Helium-like Research Articles

Analysis of circular polarization of the emitted photon in plasmas

In this paper, a new relativistic distorted wave approximation for cross section describing longitudinally-polarized electron impact excitation of ions in the presence of a strongly coupled plasma is proposed and implemented based on the flexible atomic code. The uniform electron gas model is used to incorporate the plasma effects, which includes the effect of static plasma screening on nuclear charge and the effect of confinement due to the neighbouring ions. For the bound and continuum wavefunctions, the Dirac equation with the ion-sphere potential is solved to account for relativistic effects. In order to check the consistency of the modifications, a self consistent theoretical attempt using the multiconfiguration Dirac-Fock method is also carried out to get the energy levels accounting for the plasma environment within the ion-sphere model. For the collision dynamics, as examples of applications, we investigate the total cross sections, the magnetic sublevels cross sections, and the degrees of circular polarizations of fluorescence radiation after 1s2s01→1s2ss03 and 1s2s01→1s2pp13,1 excitations of Helium-like Fe XXV ion by longitudinally-polarized electron impact with the ion-sphere potential. It is shown that the behavior of the cross sections for a pure Coulomb potential changes rapidly due to the plasma screening effects described by the ion-sphere potential in the process of calculating scattering matrix elements. A detailed, quantitative, comparison between the present results and other available theoretical values is presented for the plasma-free cases, showing good agreement. Our results contribute to a better understanding of the fundamentals of fluorescence polarization in strongly coupled plasmas. Results obtained in this work are beneficial for the diagnostic determination of laboratory plasmas.

Efficient two-electron ansatz for benchmarking quantum chemistry on a quantum computer

Quantum chemistry provides key applications for near-term quantum computing, but these are greatly complicated by the presence of noise. In this work we present an efficient ansatz for the computation of two-electron atoms and molecules within a hybrid quantum-classical algorithm. The ansatz exploits the fundamental structure of the two-electron system, and treating the nonlocal and local degrees of freedom on the quantum and classical computers, respectively. Here the nonlocal degrees of freedom scale linearly with respect to basis-set size, giving a linear ansatz with only $\mathcal{O}(1)$ circuit preparations required for reduced state tomography. We implement this benchmark with error mitigation on two publicly available quantum computers, calculating accurate dissociation curves for 4- and 6- qubit calculations of ${\rm H}_\textrm{2}^{}$ and ${\rm H}_\textrm{3}^+$.

Ground state energy of hydrogen-like ions in quantum plasmas

Using the asymptotic iteration method (AIM), we investigate the variation in the 1s energy levels of hydrogen and helium-like static ions in fully degenerate electron gas. The semiclassical Thomas–Fermi (TF), Shukla–Eliasson (SE), and corrected Shukla–Eliasson (cSE) models are compared. It is noted that these models merge into the vacuum level for hydrogen and helium-like ions in the dilute classical electron gas regime. While in the TF model, the hydrogen ground state level lifts monotonically toward the continuum limit with an increase in the electron concentration; in the SE and cSE models, a universal bound stabilization valley through the energy minimization occurs at a particular electron concentration range for the hydrogen-like ion which for the cSE model closely matches the electron concentrations in typical metals. The latter stabilizing mechanism appears to be due to the interaction between plasmon excitations and the Fermi length scales in the metallic density regime. In the case of helium-like ions, however, no such stability mechanism is found. The application of the cSE model with electron exchange and correlation effects reveals that the cSE model qualitatively accounts for the number density and lattice parameters of elemental metals within the framework of free electron assumption. According to the cSE model of static charge, screening a simple metal–insulator transition criterion is defined. The effect of the relativistic degeneracy effect on the ground state energy of the hydrogen atom is studied. It is shown that the ground state energy level of the hydrogen atom also undergoes a collapse at the well-known Chandrasekhar mass limit for white dwarf stars.

On the Two-Dimensional Hydrogen Molecule

Using analytical and numerical methods, the possibility of the existence of a planar hydrogen molecule H2 in a two-dimensional subspace of three-dimensional space has been analyzed for the first time. For this, the same mathematical formalism was used, with the help of which one of the authors of this work already demonstrated theoretically the possibility of the existence of a two-dimensional one-electron and a two-dimensional two-electron atom and the impossibility of the existence of such a many-electron atom in one of his previous works. It is shown that two-dimensional hydrogen molecules H2 can exist with characteristics similar to ordinary three-dimensional molecules. The conclusions of this work can, in principle, be verified experimentally since two-dimensional atoms were obtained experimentally quite a long time ago.

