Compact Correlated Hylleraas Wave Functions for the Lithium Isoelectronic Sequence: Ground-State Energies and a Complex-Rotation Assessment

Youssou Gning; Mame Diarra Dieng; Moussa Toure; Demba Sylla; Mamadou Lamine Samb

doi:doi:10.11648/j.ajmp.20261505.12

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Compact Correlated Hylleraas Wave Functions for the Lithium Isoelectronic Sequence: Ground-State Energies and a Complex-Rotation Assessment

Youssou Gning

, Mame Diarra Dieng, Moussa Toure

, Demba Sylla

, Mamadou Lamine Samb^*

Published in American Journal of Modern Physics (Volume 15, Issue 5)

Received: 3 September 2026 Accepted: 11 September 2026 Published: 27 September 2026

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Abstract

The nonrelativistic ground-state energy of the (1s²2s)²S term of atomic lithium and five members of its isoelectronic sequence (Be⁺, B²⁺, C³⁺, N⁴⁺, O⁵⁺; Z = 3–8) is computed with a correlated Hylleraas-type trial function combined with the complex-rotation (complex-scaling) transformation of the radial coordinates, implemented symbolically and numerically in the computer-algebra system Maxima. Two variational basis sizes, Ω = 9 and Ω = 12, are compared for neutral lithium as a sensitivity test of the complex-scaled stabilization procedure; Ω = 9 gives the closer benchmark agreement and is used as the working basis for the ionic members of the sequence. The resulting energies agree with the multiple-basis-set Hylleraas benchmark of Yan, Tambasco and Drake with relative deviations below 6.1 × 10^-3% over Z = 3–8, and with several independent Hylleraas, full-core-plus-correlation, and screening-constant calculations reported in the literature. The evolution of the optimized non-linear parameters with nuclear charge is examined and interpreted in terms of the progressive contraction of the 1s² core relative to the comparatively stable 2s valence orbital. Because the (1s²2s)²S term lies below the first ionization threshold rather than embedded in the continuum, the small imaginary parts produced by the complex-rotation procedure cannot be interpreted as a physical autoionization width; this limitation of applying a resonance-oriented technique to a genuine bound state is discussed explicitly. The present implementation is placed in the context of an earlier complex-rotation study of the same states by Diop et al. (2020), to which it is methodologically related but numerically independent.

Published in	American Journal of Modern Physics (Volume 15, Issue 5)
DOI	10.11648/j.ajmp.20261505.12
Page(s)	148-161
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Lithium Isoelectronic Sequence, Complex Rotation, Hylleraas Wave Functions, Correlated Variational Calculation, Three-Electron Atoms, Ground-State Energy, Maxima

References

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APA Style

Gning, Y., Dieng, M. D., Toure, M., Sylla, D., Samb, M. L. (2026). Compact Correlated Hylleraas Wave Functions for the Lithium Isoelectronic Sequence: Ground-State Energies and a Complex-Rotation Assessment. American Journal of Modern Physics, 15(5), 148-161. https://doi.org/10.11648/j.ajmp.20261505.12

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ACS Style

Gning, Y.; Dieng, M. D.; Toure, M.; Sylla, D.; Samb, M. L. Compact Correlated Hylleraas Wave Functions for the Lithium Isoelectronic Sequence: Ground-State Energies and a Complex-Rotation Assessment. Am. J. Mod. Phys. 2026, 15(5), 148-161. doi: 10.11648/j.ajmp.20261505.12

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AMA Style

Gning Y, Dieng MD, Toure M, Sylla D, Samb ML. Compact Correlated Hylleraas Wave Functions for the Lithium Isoelectronic Sequence: Ground-State Energies and a Complex-Rotation Assessment. Am J Mod Phys. 2026;15(5):148-161. doi: 10.11648/j.ajmp.20261505.12

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@article{10.11648/j.ajmp.20261505.12,
  author = {Youssou Gning and Mame Diarra Dieng and Moussa Toure and Demba Sylla and Mamadou Lamine Samb},
  title = {Compact Correlated Hylleraas Wave Functions for the Lithium Isoelectronic Sequence: Ground-State Energies and a Complex-Rotation Assessment},
  journal = {American Journal of Modern Physics},
  volume = {15},
  number = {5},
  pages = {148-161},
  doi = {10.11648/j.ajmp.20261505.12},
  url = {https://doi.org/10.11648/j.ajmp.20261505.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20261505.12},
  abstract = {The nonrelativistic ground-state energy of the (1s22s)2S term of atomic lithium and five members of its isoelectronic sequence (Be+, B2+, C3+, N4+, O5+; Z = 3–8) is computed with a correlated Hylleraas-type trial function combined with the complex-rotation (complex-scaling) transformation of the radial coordinates, implemented symbolically and numerically in the computer-algebra system Maxima. Two variational basis sizes, Ω = 9 and Ω = 12, are compared for neutral lithium as a sensitivity test of the complex-scaled stabilization procedure; Ω = 9 gives the closer benchmark agreement and is used as the working basis for the ionic members of the sequence. The resulting energies agree with the multiple-basis-set Hylleraas benchmark of Yan, Tambasco and Drake with relative deviations below 6.1 × 10-3% over Z = 3–8, and with several independent Hylleraas, full-core-plus-correlation, and screening-constant calculations reported in the literature. The evolution of the optimized non-linear parameters with nuclear charge is examined and interpreted in terms of the progressive contraction of the 1s2 core relative to the comparatively stable 2s valence orbital. Because the (1s22s)2S term lies below the first ionization threshold rather than embedded in the continuum, the small imaginary parts produced by the complex-rotation procedure cannot be interpreted as a physical autoionization width; this limitation of applying a resonance-oriented technique to a genuine bound state is discussed explicitly. The present implementation is placed in the context of an earlier complex-rotation study of the same states by Diop et al. (2020), to which it is methodologically related but numerically independent.},
 year = {2026}
}

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TY  - JOUR
T1  - Compact Correlated Hylleraas Wave Functions for the Lithium Isoelectronic Sequence: Ground-State Energies and a Complex-Rotation Assessment
AU  - Youssou Gning
AU  - Mame Diarra Dieng
AU  - Moussa Toure
AU  - Demba Sylla
AU  - Mamadou Lamine Samb
Y1  - 2026/09/27
PY  - 2026
N1  - https://doi.org/10.11648/j.ajmp.20261505.12
DO  - 10.11648/j.ajmp.20261505.12
T2  - American Journal of Modern Physics
JF  - American Journal of Modern Physics
JO  - American Journal of Modern Physics
SP  - 148
EP  - 161
PB  - Science Publishing Group
SN  - 2326-8891
UR  - https://doi.org/10.11648/j.ajmp.20261505.12
AB  - The nonrelativistic ground-state energy of the (1s22s)2S term of atomic lithium and five members of its isoelectronic sequence (Be+, B2+, C3+, N4+, O5+; Z = 3–8) is computed with a correlated Hylleraas-type trial function combined with the complex-rotation (complex-scaling) transformation of the radial coordinates, implemented symbolically and numerically in the computer-algebra system Maxima. Two variational basis sizes, Ω = 9 and Ω = 12, are compared for neutral lithium as a sensitivity test of the complex-scaled stabilization procedure; Ω = 9 gives the closer benchmark agreement and is used as the working basis for the ionic members of the sequence. The resulting energies agree with the multiple-basis-set Hylleraas benchmark of Yan, Tambasco and Drake with relative deviations below 6.1 × 10-3% over Z = 3–8, and with several independent Hylleraas, full-core-plus-correlation, and screening-constant calculations reported in the literature. The evolution of the optimized non-linear parameters with nuclear charge is examined and interpreted in terms of the progressive contraction of the 1s2 core relative to the comparatively stable 2s valence orbital. Because the (1s22s)2S term lies below the first ionization threshold rather than embedded in the continuum, the small imaginary parts produced by the complex-rotation procedure cannot be interpreted as a physical autoionization width; this limitation of applying a resonance-oriented technique to a genuine bound state is discussed explicitly. The present implementation is placed in the context of an earlier complex-rotation study of the same states by Diop et al. (2020), to which it is methodologically related but numerically independent.
VL  - 15
IS  - 5
ER  -

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Author Information

Youssou Gning

Department of Physics and Chemistry, Iba Der Thiam University of Thies, Thies, Senegal

http://orcid.org/0000-0003-2486-6944
Mame Diarra Dieng

Department of Physics and Chemistry, Iba Der Thiam University of Thies, Thies, Senegal
Moussa Toure

Department of Physics and Chemistry, Iba Der Thiam University of Thies, Thies, Senegal

http://orcid.org/0009-0005-4663-9533
Demba Sylla

Department of Physics and Chemistry, Iba Der Thiam University of Thies, Thies, Senegal

http://orcid.org/0009-0009-3230-9129
Mamadou Lamine Samb

Department of Physics and Chemistry, Iba Der Thiam University of Thies, Thies, Senegal

Contact Email

http://orcid.org/0000-0003-0695-553X

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APA Style

Gning, Y., Dieng, M. D., Toure, M., Sylla, D., Samb, M. L. (2026). Compact Correlated Hylleraas Wave Functions for the Lithium Isoelectronic Sequence: Ground-State Energies and a Complex-Rotation Assessment. American Journal of Modern Physics, 15(5), 148-161. https://doi.org/10.11648/j.ajmp.20261505.12

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ACS Style

Gning, Y.; Dieng, M. D.; Toure, M.; Sylla, D.; Samb, M. L. Compact Correlated Hylleraas Wave Functions for the Lithium Isoelectronic Sequence: Ground-State Energies and a Complex-Rotation Assessment. Am. J. Mod. Phys. 2026, 15(5), 148-161. doi: 10.11648/j.ajmp.20261505.12

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AMA Style

Gning Y, Dieng MD, Toure M, Sylla D, Samb ML. Compact Correlated Hylleraas Wave Functions for the Lithium Isoelectronic Sequence: Ground-State Energies and a Complex-Rotation Assessment. Am J Mod Phys. 2026;15(5):148-161. doi: 10.11648/j.ajmp.20261505.12

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@article{10.11648/j.ajmp.20261505.12,
  author = {Youssou Gning and Mame Diarra Dieng and Moussa Toure and Demba Sylla and Mamadou Lamine Samb},
  title = {Compact Correlated Hylleraas Wave Functions for the Lithium Isoelectronic Sequence: Ground-State Energies and a Complex-Rotation Assessment},
  journal = {American Journal of Modern Physics},
  volume = {15},
  number = {5},
  pages = {148-161},
  doi = {10.11648/j.ajmp.20261505.12},
  url = {https://doi.org/10.11648/j.ajmp.20261505.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20261505.12},
  abstract = {The nonrelativistic ground-state energy of the (1s22s)2S term of atomic lithium and five members of its isoelectronic sequence (Be+, B2+, C3+, N4+, O5+; Z = 3–8) is computed with a correlated Hylleraas-type trial function combined with the complex-rotation (complex-scaling) transformation of the radial coordinates, implemented symbolically and numerically in the computer-algebra system Maxima. Two variational basis sizes, Ω = 9 and Ω = 12, are compared for neutral lithium as a sensitivity test of the complex-scaled stabilization procedure; Ω = 9 gives the closer benchmark agreement and is used as the working basis for the ionic members of the sequence. The resulting energies agree with the multiple-basis-set Hylleraas benchmark of Yan, Tambasco and Drake with relative deviations below 6.1 × 10-3% over Z = 3–8, and with several independent Hylleraas, full-core-plus-correlation, and screening-constant calculations reported in the literature. The evolution of the optimized non-linear parameters with nuclear charge is examined and interpreted in terms of the progressive contraction of the 1s2 core relative to the comparatively stable 2s valence orbital. Because the (1s22s)2S term lies below the first ionization threshold rather than embedded in the continuum, the small imaginary parts produced by the complex-rotation procedure cannot be interpreted as a physical autoionization width; this limitation of applying a resonance-oriented technique to a genuine bound state is discussed explicitly. The present implementation is placed in the context of an earlier complex-rotation study of the same states by Diop et al. (2020), to which it is methodologically related but numerically independent.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Compact Correlated Hylleraas Wave Functions for the Lithium Isoelectronic Sequence: Ground-State Energies and a Complex-Rotation Assessment
AU  - Youssou Gning
AU  - Mame Diarra Dieng
AU  - Moussa Toure
AU  - Demba Sylla
AU  - Mamadou Lamine Samb
Y1  - 2026/09/27
PY  - 2026
N1  - https://doi.org/10.11648/j.ajmp.20261505.12
DO  - 10.11648/j.ajmp.20261505.12
T2  - American Journal of Modern Physics
JF  - American Journal of Modern Physics
JO  - American Journal of Modern Physics
SP  - 148
EP  - 161
PB  - Science Publishing Group
SN  - 2326-8891
UR  - https://doi.org/10.11648/j.ajmp.20261505.12
AB  - The nonrelativistic ground-state energy of the (1s22s)2S term of atomic lithium and five members of its isoelectronic sequence (Be+, B2+, C3+, N4+, O5+; Z = 3–8) is computed with a correlated Hylleraas-type trial function combined with the complex-rotation (complex-scaling) transformation of the radial coordinates, implemented symbolically and numerically in the computer-algebra system Maxima. Two variational basis sizes, Ω = 9 and Ω = 12, are compared for neutral lithium as a sensitivity test of the complex-scaled stabilization procedure; Ω = 9 gives the closer benchmark agreement and is used as the working basis for the ionic members of the sequence. The resulting energies agree with the multiple-basis-set Hylleraas benchmark of Yan, Tambasco and Drake with relative deviations below 6.1 × 10-3% over Z = 3–8, and with several independent Hylleraas, full-core-plus-correlation, and screening-constant calculations reported in the literature. The evolution of the optimized non-linear parameters with nuclear charge is examined and interpreted in terms of the progressive contraction of the 1s2 core relative to the comparatively stable 2s valence orbital. Because the (1s22s)2S term lies below the first ionization threshold rather than embedded in the continuum, the small imaginary parts produced by the complex-rotation procedure cannot be interpreted as a physical autoionization width; this limitation of applying a resonance-oriented technique to a genuine bound state is discussed explicitly. The present implementation is placed in the context of an earlier complex-rotation study of the same states by Diop et al. (2020), to which it is methodologically related but numerically independent.
VL  - 15
IS  - 5
ER  -

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