its spin structure. With the availability of more precise theory, further tests of relativistic and QED effects in this simple three-electron atom will be possible. Such tests could shed light on the discrepancies found between similar high-precision measurement and theory in helium. The measured values are 338.4778(33), 341.4027(35), 108.4819(51), and 110.1566(54) MHz for the intervals 10${\mathit{G}}_{9/2}$ to 10${\mathit{H}}_{11/2}$, 10${\mathit{G}}_{7/2}$ to 10${\mathit{H}}_{9/2}$, 10${\mathit{H}}_{11/2}$ to 10${\mathit{I}}_{13/2}$, and 10${\mathit{H}}_{9/2}$ to 10${\mathit{I}}_{11/2}$, respectively.

High precision calculations and QED effects for two- and three-electron atoms

Essentially exact results can now be obtained for the nonrelativistic energy and lowest-order relativistic corrections of helium. A combination of variational calculations and asymptotic expansion methods covers the entire singly-excited spectrum. Comparisons with recent high precision experiments allow the Lamb shift for the 1s2s1S state to be extracted with an accuracy approaching that in one-electron He+(2s). Techniques for calculating the two-electron Lamb shift are discussed, including recent asymptotic expansion methods for the Bethe logarithm. Extensions to lithium and other three-electron ions up to U89+ are briefly mentioned.

Three-electron systems with inner-shell vacancies

A computational method is described for obtaining inner-shell-vacancy states of three-electron atoms which combines a block-diagonalization procedure with generalized Feshbach projection operators applicable to systems with three or more electrons. Typically, the accuracy is about 1.5 parts per thousand (\ensuremath{\delta}E/E\ensuremath{\approxeq}1.5\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$). The strength of the method is that it provides many energy levels for each Rydberg series. A quantum-defect analysis is then applied that identifies the members of each series and yields reliable quantum defects and series limits. The method is particularly important in symmetries for which multiple Rydberg series exist. The present work reports on $^{4}$${\mathit{P}}^{\mathit{e}}$ states of three-electron systems with 3\ensuremath{\le}Z\ensuremath{\le}10, which are compared with other calculations. The energies of $^{2}$${\mathit{S}}^{\mathit{e}}$ states of ${\mathrm{C}}^{3+}$ are also presented as an example requiring both the multielectron Feshbach projection operators and the quantum-defect analysis developed here. Four distinct Rydberg series are found for this case and their series limits obtained.

Multiphoton ionization of a three-electron atom: Studies with 25-keV Al beams.

An alternative technique to obtain neutral-atom beams for multiphoton-ionization studies, via wavelength-selective photodetachment of negative ions, is descibed. This approach is illustrated by multiphoton ionization of Al(3s 2 3p 2 P) atoms with selected visible and uv laser pulses of ≤300 ps duration, and with intensities ∼10 9 -6×10 10 W/cm 2 . For visible wavelengths, single-electron ionization dominates, while excitation of doubly excited states over the ionization limit is observed under uv irradiation

Three-electron atoms from a Bethe-Salpeter equation approach.

The two-electron Bethe-Salpeter equation is reviewed. The transition to a Hamiltonian form is discussed, and the criterion that the first-order energy shift vanish is used to determine the electron-electron potential that appears in the Hamiltonian. A similar program is carried through for the three-electron case. The fundamental idea of Feldman and Fulton is used for the transition to the Hamiltonian form, although some changes are necessary. The optimum two-electron interaction is the same as that in the two-electron case.

Note on four-particle wave functions

Dougall's expansion is used in the expression for the volume element in tetrahedral coordinates, thereby lifting the spherical triangle restriction on the angles at the tetrahedral vertex. Application is made to the evaluation of matrix elements for the ground state of three-electron atoms.

Quartet spectra in three-electron atoms

Routine calculations have been performed on the quartet terms of configurations 1s2snl and 1s2pnl, n=2, 3, 4, 5; l=0, 1, 2, 3 for the isoelectronic series Li-Ne+7 in the one-configuration approximation by direct minimization. The conditions for orthogonality between wave functions identical in symmetry are provided by the frozen excited core approximation. The errors of this approximation are examined, and it is shown that the results are quite satisfactory for qualitative analysis of quartet spectra in the z sequence. A study is made of the changes in relative position of the levels as the nuclear charge alters; the intersection of terms identical in symmetry as functions of z is discussed. The z dependence of the allowed transitions is examined.