Asymptotic quantum defect of singly excited two-electron atoms at the critical charge

The validity of a conjecture concerning the asymptotic behavior of a singly-occupied atomic orbital of type nℓ outside an N-electron core, upon approaching the critical charge (below which this electron is not bound), is definitively demonstrated. According to this conjecture, this orbital becomes an infinitely diffuse hydrogenic orbital of type (n − nℓ)ℓ, where nℓ is the number of occupied (or partially occupied) core subshells with angular momentum ℓ. An indicator of this asymptotic behavior is the asymptotic value of the corresponding quantum defectlimZ→Nδnℓ=nℓ.To establish this limiting behavior for several singly-excited states of the two-electron atom, we apply the explicitly correlated Hylleraas-type variational exponential expansion to evaluate highly precise energies, that allow the critical charge (Zc = 1) to be approached much more closely than ever before. The results suggest a refinement of the asymptotic form of the quantum defects in the various states considered, exhibiting the distinct origins of s-p splitting versus singlet-triplet splitting.

A Simple Numerical Matrix Method for Accurate Triplet-1s2s 3S1 Energy Levels of Some Light Helium-like Ions

Triplet-1s2s 3S1 energy levels of some light Helium-like ions (Li+ to Ne8+) were numerically calculated using a truncated-matrix method implemented in a simple Mathematica code. The two-electron wave functions were expanded in finite number of basis states consisting of hydrogenic s-orbitals. Here, the Hamiltonian matrix used was of size 15×15, 25×25 and 35×35. From the results, triplet-1s2s 3S1 energies of the ions were in good agreement with other advanced and highly accurate calculations in the literature. Comparisons with one of the most accurate calculations in the literature showed that our results were very accurate with the largest error in energy being only 0.135 % for Li+ ion and the smallest one being approximately 0.018 % for Ne8+ ion energy, when 35 basis states were used in our calculations. Errors from our calculations were also much lower than those from the geometrical model.

QED theory of elastic electron scattering on hydrogen-like ions involving formation and decay of autoionizing states

We develop {\it ab initio} relativistic QED theory for elastic electron scattering on hydrogen-like highly charged ions for impact energies where, in addition to direct (Coulomb) scattering, the process can also proceed via formation and consequent Auger decay of autoionizing states of the corresponding helium-like ions. Even so the primary goal of the theory is to treat electron scattering on highly charged ions, a comparison with experiment shows that it can also be applied for relatively light ions covering thus a very broad range of the scattering systems. Using the theory we performed calculations for elastic electron scattering on B$^{4+}$, Ca$^{19+}$, Fe$^{25+}$, Kr$^{35+}$, and Xe$^{53+}$. The theory was also generalized for collisions of hydrogen-like highly charged ions with atoms considering the latter as a source of (quasi-) free electrons.

Semianalytical wavefunctions and Kohn–Sham exchange-correlation potentials for two-electron atomic systems in two-dimensions

We propose accurate semianalytical wavefunctions for two-electron atomic systems in two-dimensions. These wavefunctions take into account the cusp conditions arising due to the Coulombic nature of external potential and electron–electron interaction, and the screening effects. The first wavefunction is motivated by the Le Sech wavefunction for two-electron atoms and ions in three-dimensions while the second one is constructed to give more variational freedom to the Le Sech wavefunction and is obtained by solving a Schrödinger like equation self-consistently. The results for energies, densities and other quantities are highly accurate and therefore may play an essential role in developing and testing density functional methods for two-dimensional systems.

A Novel Super Current Based on Dichromatic Potential Surface Deformation in Two-Electron Atoms

Slow electron scattering from a highly excited Rydberg atom is shown to be controlled by deformed potential surfaces. More over, that deformation shows a dichroism. The potentials are different for shrinking and expanding modes of motion, respectively. We attribute this effect to a novel fictitious force in atomic structure..

Dominant Correlation Effects in Two-Electron Atoms

Two-electron atoms have been investigated near threshold of double escape within the framework of hyperspherical coordinates. A particularly useful set of hyperspherical angles has been used. It is well known for many years that the hyperradial motion is nearly separable from the hyperspherical angular motion. Therefore, the Born-Oppenheimer separation method should be useful. However, the success of that method in molecular physics is based on the small mass ratio, electron mass to nuclear mass. In the atomic application such a small parameter does not exist. Nevertheless the method works surprisingly well in the lower part of the spectrum. For increasing excitation energy the method becomes shaky. Near ionization threshold, it breaks even down. The author will present elsewhere an improved Born-Oppenheimer method. First pilot developments and comparison with the experimental situation are presented already here. Inclusion of a momentum-momentum radial coupling delivers an improved basis. We show that our extended Born-Oppenheimer approach leads to a deformation of the whole potential energy surface during the collision. In consequence of this deformation we outline a quantum derivation of the Wannier threshold cross section law, and we show that (e, 2e) angular distribution data are strongly influenced by that surface deformation. Finally, we present a mechanism for electron pair formation and decay leading to a supercurrent independent of the temperature. Our framework can be extended to more than two electrons, say 3 or 4. We conclude that our improved Born-Oppenheimer method [1] is expected not only to deliver better numerical data, but it is expected to describe also the Wannier phenomenon. The idea of the new theory together with first qualitative results is presented in this paper.

