Analytic matrix elements for the two-electron atomic basis with logarithmic terms

Liverts, Evgeny Z.; Barnea, Nir

doi:10.1063/1.4890562

Cited by 2 publications

(1 citation statement)

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“…For example, a simple analytical calculation using the uncertainty principle was successfully applied to estimate the ground state energy of helium and helium-like atoms [12]. Some other methods include the geometrical model [13], the matrix method with logarithmic terms [14], the variational principle [15], and the matrix method with hydrogenic basis states [16,17].…”

Section: Introductionmentioning

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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Section: Introductionmentioning

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

Pingak

Deta

2020

J. Phys.: Conf. Ser.

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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.

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Analytic matrix elements for the two-electron atomic basis with logarithmic terms

Cited by 2 publications

References 20 publications

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

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

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

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Analytic matrix elements for the two-electron atomic basis with logarithmic terms

Cited by 2 publications

References 20 publications

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

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

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

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