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Critical stability of the metastable bound 2 p 2 3 P e state of two-electron atoms near the critical nuclear charge Z c (smaller than which the bound state ceases to exist) is investigated by employing the explicitly correlated Hylleraas-configuration interaction basis function. Our numerical calculation shows that, when the nuclear charge decreases to Z c , the ionization energy of the system approaches zero linearly and this behavior is fully consistent with the Hellmann–Feynman theorem. Radial and angular physical quantities as well as the inner and outer electron radial density distributions are calculated to reveal the classical geometric structure of the two-electron atom. The inner electron can be well modeled by a hydrogenic model in the 2 p state and its density distribution does not change much as the nuclear charge is changed. The outer electron, which is still localized in a finite region near the nucleus before the system transforms into a well-defined resonance, is shifted notably into a far-distance region. The critical stability of the system near the bound-continuum limit is therefore dominated by the outer electron.

We develop an efficient numerical method to directly calculate the critical screening parameters for one-electron systems with Hulthén and Debye–Hückel screened Coulomb potentials (SCPs). Compared to indirect methods, which locate the critical screening parameters via searching the potential parameters with near-zero energy, the method developed in this work directly calculates the critical screening parameters as eigenvalues of a generalized eigenvalue problem. This feature allows us to simultaneously determine the critical parameters for bound states from low-lying excitation to high-lying Rydberg limit with high accuracy. The method is applied to SCPs to investigate the asymptotic behavior of critical screening parameters as the principal quantum number n approaches infinity. It has been shown that the critical screening parameters in Hulthén and Debye–Hückel potentials follow the 2n −2 and 4n −2/π asymptotic laws, respectively, and that the orbital angular momentum affects the higher-order coefficients linearly.

The kinetically and atomically balanced conditions have been extensively used in various basis-set expansion methods to remove the nonphysical spurious states appearing in the numerical calculation of the Dirac equation; however, they are generally not applicable for methods in discrete variable representation. In this paper we show that these conditions can be conveniently introduced into the generalized pseudospectral (GPS) method. Four types of balanced condition, the mono kinetic balance (MKB), dual kinetic balance, mono atomic balance (MAB), and dual atomic balance, are incorporated into the GPS method to eliminate the spurious states. Numerical calculations for a variety of bound states of H-like ions in point-charge models are compared with the analytical solutions to demonstrate the accuracy and efficiency of the developed methods. The application to highly charged ions with extended nuclear models is performed to show the flexibility of the balanced GPS methods in practical atomic structure calculations. It is concluded that the MAB-GPS and MKB-GPS methods, which are both free of any spurious states, show better performance and simpler implementation than the others in solving the Dirac equation with the potential in point-charge and extended nuclear models, respectively. The balanced GPS methods developed in this paper provide a useful tool for accurately solving the one-electron Dirac equation and efficiently constructing the multielectron relativistic wave functions.

The multipole polarizabilities for hydrogen-like atoms with Hulthen potential are calculated with the sum-over-states formalism where the system bound and pseudocontinuum states are produced by employing the generalized pseudospectral method. Energies for nonzero angular momentum states, transition oscillator strengths, and dipole polarizabilities for the ground state of H atom with HP are calculated with high accuracy and compared with previous calculations. Quadrupole, octupole, and hexadecapole polarizabilities for both the ground and excited states are reported for the first time. The combined effect of Hulthen potential and impenetrable sphere on the polarizabilities is investigated, and the relation of such system with the confined H atom and particle in a box is discussed. Further comparison among Hulthen, screened Coulomb, and exponential cosine screened Coulomb potentials reveals that although the ground state energies follow the comparison theorem very well, their first-order energy corrections show complicated variations.

The first-order relativistic corrections to the non-relativistic energies of hydrogen-like atom embedded in plasma screening environments are calculated in the framework of direct perturbation theory by using the generalized pseudospectral method. The standard Debye-Huckel potential, exponential cosine screened Coulomb potential, and Hulthen potential are employed to model different screening conditions and their effects on the eigenenergies of hydrogen-like atoms are investigated. The relativistic corrections which include the relativistic mass correction, Darwin term, and the spin-orbit coupling term for both the ground and excited states are reported as functions of screening parameters. Comparison with previous theoretical predictions shows that both the relativistic mass correction and spin-orbit coupling obtained in this work are in good agreement with previous estimations, while significant discrepancy and even opposite trend is found for the Darwin term. The overall relativistic-corrected system energies predicted in this work, however, are in good agreement with the fully relativistic calculations available in the literature. We finally present the scaling law of the first-order relativistic corrections and discuss the validity of the direct perturbation theory with respect to both the nuclear charge and the screening parameter.

The first-order relativistic corrections to the non-relativistic energies of hydrogen-like atom embedded in plasma screening environments are calculated in the framework of direct perturbation theory by using the generalized pseudospectral method. The standard Debye-Hückel potential, exponential cosine screened Coulomb potential, and Hulthén potential are employed to model different screening conditions and their effects on the eigenenergies of hydrogen-like atoms are investigated. The relativistic corrections which include the relativistic mass correction, Darwin term, and the spin-orbit coupling term for both the ground and excited states are reported as functions of screening parameters. Comparison with previous theoretical predictions shows that both the relativistic mass correction and spin-orbit coupling obtained in this work are in good agreement with previous estimations, while significant discrepancy and even opposite trend is found for the Darwin term. The overall relativistic-corrected system energies predicted in this work, however, are in good agreement with the fully relativistic calculations available in the literature. We finally present the scaling law of the first-order relativistic corrections and discuss the validity of the direct perturbation theory with respect to both the nuclear charge and the screening parameter.

The angular quantities 〈 θ 12 〉 and 〈 cos θ 12 〉 for helium atom in both ground and singly-excited states are calculated in the framework of full configuration interaction. The two-electron system wave functions are expanded in terms of the product of Slater-type orbitals. The convergence and accuracy of reported results are examined in detail. From the comparison with previous work, it is shown that the present work establishes the most accurate prediction of the quantity 〈 θ 12 〉 for helium atom up to the present. Radial quantities of 〈 r 〉 , 〈 r > 〉 , 〈 r 〉 , and 〈 r 12 〉 are also reported and compared with previous calculations.

The critical screening parameters for one-electron systems screened by Hulthén, Debye–Hückel, and exponential cosine screened Coulomb potentials are calculated with an accuracy close to the precision of numerical arithmetic. The results for a H atom with an infinitely heavy nucleus are reported from the ground to high-lying excited states, and those for arbitrary two-body charged systems are derived from the Zm-scaling law. A thorough comparison of the critical screening parameters for the ground and the first p-wave excited states with previous predictions is made to demonstrate the accuracy of our calculations. The critical behaviors of system-bound and pseudo-continuum eigenenergies for s- and non-s-wave states are shown to follow the quadratic and linear laws, respectively. The variation of the corresponding wave functions is analyzed in detail. For systems with non-zero orbital angular momenta, the bound states convert into shape-type resonances when the screening parameter exceeds the critical value. The resonance energy shares the same linear law as the pseudo-continuum state, while the resonance width varies by an l-dependent power law. It is further shown that the different asymptotic behaviors of the resonance energy and width are consistent with the complex analog of the Hellmann–Feynman theorem.

The expectation values of radial geometric quantities $\ensuremath{\langle}{r}_{l}\ensuremath{\rangle}, \ensuremath{\langle}{r}_{g}\ensuremath{\rangle}, \ensuremath{\langle}r\ensuremath{\rangle}$, and $\ensuremath{\langle}{r}_{12}\ensuremath{\rangle}$ and angular ones $\ensuremath{\langle}{\ensuremath{\theta}}_{12}\ensuremath{\rangle}$ and $\ensuremath{\langle}cos{\ensuremath{\theta}}_{12}\ensuremath{\rangle}$ are investigated in detail for two-electron systems. Although these quantities can be calculated straightforwardly in the framework of Hartree-Fock or configuration-interaction methods, their computations based on explicitly correlated type wave functions are nontrivial tasks. In this work we employ the Hylleraas configuration-interaction basis functions to produce accurate system energies and wave functions for the He atom and He-like ions. Computational methods are developed to accurately and efficiently calculate the geometric quantities, especially for inner and outer electron radii, i.e., $\ensuremath{\langle}{r}_{l}\ensuremath{\rangle}$ and $\ensuremath{\langle}{r}_{g}\ensuremath{\rangle}$, and the average interelectronic angle $\ensuremath{\langle}{\ensuremath{\theta}}_{12}\ensuremath{\rangle}$. Compared to previous Hartree-Fock and multiconfiguration Hartree-Fock predictions and the configuration-interaction calculations based on Slater-type orbitals, our present work improve significantly the accuracy of all geometric quantities for He atom in the ground and singly excited states. The application of the present method to He-like ions with different nuclear charge reveals asymptotic power laws for both radial and angular quantities as they approach to the high-$Z$ limit. Further extensions of the present work to other systems are discussed.

The screening effect of dense quantum plasmas on the ground and singly excited states of two-electron atoms are investigated by employing the explicitly correlated Hylleraas configuration-interaction wave functions in the framework of Ritz variational principle. Exponential cosine screened Coulomb potential is used to model the electron–nucleus​ and interelectronic interactions in the atomic systems. Energy levels for the ground and 1 s n l 1 , 3 L (n ≤ 5 and l ≤ 4 ) singly excited states of H − , He, Li + , Be 2 + , and B 3 + are reported for screening parameters varying from zero to the value nearing corresponding critical screening parameter. Convergence of our calculations are estimated by systematically increasing the number of basis functions. A thorough comparison with available theoretical predictions on the ground state energies at various screening parameters is made, while for the excited states a comparison with the well-established energies of unscreened atoms is performed. The variation of ionization energy and the validity of Hund’s rule for two-electron systems in dense quantum plasmas are analyzed.

Binding energies, oscillator strengths, and polarizabilities of hydrogen-like atoms embedded in dense quantum plasmas are calculated in high precision by employing the generalized pseudospectral method. Benchmark predictions of these quantities are obtained for both the ground and excited states including high-order transitions. The critical behaviour of system eigenenergies, radial mean values, oscillator strengths, and multipole polarizabilities for bound states when they approach to the corresponding continuum limit as increasing the plasma screening strength are investigated in detail. It is shown that near the critical screening parameters the quantities in s-wave states show distinct behaviour from those in non-s states and such phenomenon is attributed to the divergent asymptotic form of wave functions caused by the centrifugal potential of orbital angular momentum. Based on the power laws of energies and radial mean values extracted from numerical calculations and the approximate formulae for dipole polarizabilities, the critical behaviour of the 2 k -pole polarizability of s-wave states is explicitly derived and numerically validated to follow a power law with the exponent −2(k + 1).

The electric multipole polarizabilities of one-electron atoms embedded in weakly coupled Debye plasmas are calculated in the non-relativistic framework. The static dipole, quadrupole, octopole, and hexadecapole polarizabilities for hydrogen atoms in both ground and excited states at a variety of Debye screening parameters are calculated in high precision based on the sum-over-states method, where the system bound and continuum states are produced by employing the generalized pseudospectral method. It is shown that the contribution of bound states to the polarizability decreases with increasing the plasma screening strength, whereas the contribution of continuum states is enhanced. At very small screening parameters where the plasma environment starts to take effect, it is found that the 2 l -pole polarizability for s -wave states with principle quantum number n ≥ l + 1 has an abrupt change from its non-screening value to infinity. We attribute such a phenomenon to the sudden non-degeneracy of different angular momentum states in the n shell. With continuously increasing the screening strength, the polarizabilities for n ≥ l + 1 states decrease to certain values and, eventually, they approach to infinity at the critical screening parameter. For states with n ≤ l, the 2 l -pole polarizabilities show regular enhancement from the non-screening value to infinity. The present results are compared with other theoretical calculations available in the literature and it is shown that our work has established by now the most accurate predictions of multipole oscillator strengths and polarizabilities for one-electron atoms in Debye plasmas.

Spatially confined atoms have been extensively investigated to model atomic systems in extreme pressures. For the simplest hydrogen-like atoms and isotropic harmonic oscillators, numerous physical quantities have been established with very high accuracy. However, the expectation value of $$\langle r^{-2} \rangle $$ which is of practical importance in many applications has significant discrepancies among calculations by different methods. In this work we employed the basis expansion method with cut-off Slater-type orbitals to investigate these two confined systems. Accurate values for several low-lying bound states were obtained by carefully examining the convergence with respect to the size of basis. A scaling law for $$\langle r^{n} \rangle $$ was derived and it is used to verify the accuracy of numerical results. Comparison with other calculations show that the present results establish benchmark values for this quantity, which may be useful in future studies.

Synopsis Positronium formation processes in positron-helium collisions with Debye potentials are studied using the screening approximation model in incident energy range from the threshold to 100 eV for screening parameters μ=0.00-0.25.

The screening effects of Debye plasmas on the 2p2 3Pe state of H− ion and on its radiative transition to the H−(3Po) states are investigated. The plasma conditions are modeled by screened Coulomb potentials characterized by the Debye length. It is shown that the metastable bound 2p2 3Pe state in the absence of plasma screenings would transform to a shape resonance lying above the H(2p) threshold when the Debye length decreases to a critical value. The complex-scaling method together with the configuration interaction-type wave functions is used to extract the resonance energy and width. The transition wavelengths of the 2p2 3Pe state to the 2s2p 3Po resonant state and the 1skp 3Po continuum are calculated in a wide range of Debye lengths. It is shown that wavelengths are prolonged significantly by increasing the screening effects.

The doubly excited 1Se, 1Po, and 1De resonance states of helium atom with screened Coulomb potentials are investigated. The complex scaling method with configuration interaction type basis functions are employed to extract the resonances associated with the He+(N = 2, 3, and 4) thresholds. 18 resonances (six below each of the He+ threshold) for each angular-momentum state are calculated. The results lying below the He+(N = 2) threshold are in good agreement with previous calculations by the stabilization method with correlated basis wave functions. The 1Po and 1De resonance states lying below the He+(N = 3 and 4) thresholds in the screening environment are reported for the first time. © 2013 Wiley Periodicals, Inc.

We have investigated the S-wave shape resonance states of the positronium negative ion (Ps−) system. The resonance poles are traced from H− system to Ps− by systematically varying the mass of the positively charged particle from infinitely heavy to one unit of the electron mass. The shape resonances that associated with and lying above the Ps(N = 2, 3, 4 and 5) thresholds are located by employing the complex-coordinate rotation method with highly correlated Hylleraas-type wave functions. It has been shown that the Ps−(N = 3) shape resonance lies at an energy which is higher than the Ps(N = 4) thresholds and even the Ps−(N = 4) shape resonance. An explanation was given to shed light on such phenomena.

The second-order distorted wave Born approximation (DWBA) method is employed to investigate the triple differential cross sections (TDCS) of coplanar doubly symmetric ( e , 2 e ) collisions for alkali target sodium at excess energies of 6–60 eV. Comparing with the first-order DWBA calculations, the inclusion of second-order Born term in the scattering amplitude improves the degree of agreement with experiments, especially for backward scattering region of TDCS. This indicates the present second-order Born term is capable to give a reasonable correction to DWBA model in studying coplanar symmetric ( e , 2 e ) problems in low and intermediate energy range.

Recent development of information theory provides researchers an alternative and useful tool to quantitatively investigate the variation of the electronic structure when atoms interact with the external environment. In this work, we make systematic studies on the information-theoretic measures for hydrogen-like ions immersed in weakly coupled plasmas modeled by Debye-Huckel potential. Shannon entropy, Fisher information, and Fisher-Shannon complexity in both position and momentum spaces are quantified in high accuracy for the hydrogen atom in a large number of stationary states. The plasma screening effect on embedded atoms can significantly affect the electronic density distributions, in both conjugate spaces, and it is quantified by the variation of information quantities. It is shown that the composite quantities (the Shannon entropy sum and the Fisher information product in combined spaces and Fisher-Shannon complexity in individual space) give a more comprehensive description of the atomic structure ...

In a recent paper [S. Lopez-Rosa et al., J. Math. Phys. 54, 052109 (2013)], the authors made detailed investigations on the information-theoretic quantities for hydrogenic atoms in Rydberg states. Asymptotic forms of different types of complexity measures (Cramer-Rao, Fisher-Shannon, and LMC types) in both position and momentum spaces were obtained for hydrogenic states (n,l,m) with fixed l and increasingly large n. However, the expressions for the Cramer-Rao complexity are incorrect due to the wrong definition of variance. The correct form of this complexity in position space is presented exactly in this work. In momentum space, two different definitions of variance are provided and corresponding Cramer-Rao complexities are discussed separately. In either case, the momentum complexity increases much faster than that in position space when increasing principal quantum number n.

Positronium (Ps) formation processes in positron-lithium collisions in Debye plasma environments are investigated using the screening approximation model. We present the Ps(1s) formation cross sections in the incident energy range 2-40eV for screening parameters μ=0.00-0.25.

We present two analytical methods, Taylor expansion and Gegenbauer expansion, to efficiently and accurately calculate the two-electron screened Coulomb potential matrix elements with Slater-type configuration-interaction basis functions. The former permits great advantages in fast computation of the potential matrices at small screening parameters and the latter allows accurate calculation of the matrices at all screening parameters. The bound and resonant states of a He atom embedded in the screening environment are calculated by employing the variational and complex-scaling methods, respectively, and the results are compared with other theoretical predictions. The expectation values of some physical quantities for He ground state are compared with the recent calculation of Ancarani and Rodriguez [Phys. Rev. A 89, 012507 (2014)] and extended to stronger screening environment. The energies and widths for the doubly excited resonant states are in good agreement with previous calculations, while the interelectronic angle $arccos\ensuremath{\langle}cos({\ensuremath{\theta}}_{12})\ensuremath{\rangle}$ show significant discrepancies with the Feshbach projection calculation of Ord\'o\~nez-Lasso et al. [Phys. Rev. A 88, 012702 (2013)]. The expectation values of $\ensuremath{\langle}{\mathbf{p}}_{\mathbf{1}}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{p}}_{\mathbf{2}}\ensuremath{\rangle}$ are also calculated for the resonant states investigated here. We conclude that the present methods in the framework of complex scaling enable us to get reliable energy, width, and other physical quantities of the resonant states in a variety of screening conditions.

We present a detailed investigation of the supermultiplet structures for the intrashell resonances of ${\mathrm{H}}^{\ensuremath{-}}$, classified according to their approximate quantum numbers. The resonance energies and widths are calculated by employing the complex-scaling method with configuration-interaction basis functions, and our results are compared with previous available calculations. The complete $I$-supermultiplet structures are constructed for the intrashell resonances associated with the H ($N=2,$ 3, and 4) thresholds. They include the doubly excited states of ${\mathrm{H}}^{\ensuremath{-}}$ with a variety of configuration symmetries showing both Feshbach and shape types. It is found that the ${\mathrm{H}}^{\ensuremath{-}}$ ($N=4$) supermultiplet has upper portions exceeding the H ($N=5$) threshold. This phenomenon should be useful in predicting the supermultiplet structures associated with higher-lying thresholds.

The direct single ionization cross sections for electron scattering with magnesium are calculated by applying a complex equivalent-local polarization potential in momentum space from the ionization threshold (7.64 eV) to 500 eV. The present results are in excellent agreement with the recent experimental measurements in both shape and magnitude. The present results are also compared with other theoretical calculations.

The structures and spectral properties of confined atoms differ from those in vacuum due to the potentials felt by charged particles are modified depending on the nature of the environment. In this work, we investigate the one- and two-electron atoms confined in a quantum dot described by a finite oscillator potential. The bound states of one- and two-electron atoms and resonances of two-electron atoms are calculated by using the variational and complex-scaling methods, respectively. The energies of the singly-excited states for a two-electron atom decrease monotonously with increasing dot radius and show two distinct plateaus, which corresponds to the confining effects on two electrons in different shells. The resonance energies have no obvious plateaus, while the widths show startling oscillations when the dot radius is varied. Such phenomenon stems from the destructive or constructive interferences between the confining potential and the nodal or antinodal structures of the resonance wave function. Enhancement of the confining potential would increase the oscillatory amplitudes and produce further interferences between the nearly degenerate resonances.

We investigate the shape resonances of the electron-hydrogen system embedded in Debye plasma environments modeled by screened Coulomb potentials. The complex-coordinate rotation method is employed to calculate the resonance parameters and the Hylleraas-type wave functions are used to represent the correlation effects of the two electrons. The 1Se resonance states of H− above the N = 2–5 thresholds of H atom are investigated and the resonance parameters for a Debye length ranging from infinity to small values are reported. It is shown that the widths of the shape resonances lying above the H(N = 4,5) thresholds exhibit some interesting behaviors in relatively strong screening situations. The possible reason for such phenomena is discussed.

Ground- and excited-state positronium (Ps) formation from the hydrogen atom by positron impact at low and intermediate energies have been studied within the framework of two-center two-channel eikonal final state-continuum initial distorted wave model. A general method for obtaining Ps formation into arbitrary final states has been derived. The present results of Ps (1$s$), Ps $(n=2),$ and total Ps formation cross sections agree very well with previous close-coupling calculations and experimental measurements. The maximum positions of Ps($n$) formation cross sections are found to be in good agreement with the prediction of wave vector matching model introduced by Charlton [J. Phys. B: At. Mol. Opt. Phys. 39, 4575 (2006)]. We also present a scaling law for Ps ($n$) formation cross sections that is valid for all $n$ in the entire energy range.

Energy-dependent phenomena for positron-lithium scattering in the energy region 1.45-4.5 eV have been investigated using the momentum-space coupled-channels optical method. Positions and widths of resonance states with angular momenta of J=0-2 and several threshold behaviors have been obtained. The present results are compared with other theoretical calculations.

The interaction of electrons and atomic sodium was studied by using the momentum space coupled-channel optical method at low-impact energies. Total scattering cross sections have been calculated, and negative-ion resonances have been located. The present calculation results are in good agreement with available experimental data and previous theoretical results. We have obtained resonance features at 4.166 and 4.502 eV and also found possible resonance structures near the ionization threshold in the total cross sections.

Positronium (Ps) formation processes from helium ions by positron impact are studied using the two-channel two-center eikonal final state–continuum initial distorted wave method. The Ps(n = 1 and 2) and total Ps formation cross sections are calculated from the threshold to the high energy region, and the results are compared with other theoretical calculations available in the literature. It is found that the present results agree reasonably well with the close-coupling calculations, while other predictions, such as the Coulomb–Born approximation, the optical potential method, and the recent classical trajectory Monte Carlo method (Naginey 2014 Phys. Rev. A 89, 062704), are all much higher in the entire energy region. The maximum positions of the Ps(n) formation cross sections in our present work are in good agreement with the wave vector matching model of (Charlton 2014 J. Phys. B: At. Mol. Opt. Phys. 39, 4575). Finally, we discuss the scaling law of the Ps(n) formation cross sections with respect to the principal quantum number n of the Ps atom.

We investigate the doubly excited P-wave resonance states of H− system in Debye plasmas modeled by static screened Coulomb potentials. The screening effects of the plasma environment on resonance parameters (energy and width) are investigated by employing the complex-scaling method with Hylleraas-type wave functions for both the shape and Feshbach resonances associated with the H(N = 2 to 6) thresholds. Under the screening conditions, the H(N) threshold states are no longer l degenerate, and all the H− resonance energy levels are shifted away from their unscreened values toward the continuum. The influence of Debye plasmas on resonance widths has also been investigated. The shape resonance widths are broadened with increasing plasma screening strength, whereas the Feshbach resonance widths would generally decrease. Our results associated with the H(N = 2) and H(N = 3) thresholds are compared with others in the literature.

S-, P- and D-wave Feshbach resonances in positron-sodium scattering have been investigated by using the momentum space coupled-channels optical method. The target continuum and positronium (Ps) formation channels are included via an optical potential. Feshbach resonances below the target excitation and Ps (n = 2) formation thresholds are predicted and the effects of channel-coupling scheme, target continuum and Ps formation channels on the resonance energy and resonance width are discussed. We have also found the Wigner cusp structures at the inelastic channel-opening thresholds in positron-sodium scattering cross sections.