Calculations of Relativistic, Correlation, Nuclear, and Quantum-Electrodynamics Corrections to Energy and Ionization Potential of the Ground State of Helium-Like Ions

Nonrelativistic and relativistic variational calculations of energies and ionization potentials of the ground state of helium-like ions with nuclear number Z = 2−20 have been performed. All leading corrections to the total energy have been calculated, including the contribution of electron correlations, relativistic and quantum-electrodynamics corrections, and contributions due to the finite nuclear size (field shift) and finite nuclear mass (nuclear recoil) effects. Relativistic wave functions were calculated using the Dirac–Coulomb–Breit Hamiltonian.

Energy Positions of the <sup>1,3</sup>D<sup>e</sup> Rydberg Series of the He-Like Ions Up to the <i>N</i> = 9 Threshold

The doubly excited states of helium-like ions are investigated using a combination of the no-linear parameters of Hylleraas and the β-parameters of screening constant by unit nuclear charge. Calculations are performed for total energies of low-lying doubly excited states (N = 2 - 9) in He-like ions up to Z = 10. The results obtained from the novel method are in good agreement with the available theoretical calculations and experimental observations.

Расчеты релятивистских, корреляционных, ядерных и квантово-электродинамических поправок к энергии и потенциалу ионизации основного состояния гелиеподобных ионов

In this work, nonrelativistic and relativistic variational calculations of the energies and ionization potentials of the ground state of helium-like ions for the nuclear charges in the range Z = 2 − 20 were performed.the leading corrections to the total energy were calculated including the contribution of electronic correlations, relativistic and quantum-electrodynamic (QED) corrections, and the contributions of the finite size ofnucleus (field shift) and the finite mass of the nucleus (recoil effect). Relativistic сalculations of the wave functions were performed using the Dirac-Coulomb-Breit (DCB) Hamiltonian.

QED calculation of electron-electron correlation effects in heliumlike ions

Fully relativistic approach to evaluate the correlation effects in highly charged ions is presented. The interelectronic-interaction contributions of first and second orders in $1/Z$ are treated rigorously within the framework of bound-state quantum electrodynamics, whereas the calculations of the third- and higher-order contributions are based on the Dirac-Coulomb-Breit Hamiltonian. The developed approach allows one to deal with single as well as degenerate or quasi-degenerate states. We apply this approach to the calculations of the correlation contributions to the $n=1$ and $n=2$ energy levels in heliumlike ions. The obtained contributions are combined with the one-electron and screened QED corrections, nuclear recoil and nuclear polarization corrections to get the total theoretical predictions for the ionization and transition energies in high-$Z$ heliumlike ions.

Universalities among natural orbitals and occupation numbers pertaining to ground states of two electrons in central potentials.

Although both the natural orbitals (NOs) {ψnlm(r→)} and their occupation numbers {νnl} pertaining to the ground state of two electrons confined by a central potential are completely determined by the spatial component Ψ(r→1,r→2) of the underlying wavefunction through a homogeneous Fredholm equation of the second kind in which Ψ(r→1,r→2) plays the role of the kernel, for the species with a single positive-valued natural amplitude that corresponds to the strongly occupied NO ("the normal sign pattern"), these quantities turn out to depend almost entirely on the "on-top" wavefunction Ψ(r→,r→). For such species, for which the occupation numbers are found to have the large-n asymptotics of n-8, a universal expression involving only Ψ(r→,r→) that reproduces the weakly occupied NOs with remarkable accuracy is inferred from the electron-electron coalescence cusp in Ψ(r→1,r→2). These theoretical predictions are fully confirmed by comparisons among the benchmark-quality NOs computed for the helium atom, the isoelectronic cations with the nuclear charges ranging from 3 to 5, and the two-electron harmonium atom at the limit of an infinitely strong confinement.

Description of two-electron atoms with correct cusp conditions

New sets of functions with arbitrarily large finite cardinality are constructed for two-electron systems. Functions from these sets exactly satisfy Kato’s cusp conditions. The new functions are particular linear combinations of Hylleraas- and Kinoshita-type terms. The energies of the system can be calculated by a standard variational calculation, leading to a matrix eigenvalue problem. There is no need for a constrained optimization to satisfy the cusp conditions. In the numerical examples, the ground state energies of the He atom and the H− ion are considered and several expectation values are calculated for the He.

