[Modèle semi-classique et évaluation de l’énergie du système atomique à deux électrons dans les orbitales s]
For physics and engineering undergraduate students, the semi-classical model formulation is a convenient approach to understanding electron behavior, which in turn facilitates the comprehension of more elaborate quantum mechanical formulations. In this line of thought, a two-electron atom semi-classical model was developed for the ground-state energy and excited states of He and ions. The two electrons are in a state occupying the same s orbital; thus, the average position distance was worked out from the classical orbital radius. Utilizing the Coulombic repulsion force between the electrons and the virial theorem, a relationship is established. Numerical results against optical spectroscopy yield an average energy relative error $< 0.23 \%$. When compared against ab initio calculations, the average energy relative error is $< 2.15 \%$. Exact energy values of the system for helium using ICI, QD Monte Carlo, variational and CCSD methods assessed the model. These latter results comparison can serve the purpose of illustrating to the undergraduate student that semi-classical models, although not entirely accurate, can still provide a close agreement with the scientific fundamentals to comprehend the physics behind electrons’ behavior.
Pour les étudiants de premier cycle en physique et en ingénierie, la formulation du modèle semi-classique est une approche pratique pour comprendre le comportement des électrons, ce qui facilite une compréhension plus approfondie des formulations plus élaborées de la mécanique quantique. Dans cette optique, un modèle semi-classique d’atome à deux électrons a été développé pour l’énergie de l’état fondamental et les états excités de He et des ions. Les deux électrons occupent la même orbitale s ; la distance de position moyenne a donc été calculée à partir du rayon de l’orbite classique. En utilisant la force de répulsion coulombienne entre les électrons et le théorème du viriel, une relation est établie. Les résultats numériques comparés à la spectroscopie optique donnent une erreur énergétique relative moyenne inférieure à 0,23%. Comparés aux calculs ab initio, l’erreur énergétique relative moyenne est inférieure à 2,15%. Les valeurs énergétiques exactes du système pour l’hélium par les méthodes ICI, QD Monte Carlo, variationnelle et CCSD ont permis d’évaluer le modèle. La comparaison de ces derniers résultats peut servir à illustrer à l’étudiant de premier cycle que les modèles semi-classiques, bien que pas tout à fait exacts, peuvent néanmoins fournir un accord étroit avec les fondamentaux scientifiques pour comprendre la physique derrière le comportement des électrons.
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Mots-clés : Distance moyenne entre deux points sur une circonférence, théorème du viriel, forces de répulsion des électrons, énergie de répulsion, orbitales s
Jorge Andrés Ramos-Grez 1 ; Vicente Bañados-BaySchmith 1 ; Joaquín Meyer-Larraín 2
CC-BY 4.0
@article{CRPHYS_2025__26_G1_681_0,
author = {Jorge Andr\'es Ramos-Grez and Vicente Ba\~nados-BaySchmith and Joaqu{\'\i}n Meyer-Larra{\'\i}n},
title = {Semi-classical model and assessment of the two-electron atomic system energy in s~orbitals},
journal = {Comptes Rendus. Physique},
pages = {681--695},
year = {2025},
publisher = {Acad\'emie des sciences, Paris},
volume = {26},
doi = {10.5802/crphys.267},
language = {en},
}
TY - JOUR AU - Jorge Andrés Ramos-Grez AU - Vicente Bañados-BaySchmith AU - Joaquín Meyer-Larraín TI - Semi-classical model and assessment of the two-electron atomic system energy in s orbitals JO - Comptes Rendus. Physique PY - 2025 SP - 681 EP - 695 VL - 26 PB - Académie des sciences, Paris DO - 10.5802/crphys.267 LA - en ID - CRPHYS_2025__26_G1_681_0 ER -
%0 Journal Article %A Jorge Andrés Ramos-Grez %A Vicente Bañados-BaySchmith %A Joaquín Meyer-Larraín %T Semi-classical model and assessment of the two-electron atomic system energy in s orbitals %J Comptes Rendus. Physique %D 2025 %P 681-695 %V 26 %I Académie des sciences, Paris %R 10.5802/crphys.267 %G en %F CRPHYS_2025__26_G1_681_0
Jorge Andrés Ramos-Grez; Vicente Bañados-BaySchmith; Joaquín Meyer-Larraín. Semi-classical model and assessment of the two-electron atomic system energy in s orbitals. Comptes Rendus. Physique, Volume 26 (2025), pp. 681-695. doi: 10.5802/crphys.267