Semiclassical model for atoms

The energies of several two- and three-electron atoms, in both ground states and excited states, are calculated by a very simple semiclassical model. The only change from Bohr's original method is to replace definite orbits by probability distribution functions based on classical dynamics. The energies are better than Hartree-Fock values. There is still a need for an exchange-energy correction.

Theoretical studies of core-excited states of two- and three-electron atoms

We have carried out configuration-interaction calculations for $^{4}F^{e,o}$ states of Li and ${\mathrm{Be}}^{+}$ and for the lowest $^{4}D^{e,o}$ states of ${\mathrm{Be}}^{+}$ in an effort to test the accuracy that can be achieved in transition-energy calculations by assuming transferability of relativistic, radiative, and mass-polarization corrections among similar electron cores. He and ${\mathrm{Li}}^{+}$ $2p4f^{1,3}F$ states have also been considered. The nonrelativistic result for the Li $1s2p4f^{4}F\ensuremath{\rightarrow}1s2s4f^{4}F^{o}$ transition is ${\ensuremath{\lambda}}_{\mathrm{air}}=5474.3\ifmmode\pm\else\textpm\fi{}0.4$ \AA{}. Including relativistic, radiative, and mass-polarization corrections for the $1s2p^{3}P$ and $1s2s^{3}S$ ${\mathrm{Li}}^{+}$ cores we get $\ensuremath{\lambda}=5471.7\ifmmode\pm\else\textpm\fi{}0.4$ \AA{}, in excellent agreement with a $\ensuremath{\lambda}=5471.9\ifmmode\pm\else\textpm\fi{}1.4$ \AA{} recently observed in the beam-foil source. Stronger support for our basic assumption comes from three calculated (experimental) wavelengths $\ensuremath{\lambda}=3406.0\ifmmode\pm\else\textpm\fi{}0.3(3405.6\ifmmode\pm\else\textpm\fi{}0.6), 4330.1\ifmmode\pm\else\textpm\fi{}0.5(4330.2\ifmmode\pm\else\textpm\fi{}0.5), \mathrm{and} 3510.8\ifmmode\pm\else\textpm\fi{}0.5(3510.8\ifmmode\pm\else\textpm\fi{}0.5)$ \AA{} in the Be ${\mathrm{ii}}^{**}$ spectrum. Furthermore, three lines (the first two above and $\ensuremath{\lambda}=981$ \AA{}) have been reassigned and three new wavelengths are predicted for this spectrum. The $1s2s4f\ifmmode\pm\else\textpm\fi{}1s2p3d$ nature of the $^{4}F^{o}(1)$ and $^{4}F^{o}(2)$ states in ${\mathrm{Be}}^{+}$ is responsible for the uncommon situation that $^{4}F^{o}(1)$ is below $1s2s4d ^{4}D^{o}(2)$, thus explaining the failure of some of the earlier assignments. The lowest ${\mathrm{Be}}^{+}$ $1s2s2p^{4}P^{o}$ state is found 115.845\ifmmode\pm\else\textpm\fi{}0.016 eV above the ${\mathrm{Be}}^{+}$ $1{s}^{2}2s$ ground state, in fair agreement with a recent experimental value of 115.7\ifmmode\pm\else\textpm\fi{}0.1 eV. The He and ${\mathrm{Li}}^{+}$ $2p4f^{1,3}F$ states decay to $1s4f^{1,3}F$ states with wavelengths too close to the resonance transition $2p\ensuremath{\rightarrow}1s$ of the corresponding one-electron system to be observed in beam-foil experiments. The ${\mathrm{Li}}^{+}$ $2p4f ^{1,3}F$ states, however, decay to $2p3d ^{1,3}D$ states with $\ensuremath{\lambda}=3825.4\ifmmode\pm\else\textpm\fi{}0.1 \mathrm{and} 4352.2\ifmmode\pm\else\textpm\fi{}0.1$ \AA{}, respectively, and they should be susceptible of experimental observation.

