S-states of helium-like ions

Liverts, Evgeny Z.; Barnea, Nir

doi:10.1016/j.cpc.2010.11.031

Cited by 15 publications

(35 citation statements)

References 32 publications

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“…It is worth noting that the Laguerre function In Fig.1 we have presented functions RΨ av (R) and RΨ appr (R) for the ground state of helium. The function Ψ av (R) was obtained by averaging the Pekeris-like wave function [2] with Ω = 25 and Ψ av (0) = 1, whereas function Ψ appr (R) was calculated by analytic representation (19) with parameters presented in Table I. Using these parameters it is easy to verify that parameter ζ introduced by Eq.…”

Section: Averaging Over the Hyperspherical Anglesmentioning

confidence: 99%

“…Using the Pekeris-like method and codes [2] we can calculate the "actual" WFs (Ψ) of the helium-like atom/ions in the S-state. This enables us to calculate numerically the averaged WF, Ψ av , according to definition (7).…”

Section: Averaging Over the Hyperspherical Anglesmentioning

confidence: 99%

“…The two remaining parameters (together with the first two, of course) can be calculated by fitting the averaged "actual" WF [2] in the specific range [0, R m ] of the substantial decrease of this function. The sets of parameters calculated by the method described above are presented in Table I.…”

Section: Averaging Over the Hyperspherical Anglesmentioning

confidence: 99%

“…To get deeper insight into the analytical properties of the one-electron density ρ(r) > 0 of manyelectron systems [1], it is desirable to investigate those features of this density that carry over to the many-electron case. Using the Wolfram Mathematica codes [2] we investigate the properties of the one-electron densities/wavefunctions of the two-electron atom/ions.…”

Section: Introductionmentioning

confidence: 99%

“…These methods of averaging can be described as follows (see, e.g., [5]). The first step consists in introducing the so called diagonal of the spinless one-electron density matrix ρ(r 1 , r 1 ) = Ψ * (r 1 , r 2 , ..., r n )Ψ(r 1 , r 2 , ..., r n )d 3 r 2 ...d 3 r n , (2) to within a constant multiplicative factor. Note that the radius-vector r 1 can represent any of the n electrons of the considered atomic system.…”

Section: Introductionmentioning

confidence: 99%

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Averaged electron densities of the helium-like atomic systems

Liverts

Krivec

2020

Journal of Mathematical Physics

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Different kinds of averaging of the wavefunctions/densities of the two-electron atomic systems are investigated. Using the Pekeris-like method [2], the ground state wave functions Ψ of the heliumlike atoms with nucleus charge 1 ≤ Z ≤ 5 are calculated in a few coordinate systems including the hyperspherical coordinates {R, α, θ}. The wave functions Ψ av (R) of the hyperspherical radius R are calculated numerically by averaging Ψ over the hyperspherical angles α and θ. The exact analytic representations for the relative derivatives Ψ av (0)/Ψ av (0) and Ψ av (0)/Ψ av (0) are derived. Analytic approximations very close to the actual Ψ av (R) are obtained. Using actual wave functions Ψ, the one-electron densities ρ(r) are calculated as functions of the electron-nucleus distance r. The relevant derivatives ρ (0)/ρ(0) and ρ (0)/ρ(0) characterizing the behavior of ρ(r) near the nucleus are calculated numerically. Very accurate analytical approximations, representing the actual one-electron density both near the nucleus and far away from it, are derived. All the analytical and numerical results are supplemented with tables and graphs.

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Section: Averaging Over the Hyperspherical Anglesmentioning

confidence: 99%

Section: Averaging Over the Hyperspherical Anglesmentioning

confidence: 99%

Section: Averaging Over the Hyperspherical Anglesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Averaged electron densities of the helium-like atomic systems

Liverts

Krivec

2020

Journal of Mathematical Physics

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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

Turbiner

Vieyra

Valle

et al. 2022

Int J of Quantum Chemistry

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As 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 charges Z ≤ 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 2 1 S and lowest spin-triplet 1 3 S state. For both states, it provides absolute accuracy for energy $10 À3 a.u., exact values for cusp parameters and also for six expectation values the relative accuracy $10 À2 . Bressanini-Reynolds observation about the special form of nodal surface of 2 1 S state of helium is confirmed and extended to He-like ions with Z > 2. Critical charges Z = Z B , where ultra-compact trial functions lose their square-integrability, are estimated: Z B (1 1 S) ≈ Z B (2 1 S) $ 0.905 and Z B (1 3 S) $ 0.902. For both excited states, the Majorana formula-the energy as the second degree polynomial in Z-provides accurately the 4-5 significant digits for Z ≤ 20.

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