[1] Die Ionisierungsspannung des Heliums nach der Schrödingerschen Theorie, Z. Phys., Volume 44 (1927), pp. 91-109 | DOI | Zbl
[2] The structure of the Helium atom: I, Proc. Natl. Acad. Sci. USA, Volume 13 (1927), pp. 423-430 | DOI
[3] Beiträge zur Quantenmechanik der Atome, Ann. Phys. (Berlin), Volume 82 (1927), pp. 355-393 | DOI | Zbl
[4] The wave mechanics of an atom with a non-Coulomb central field. Part I: Theory and Methods, Proc. Camb. Philos. Soc., Volume 24 (1928), pp. 89-111 | DOI
[5] Neue Berechnung der Energie des Heliums im Grundzustande, sowie des tiefsten Terms von Ortho-Helium, Z. Phys., Volume 54 (1929), pp. 347-366 | DOI | Zbl
[6] Majorana solutions to the two-electron problem, Found. Phys., Volume 42 (2012), pp. 1586-1608 | DOI | Zbl
[7] Experiment and theory in computations of the He atom ground state, Int. J. Mod. Phys. E, Volume 15 (2006), pp. 877-888 | DOI
[8] Solving the Schrödinger equation for helium atom and its isoelectronic ions with the free iterative complement interaction (ICI) method, J. Chem. Phys., Volume 127 (2007) no. 22, 224104 | DOI
[9] A diffusion Monte Carlo algorithm with very small time-step errors, J. Chem. Phys., Volume 99 (1993) no. 4, pp. 2865-2890 | DOI
[10] An explicitly correlated coupled cluster calculation of the helium–helium interatomic potential, J. Chem. Phys., Volume 103 (1995) no. 14, pp. 6127-6132 | DOI
[11] and states of Helium, Phys. Rev., Volume 115 (1959), pp. 1216-1221 | DOI
[12] The semiclassical two-electron atom and the old quantum theory, J. Phys. B. At. Mol. Opt. Phys., Volume 13 (1980) no. 6, 1037 | DOI
[13] Semiclassical theory of the helium atomic spectrum, J. Chem. Phys., Volume 81 (1984) no. 12, pp. 5976-5985 | DOI
[14] Semiclassical approach to few-body problems: the helium atom, Prog. Theor. Phys., Suppl., Volume 116 (1994), pp. 121-142 | DOI
[15] The theory of two-electron atoms: between ground state and complete fragmentation, Rev. Mod. Phys., Volume 72 (2000) no. 2, pp. 497-544 | DOI
[16] Semiclassic approach within the He atom sphere model, Int. J. Quantum Chem., Volume 96 (2004) no. 5, pp. 507-511 | DOI
[17] Three-body configuration space calculations with hard-core potentials, J. Phys. B. At. Mol. Opt. Phys., Volume 31 (1998) no. 6, 1279 | DOI
[18] Coulomb oscillations as a remedy for the helium atom (2007) | arXiv
[19] A study of confined helium atom, Commun. Theor. Phys., Volume 48 (2007) no. 2, 331 | DOI
[20] Potential energy surfaces in atomic structure: the role of Coulomb correlation in the ground state of helium, Phys. Rev. A, Volume 95 (2017) no. 2, 022502, 8 pages | DOI
[21] Ground state of two-electron atoms, Phys. Rev., Volume 112 (1958), pp. 1649-1658 | DOI | Zbl
[22] Ground state of the helium atom, Phys. Rev., Volume 105 (1957) no. 5, pp. 1490-1502 | DOI | Zbl
[23] Roothaan–Hartree–Fock ground-state atomic wave functions: Slater-type orbital expansions and expectation values for , At. Data Nucl. Data Tables, Volume 1 (1993) no. 53, pp. 113-162 | DOI
[24] Ultralow energy scattering of a He atom off a He dimer, Phys. Rev. A, Volume 56 (1997) no. 3, p. R1686-R1689 | DOI
[25] Nonrelativistic ionization energy for the helium ground state, Phys. Rev. A, Volume 66 (2002) no. 2, 024501, 2 pages | DOI
[26] Energy levels for the stable isotopes of atomic helium (4He I and 3He I), Can. J. Phys., Volume 84 (2006) no. 2, pp. 83-105 | DOI
[27] Précis de géométrie analytique, Librairie Vuibert, Paris, 1957
[28] Ionization potentials and ionization limits derived from the analyses of optical spectra (1970) no. NSRDS-NBS 34 (Technical report)
[29] Perturbation theory in calculations of atomic energy levels, J. Phys. B. At. Mol. Opt. Phys., Volume 8 (1975) no. 10, 1591 | DOI
[30] Variational calculations, Phys. Rev. B, Volume 49 (1994) no. 21, pp. 15253-15257 | DOI
[31] Quantum Monte Carlo studies of two-electron atoms constrained in spherical boxes, J. Phys. B. At. Mol. Opt. Phys., Volume 25 (1992) no. 9, 1965 | DOI
[32] Hylleraas-configuration interaction calculations on the S state of the helium atom, J. Coord. Chem., Volume 68 (2015) no. 17–18, pp. 3340-3361 | DOI
[33] Accurate exchange correlation potentials and total energy components for the helium isoelectronic series, Phys. Rev. A, Volume 50 (1994), pp. 3827-3837 | DOI
[34] Calculation of ground state energy of helium using Hylleraas trial function expansion, Int. J. Eng. Res. Technol., Volume 12 (2019) no. 8, pp. 1178-1182
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