Cluster expansion of the wavefunction. Pseudo-orbital theory based on the SAC expansion and its application to the spin density of open-shell systems

We have studied the pseudo-orbital theory, which is based on the symmetry-adapted-cluster (SAC) expansion of an exact wavefunction proposed previously, in comparison with the conventional open-shell orbital theories, and applied it to the calculations of the spin densities of the first-row atoms, Li(2S), Li(2P), Be+(2S), B2+(2S), B(2P), C(3P), N(4S), O(3P), and F(2P). We have started from the RHF reference wavefunction and considered mainly the spin-polarization excitation operator and its self-consistency effect. This pseudo-orbital theory corresponds to an extension of the UHF and spin-extended HF (SEHF) theories, and yet it is free from the theoretical defects found previously for these theories. The relative magnitudes of the calculated spin densities are predicted to be in the order of the UHF or SEHF, present, and first-order (FO) CI in the decreasing order. This sequence has been confirmed in the calculated spin densities for the first-row atoms. For the three-electron atoms the present theory gives an excellent agreement with experiment, and for boron through fluorine the present results are fairly better than those of the UHF and SEHF theories and reasonable within the orbital theoretic approach. The calculated energies are also satisfactory when the angular correlation is included within the pseudo-orbital theory.

Doubly-excited States in Three-electron Systems

Quantum mechanical calculations of the quartet states of the three-electron atoms are discussed with particular reference to the classification of the 4Po and 4Pe states. New assignments are proposed for the lines at 4 760 Å, 5 315 Å and 8 517 Å in Li I.

Hartree-Fock and Configuration Interaction Calculations of Hyperfine Constants in Quartet States of Three-electron Atoms

The hyperfine structure constants in the lowest 4Po and 4Pe states of the isoelectronic sequence Li-F6+ have been calculated from Hartree-Fock and accurate configuration interaction (CI) wavefunctions, in good agreement with available experimental results for Li. In addition, Hartree-Fock calculations of energies and hfs constants for several higher 4S, 4Po, 4D and 4Do states of Li are reported.

Third-order energy for the ground state of three-electron atoms inZ-dependent perturbation theory

Previous theoretical studies of the ground state of three-electron atoms based on $Z$-dependent perturbation theory have yielded values for the third-order energy coefficient which differ by as much as a factor of 2. Estimates of the third-order energy coefficient obtained via a "differencing technique" from variational calculations and from experimental data for the lithium isoelectronic sequence also disagree with each other and with the theoretical results. The present value of the third-order energy coefficient is based on a first-order wave function constructed from accurate, first-order, two-electron wave functions and appears more accurate than these earlier results.

Diagram approach to the calculation of the energy of a many-electron atom

The leading Feynman diagrams contributing to the energy of an atom are calculated. Numerical calculations of the energy matrices are carried out for the states of configurations 1s2s2 and 1s2p2 in three-electron atoms and 1s22s2 and 1s22p2 in four-electron atoms. For the case of Fe XXIV the theoretical wavelengths of the transition are compared with experimental data. The numerical calculation shows that the two-photon diagrams with two and three open ends make contributions of the same order of magnitude to the energy of the atom.

Upper and lower bounds to atomic and molecular properties. IV. Electric polarizabilities of three-electron atoms by a lower-bound procedure

Wave functions, which include interelectron coordinates ${r}_{\mathrm{ij}}$ explicitly, are employed for the $1{s}^{2}\mathrm{ns}^{2}S$ and $1{s}^{2}2p^{2}P_{1}$ states of LiI in obtaining lower bounds and Hylleraas variation-perturbation estimates of dipole polarizabilities for the four lowest $^{2}S$ states of LiI. The results of this study are polarizabilities for the four lowest $^{2}S$ states of LiI with probable accuracy 2-5%. In addition, the latest experimental result for the $2^{2}S$ state of LiI is found to be in excellent agreement with our result.