- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review A

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Self-energy corrections to the ionization energies in sodiumlike ions: Comparison of the ab initio QED and model-QED-operator approaches

P. Yang1,2,3, A. V. Malyshev3,4, E. A. Prokhorchuk3,5, I. I. Tupitsyn3, V. M. Shabaev3,4, and D. P. Usov3

PDF

Phys. Rev. A 114, 012820 – Published 28 July, 2026

DOI: https://doi.org/10.1103/2f1q-qt8c

Abstract

Calculations of the self-energy corrections to ionization energies of the 3s, 3p1/2, and 3p3/2 states in sodiumlike ions with nuclear-charge numbers Z=30, 50, 70, and 92 are presented. The calculations are performed using two approaches: the rigorous bound-state QED formalism and the model-QED-operator method. Within the first method, the first and second orders of the QED perturbation theory formulated in the Furry picture are evaluated. Various screening potentials are included into the initial approximation to partially take into account the electron-electron interaction effects already at the lowest order, thereby accelerating the convergence of perturbation series. Within the second approach, different implementations of the model-QED operator, including its incorporation into the relativistic configuration-interaction calculations, are considered. A detailed comparison of the results obtained by these two independent methods is presented, demonstrating good agreement and thus validating the accuracy and efficiency of the model-QED-operator approach for many-electron systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (71)

  1. A. M. Desiderio and W. R. Johnson, Lamb shift and binding energies of K electrons in heavy atoms, Phys. Rev. A 3, 1267 (1971).
  2. P. J. Mohr, Self-energy radiative corrections in hydrogen-like systems, Ann. Phys. (N.Y.) 88, 26 (1974).
  3. P. J. Mohr, Numerical evaluation of the 1S12-state radiative level shift, Ann. Phys. (N.Y.) 88, 52 (1974).
  4. P. Indelicato and P. J. Mohr, Coordinate-space approach to the bound-electron self-energy, Phys. Rev. A 46, 172 (1992).
  5. H. Persson, I. Lindgren, and S. Salomonson, A new approach to the electron self energy calculation, Phys. Scr. T46, 125 (1993).
  6. H. M. Quiney and I. P. Grant, Partial-wave mass renormalization in atomic QED calculations, Phys. Scr. T46, 132 (1993).
  7. L. N. Labzowsky and I. A. Goidenko, Multiple commutator expansion for the Lamb shift in a strong Coulomb field, J. Phys. B 30, 177 (1997).
  8. P. Indelicato and P. J. Mohr, Coordinate-space approach to the bound-electron self-energy: Coulomb field calculation, Phys. Rev. A 58, 165 (1998).
  9. U. D. Jentschura, P. J. Mohr, and G. Soff, Calculation of the electron self-energy for low nuclear charge, Phys. Rev. Lett. 82, 53 (1999).
  10. N. J. Snyderman, Electron radiative self-energy of highly stripped heavy atoms, Ann. Phys. (N.Y.) 211, 43 (1991).
  11. S. A. Blundell and N. J. Snyderman, Basis-set approach to calculating the radiative self-energy in highly ionized atoms, Phys. Rev. A 44, R1427(R) (1991).
  12. S. A. Blundell, Accurate screened QED calculations in high-Z many-electron ions, Phys. Rev. A 46, 3762 (1992).
  13. K. T. Cheng, W. R. Johnson, and J. Sapirstein, Lamb-shift calculations for non-Coulomb potentials, Phys. Rev. A 47, 1817 (1993).
  14. V. A. Yerokhin and V. M. Shabaev, First-order self-energy correction in hydrogenlike systems, Phys. Rev. A 60, 800 (1999).
  15. H. Persson, S. Salomonson, P. Sunnergren, and I. Lindgren, Two-electron Lamb-shift calculations on heliumlike ions, Phys. Rev. Lett. 76, 204 (1996).
  16. V. A. Yerokhin, A. N. Artemyev, and V. M. Shabaev, Two-electron self-energy contribution to the ground-state energy of helium-like ions, Phys. Lett. A 234, 361 (1997).
  17. V. A. Yerokhin, A. N. Artemyev, T. Beier, G. Plunien, V. M. Shabaev, and G. Soff, Two-electron self-energy corrections to the 2p1/2–2s transition energy in Li-like ions, Phys. Rev. A 60, 3522 (1999).
  18. P. Indelicato and P. J. Mohr, Coordinate-space approach to the bound-electron self-energy: Self-energy screening calculation, Phys. Rev. A 63, 052507 (2001).
  19. V. A. Yerokhin, K. Pachucki, and V. M. Shabaev, One-loop self-energy correction in a strong binding field, Phys. Rev. A 72, 042502 (2005).
  20. A. N. Artemyev, V. M. Shabaev, I. I. Tupitsyn, G. Plunien, and V. A. Yerokhin, QED calculation of the 2p3/2–2p1/2 transition energy in boronlike argon, Phys. Rev. Lett. 98, 173004 (2007).
  21. J. Sapirstein and K. T. Cheng, Simplified partial wave expansion of the Lamb shift, Phys. Rev. A 108, 042804 (2023).
  22. A. V. Malyshev, E. A. Prokhorchuk, and V. M. Shabaev, Convergence-acceleration approach to partial-wave expansion of two-electron self-energy contributions to the Lamb shift, Phys. Rev. A 109, 062802 (2024).
  23. V. A. Yerokhin, Z. Harman, and C. H. Keitel, Two-loop electron self-energy for low nuclear charges, Phys. Rev. Lett. 133, 251803 (2024).
  24. V. A. Yerokhin, Z. Harman, and C. H. Keitel, One-loop electron self-energy with accelerated partial-wave expansion in the Coulomb gauge, Phys. Rev. A 111, 012802 (2025).
  25. A. N. Artemyev, V. M. Shabaev, V. A. Yerokhin, G. Plunien, and G. Soff, QED calculation of the n=1 and n=2 energy levels in He-like ions, Phys. Rev. A 71, 062104 (2005).
  26. A. V. Malyshev, D. A. Glazov, Y. S. Kozhedub, I. S. Anisimova, M. Y. Kaygorodov, V. M. Shabaev, and I. I. Tupitsyn, Ab initio calculations of energy levels in Be-like xenon: Strong interference between electron-correlation and QED effects, Phys. Rev. Lett. 126, 183001 (2021).
  27. A. V. Malyshev, Y. S. Kozhedub, V. M. Shabaev, and I. I. Tupitsyn, QED calculations of intra-L-shell doubly excited states in Be-like ions, Phys. Rev. A 112, 062811 (2025).
  28. V. A. Yerokhin, V. Patkóš, and K. Pachucki, QED calculations of energy levels of heliumlike ions with 5≤Z≤30, Phys. Rev. A 106, 022815 (2022).
  29. P. Indelicato and J. P. Desclaux, Multiconfiguration Dirac-Fock calculations of transition energies with QED corrections in three-electron ions, Phys. Rev. A 42, 5139 (1990).
  30. P. Pyykkö and L.-B. Zhao, Search for effective local model potentials for simulation of quantum electrodynamic effects in relativistic calculations, J. Phys. B: At. Mol. Opt. Phys. 36, 1469 (2003).
  31. I. Draganić, J. R. Crespo López-Urrutia, R. DuBois, S. Fritzsche, V. M. Shabaev, R. Soria Orts, I. I. Tupitsyn, Y. Zou, and J. Ullrich, High precision wavelength measurements of QED-sensitive forbidden transitions in highly charged argon ions, Phys. Rev. Lett. 91, 183001 (2003).
  32. V. V. Flambaum and J. S. M. Ginges, Radiative potential and calculations of QED radiative corrections to energy levels and electromagnetic amplitudes in many-electron atoms, Phys. Rev. A 72, 052115 (2005).
  33. C. Thierfelder and P. Schwerdtfeger, Quantum electrodynamic corrections for the valence shell in heavy many-electron atoms, