The Heisenberg Model of the He Atom Revisited

Gueribah, S.; Ighezou, F.-Z.; Lombard, R.J.; Ngô, H.

doi:10.1007/s00601-010-0210-9

Cited by 3 publications

(2 citation statements)

References 14 publications

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“…one electron was kept at the ground states, and the other one in a particular excited state n was used by Hall and Siegel [37], who applied the shooting method to obtain accurate excited state energies of the He atom in coordinate space. This type of state configuration was also used as a pseudo-two-body problem by Gueribah et al [38], who used the experimental energies to determine the local equivalent potential of the He atom using the Heisenberg model. Moreover, was applied by Herschbach et al [39] applied the scaling dimension technique to obtain accurate ground state energies of two-electron atoms based on the dimensional dependence of a hydrogen atom.…”

Section: Comments On the Accuracy And Limitations Of Our Methods And Further Improvementmentioning

confidence: 99%

Ground State Energies of Helium-Like Ions Using a Simple Parameter-Free Matrix Method

2021

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This study aims to use hydrogenic orbitals within an analytic and numeric parameter-free truncated-matrix method to solve the projected Schrödinger equation of some Helium-like ions (3 ≤ Z ≤ 10). We also derived a new analytical expression of the ion ground state energies, which was simple and accurate and improved the accuracy of the analytic calculation, numerically using Mathematica. The standard matrix method was applied, where the wave function of the ions was expanded in a finite number of eigenvectors comprising hydrogenic orbitals. The Hamiltonian of the systems was calculated using the wave function and diagonalized to obtain their ground state energies. The results showed that a simple analytic expression of the ground state energies of He-like ions was successfully derived. Although the analytic expression was derived without involving any variational parameter, it was reasonably accurate with a 0.12% error for Ne8+ ion. From this method, the accuracy of the analytic energies was also numerically improved to 0.10% error for Ne8+ ion. The results clearly showed that the energies obtained using this method were more accurate than the hydrogenic perturbation theory and the uncertainty principle-variational approach. In addition, for Z > 4, our results were more accurate than those from the geometrical model.

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Section: Comments On the Accuracy And Limitations Of Our Methods And Further Improvementmentioning

confidence: 99%

Ground State Energies of Helium-Like Ions Using a Simple Parameter-Free Matrix Method

2021

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“…In this configuration, one electron is kept at the 1s state while the other one can be in any state of the systems. The same configuration was also applied in other studies including [7,[30][31][32]. To improve the accuracy of our approach, the inclusion of higher bound states and unbound states in the wavefunction of the systems is necessary.…”

Section: Spectral Decomposition Of the Ground State Energies Of The S...mentioning

confidence: 99%

Approximate Analytical Solution of the Ground State Problem of He and He-like Ions using Symmetrized-Hydrogenic States

Pingak

2024

Positron

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The matrix method using three symmetrized-hydrogenic basis states has been applied to analytically obtain an approximate solution to the Schrödinger equation of the Helium atom and some He-like ions (2<Z<10). This study aims at obtaining more accurate ground state energies of the systems compared to our previous calculation using un-symmetrized basis states and some other simple calculations in the literature. The contribution of the symmetrized basis states on the ground state energies of the systems is also investigated. The time-independent Schrödinger equation involving a 3×3 Hamiltonian matrix, formed by hydrogenic s-states, was analytically solved to obtain three energy eigenvalues of the systems as well as their corresponding eigenvectors. Results showed that the 1s2 energies of the systems were more accurate than our previous unsymmetrized basis calculations, with significant error reduction observed for He and Li+. With the same matrix size, the ground state energies of He and He-like ions obtained from three symmetrized basis states in this study were found to be closer to the exact and experimental energies than those obtained from unsymmetrized basis states. It was also demonstrated that the |100;100> state made the largest contribution to the ground state energies of the systems, i.e. about 90.9% for He and around 99.9% for Ne8+, and consequently the smallest contribution came from the other two symmetrized states (less than 1%). To conclude, the calculated ground state energies were more accurate than some other simple calculations reported in the literature.

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