Asymptotic behavior of two-electron expectation values in two-electron excited states

Abstract Singly-excited states of the two-electron atom cease being bound when Z → 1 (from above), the outer orbital becoming infinitely diffuse. The asymptotic relations lim Z → 1 ⁡ ( Z − 1 ) k 〈 ( 1 s n s ) 1 , 3 S | r 12 k | ( 1 s n s ) 1 , 3 S 〉 = 〈 ( n − 1 ) s ( 0 ) | r k | ( n − 1 ) s ( 0 ) 〉 , where k = − 1 , 1 , 2 , 3 , ⋯ , are demonstrated to hold. Here, ( n − 1 ) s ( 0 ) is a hydrogenic s orbital with principal quantum number ( n − 1 ) . New, more nuanced light is shed on the already challenged dogma that the Pauli principle keeps the electrons further apart in the triplet than in the corresponding singlet.

Understanding two-photon double ionization of helium from the perspective of the characteristic time of dynamic transitions

By using the B-spline numerical method, we investigate a two-photon double-ionization (TPDI) process of helium in a high-frequency laser field with its carrier frequency ranging from 1.6 to 3.0 a.u. and the pulse duration ranging from 75 to 160 attoseconds. We found that there exists a characteristic time tc for a TPDI process, such that the pattern of energy distribution of two ionized electrons presents a peak or two, depending respectively on whether the pulse duration is shorter or longer than tc. Especially, when the pulse duration is larger than tc, the TPDI spectrum shows a double-peak structure which is attributed to the fact that most of the electron–electron Coulomb interaction energy is acquired by a single electron during their oscillation around the nucleus before the two electrons leave, and hence the double-peak structure cannot be identified as a signal of sequential ionization. Additionally, if the carrier frequency is less than the ionization energy of He+, i.e. that carrier frequency is in the so called correlated region, tc is not a fixed value, and it increases as the carrier frequency decreases; while if the carrier frequency is greater than the ionization energy of He+, i.e. the carrier frequency is in the so called non-correlated region, tc is fixed at about 105 attoseconds. We further found that, for a helium-like ion in its ground state, the characteristic time for when the carrier frequency is larger than the ionization energy of the second electron has a key relation with the Coulomb interaction energy between the two electrons, which can be expressed as , a type of quantum mechanical uncertainty relation between time and energy. In addition, this relation can be attributed to the existence of a minimal evolution time from the ground state to a double ionization state with two electrons carrying different energies. These results may shed light on a deeper understanding of many-electron quantum dynamical processes.

High-precision calculation of the geometric quantities of two-electron atoms based on the Hylleraas configuration-interaction basis

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.

Few-electron atomic ions in non-relativistic QED: The ground state

Following detailed analysis of relativistic, QED and mass corrections for helium-like and lithium-like ions with static nuclei for Z≤20 the domain of applicability of Non-Relativistic QED (NRQED) is localized for ground state energy. It is demonstrated that for both helium-like and lithium-like ions with Z≤20 the finite nuclear mass effects do not change 4–5 significant digits (s.d.), and the leading relativistic and QED effects leave unchanged 3–4 s.d. in the ground state energy. It is shown that the non-relativistic ground state energy can be interpolated with accuracy not less than 13 s.d. for Z≤12, and not less than 12 s.d. for Z≤50 for helium-like as well as for Z≤20 for lithium-like ions by a compact meromorphic function in variable λ=Z−ZB (here ZB is the 2nd critical charge (Turbiner et al., 2016)), P9(λ)∕Q5(λ). It is found that both the Majorana formula – a second degree polynomial in Z with two free parameters – and a fourth degree polynomial in λ (a generalization of the Majorana formula) reproduce the ground state energy of the helium-like and lithium-like ions for Z≤20 in the domain of applicability of NRQED, thus, at least, 3 s.d. It is noted that ≳99.9% of the ground state energy is given by the variational energy for properly optimized trial function of the form of (anti)-symmetrized product of three (six) screened Coulomb orbitals for two-(three) electron system with 3 (7) free parameters for Z≤20, respectively. It may imply that these trial functions are, in fact, exact wavefunctions in non-relativistic QED, thus, the NRQED effective potential can be derived. It is shown that the sum of relativistic and QED effects in leading approximation - 3 s.d. - for both 2 and 3 electron systems is interpolated by 4th degree polynomial in Z for Z≤20.