Helium Sequence Research Articles

Probing Collins Conjecture with correlation energies and entanglement entropies for the ground and excited states in the helium iso‐electronic sequence

AbstractIn the present work, we present an investigation of Collins Conjecture, a hypothesis made by D. M. Collins in 1993 relating correlation energy and entanglement entropy, by calculating the ground state and singly‐excited triplet‐spin 1s2s 3S and 1s3s 3S state energies of the helium iso‐electronic sequence, with Z = 2–15. By using extensive orthonormal configuration interaction (CI) type wave functions with B‐Spline basis up to about 6000 terms, linear entropy and von Neumann entropy for the abovementioned atomic systems are determined. Together with the Hartree‐Fock energies obtained following a self‐consistent field theory, we have found that there exist linearly proportionalities between the renormalized correlation energies and entanglement entropies of both linear and von Neumann, showing a support for Collins Conjecture applicable to the ground and singly‐excited triplet‐spin states in the helium sequence, for a range of finite Z‐values.

Ultra‐compact accurate wave functions for He‐like and Li‐like iso‐electronic sequences and variational calculus:II.Spin‐singlet (excited) and spin‐triplet (lowest) states of helium sequence

AbstractAs a continuation of Part I (Int. J. Quantum Chem. 2021; 121: qua.26586), dedicated to the ground state of He‐like and Li‐like isoelectronic sequences for nuclear chargesZ ≤ 20, a few ultra‐compact wave functions in the form of generalized Hylleraas‐Kinoshita functions are constructed, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC) for energies (4–5 significant digits [s.d.]) of two excited states of He‐like ions: the spin‐singlet (first) excited state 21Sand lowest spin‐triplet 13Sstate. For both states, it provides absolute accuracy for energy ∼10−3 a.u., exact values for cusp parameters and also for six expectation values the relative accuracy ∼10−2. Bressanini‐Reynolds observation about the special form of nodal surface of 21Sstate of helium is confirmed and extended to He‐like ions withZ > 2. Critical chargesZ = ZB, where ultra‐compact trial functions lose their square‐integrability, are estimated:ZB(11S) ≈ ZB(21S) ∼ 0.905 andZB(13S) ∼ 0.902. For both excited states, the Majorana formula—the energy as the second degree polynomial inZ—provides accurately the 4–5 significant digits forZ ≤ 20.

Hot subdwarf stars in close-up view

Atmospheric parameters and helium abundances of 44 bright subdwarf B stars have been determined. More than half of our sample consists of newly discovered stars from the Edinburgh Cape survey. We showed that effective temperatures and surface gravities can be derived from high resolution echelle spectra with sufficient accuracy. Systematic uncertainties have been determined by comparing the parameters derived from the high resolution data with the ones derived from medium resolution spectra. Helium abundances have been measured with high accuracy. Besides the known correlation of helium abundance with temperature, two distinct sequences in helium abundance have been confirmed. Significant isotopic shifts of helium lines due to an enrichment in $^{3}$He have been found in the spectra of 8 sdBs. Most of these stars cluster in a small temperature range between $27\,000\,{\rm K}$ and $31\,000\,{\rm K}$ very similar to the known $^{3}$He-rich main sequence B stars, which also cluster in such a small strip, but at different temperatures. Both the helium sequences and the isotopic anomaly are discussed.

Padé approximant formulas for atomic oscillator strengths

Interpolation formulas, having functional forms inspired by Pade approximants, and containing at most three parameters, were derived for a number of atomic oscillator strengths. It is shown that very accurate results are obtained both for specific transitions in the helium and lithium isoelectronic sequences, and for certain series in hydrogen, helium, and lithium. Application of the methods to other systems is prevented only by the lack of reliable input and comparison data. 15 references.

Nonuniversality of commonly used correlation-energy density functionals

The correlation energies of the helium isoelectronic sequence and of Hooke's atom isoelectronic sequence have been evaluated using an assortment of local, gradient, and metagradient density functionals. The results are compared with the exact correlation energies, showing that while several of the more recent density functionals reproduce the exact correlation energies of the helium isoelectronic sequence rather closely, none is satisfactory for Hooke's atom isoelectronic sequence. It is argued that the uniformly acceptable results for the helium sequence can be explained through simple scaling arguments that do not hold for Hooke's atom sequence, so that the latter system provides a more sensitive testing ground for approximate density functionals. This state of affairs calls for further effort towards formulating correlation-energy density functionals that would be truly universal at least for spherically symmetric two-fermion systems.

Crossover phenomena and resonances in quantum systems

tonian that exhibit resonances. We show the existence of a crossover phenomenon for the energy spectrum; the transition from a bound state to a continuum is a ‘‘second-order phase transition’’ in one region and a ‘‘firstorder phase transition’’ in another region. As the parameters varied, the numerical value of the critical exponent of the energy levels changed from two to one. In the zone where the critical exponent equals one, the system has narrow resonances and they disappear when the exponent is two. The method has potential applicability in predicting stable and metastable atomic and molecular states. We also show that finite-size scaling methods are capable of detecting multicritical points. DOI: 10.1103/PhysRevA.64.062502 PACS number~s!: 31.15.2p The finite-size scaling method ~FSS! was formulated in statistical mechanics to extrapolate information obtained from a finite system to the thermodynamic limit @1,2# .I n quantum mechanics, when using variation methods, one encounters the same finite-size problem in studying the critical behavior of a quantum Hamiltonian H(l 1 , . .., l k )a s a function of its set of parameters $l i%. In this context, critical means the values of $l i% for which a bound-state energy is nonanalytic. In this paper, a critical point l c is defined as the point where a bound-state energy becomes absorbed or degenerate with a continuum. In the variational calculations, the finite-size corresponds to the number of elements in a complete basis set used to expand the exact wave function of a given Hamiltonian @3#. The problem of what happens when the continuum ‘‘swallows’’ a bound state has a long history with many important results and was reviewed in Ref. @3#. Recently, Neirotti, Serra, and Kais have developed the FSS method to calculate atomic and molecular critical parameters @4,5#. In particular, FSS calculations for two and three electron atoms gave very accurate results for the critical nuclear charges @6,7#. These results have shown that the analytical behavior of the energy as a function of the nuclear charge for lithiumlike atoms is completely different from that of heliumlike atoms. Analogy with standard phase transitions show that for helium, the transition from a bound state to a continuum is ‘‘first-order,’’ while lithium exhibits a ‘‘secondorder phase transition.’’ For the helium sequence, the question of whether or not the radius of convergence, of the 1/Z expansion for the energy, is the same as the variational critical energy was discussed by several authors @ 8‐1 0#. The coincidence between the critical point and the radius of convergence was conjectured by Reinhardt using arguments based on dilatation analyticity @11# and later rigorously confirmed, analytically and numerically, by Baker et al. @12#. Our results for the helium sequence using the finite-size scaling method were in complete agreement with the Reinhardt

Formation of excited states in high-Z helium like systems

High-Z helium-like ions represent the simplest multi-electron systems for studying the interplay between electron‐electron correlations, relativistic as well as quantum electrodynamical effects in strong fields. In contrast to the adjacent lithium-like ions, however, almost no experimental information is available about the excited states in the high-Z domain of the helium sequence. Here, we present a theoretical analysis of the X-ray production and decay dynamics of the excited states in helium-like uranium. Emphasis has been paid particularly to the formation of the 3 P0 and 3 P2 levels by using electron capture into hydrogen-like U 91+ . Both states are of interest for precise measurements on high-Z helium-like ions in the future.

Dipole polarizabilities of the 1, 21S and 23S states of the helium sequence

The dipole polarizabilities of the 1, 21S and 23S states of helium-like ions (2<or=Z<or=10) are calculated through a variation-perturbation approach. The polarized wavefunctions are constructed by using selecting B-spline basis functions in a CI scheme. The values of alpha d are converged to an accuracy of at least 0.01% and are in excellent agreement with those obtained from explicitly correlated wavefunctions and with experiment (Leonard and Barker, 1975).

Moments of dipole oscillator-strength distribution for the helium sequence

The moments S( mu ) for -6<or= mu <or=2 and L( mu ) for mu =0, 1 and 2 are calculated for the helium sequence for atomic numbers Z up to 30 under a screened hydrogenic model. In this model, one describes the atom by single-particle hydrogenic wavefunctions and treats the initial and the final state as characterised by two different effective charge parameters Zi and Zf, respectively. An asymptotic expansion is made of the differential oscillator strength of the screened hydrogenic model. Assuming the value 287.6 for the coefficient of the term in -7/2 for helium atom as given by Salpeter et al. (1962), the parameter Zf is determined for the helium sequence. This approach has resulted in values which are in reasonable agreement with the various moment values of other authors.

Differential oscillator strengths and dipole polarizabilities for transitions of the helium sequence

The dipole radial integral for an initial discrete 1s state and a final continuum state has been calculated under the screened hydrogenic model. In this model, single-electron hydrogenic wave functions are employed and the initial and the final states are treated by two different effective-charge parameters. Numerical values of differential oscillator strengths for transitions from 1s 21S to the continuum for the helium sequence ions are obtained. Also calculated are the dipole polarizabilities, which are found to be in excellent agreement with the results of other authors.

A lesser known one-parameter wave function for the helium sequence and the virial theorem

In this note, a one-parameter trial function for the the treatment of the helium atom and helium-like ions is introduced.

Atomic data for opacity calculations. VI. Static dipole polarisabilities of the ground states of the helium sequence

For pt.V see ibid., vol.20, p.6431-46 (1987). The variation-perturbation technique with optimised configuration-interaction expansions is used to obtain static dipole polarisabilities alpha d of the ground states of helium-like ions from Z=2 to Z=12. The values of alpha d are in excellent agreement with those obtained using Hylleraas expansions. The ground 1S and 'pseudo' 1Po states are suitable as target states in R-matrix calculations.

Limiting behavior of the ratio of 〈r〉 and 〈r〉 in the helium sequence

AbstractThe limiting of the 〈r〈/〈r〈 ratio in the He sequence is critically examined at different levels of approximations leading to some interesting results.

Complex-coordinate calculations for doubly excited states of two-electron atoms

Doubly excited states of helium sequence ($Z=1\ensuremath{-}10$) are investigated by a complex-coordinate method. Some lower-lying $^{1}S^{e}$ and $^{1,3}P^{o}$ states below the $N=2$ hydrogenic thresholds are calculated by the use of Hylleraas-type wave functions. Results are compared with recent calculations and experiments.

Electron-impact-ionization cross sections for highly ionized heliumlike atoms

Cross sections for electron-impact ionization from the ground states of four ions in the helium isoelectronic sequence have been calculated in the distorted-wave Born-exchange approximation. The results are in good agreement with available experimental data. Exchange in the scattering matrix element was found to be important in the determination of accurate cross sections. An isoelectronic plot of the scaled cross section permits ready interpolation of the nonrelativistic ionization cross section for any ion in the helium sequence in the incident electron energy range 1-5 times threshold.

Doubly excited resonances of the helium sequence associated with N=4 hydrogenic thresholds

Several lower-lying doubly excited resonances associated with N=4 hydrogenic thresholds for two electron atoms have been calculated using a method of complex rotation.

Perturbation treatment of 2p3d1,3L0doubly excited states of the helium sequence

The first-order Hartree-Fock equations for 2p3d 1,3L0 doubly excited states of the helium sequence have been solved exactly. These solutions are used to evaluate the Hartree-Fock energy up to third accuracy. The Hartree-Fock energies up to third order are compared with those obtained from experiment and other refined theoretical calculations for He and Li+. The results predict two new resonances at 63.5010 and 63.7222 eV above the ground state of He.

Radiative recombination in the helium sequence

Recombination coefficients for the reaction X +m+1(1 2 S+ e → X +m(1 1 S) + hv for the helium] sequence are calculated using recently obtained optical differential oscillator strengths for electron temperatures in the range 10 to 5 x 10 4 K. The coefficients for the entire sequence are represented by a simple formula, which requires only the ionization potential of the members of the sequence.

3P10-1S0intercombination transitions in the helium sequence

Probabilities of 23P10-11S0 electric dipole intercombination transitions in the helium sequence from He I to Ne IX are calculated. A recently proposed specific relativistic correction is included.

Variational calculation of the single-particle density matrix and momentum density for the helium ground-state isoelectronic sequence

A recently developed variational formalism for the determination of the reduced single-particle density matrix, correct to second order, is applied to the ground state of the helium isoelectronic sequence. For a Slater-determinant-- type trial wave function the method requires the initial determination of either the charge density or equivalently its cosine Fourier transform for spherically symmetric systems. The trial wave function employed is a one-parameter Hartree product of hydrogenic functions and use is made of the highly accurate analytic expressions derived elsewhere for the Fourier transform of the charge density for the helium sequence. Analytic expressions for the single-particle density matrix are obtained and the internal self-consistency of the technique with regard to the Kato cusp condition is discussed. These expressions are then employed to obtain closed form analytic expressions for the momentum density valid for the entire isoelectronic sequence. These results are subsequently employed to obtain expressions for the expectation values of the operators p$sup 4$, p$sup 2$, vertical-barpvertical-bar, and vertical-barpvertical-bar$sup -1$ and the Compton profile in the impulse approximation. Analytic Hartree-Fock calculations for these properties are also performed, and the results of the variational calculation are compared with these results and those of many-parameter correlated wave-function calculations wherevermore » possible. It is observed that the results of the single-parameter variational calculation for helium are highly accurate and improve further for each heavier element of the isoelectronic sequence.« less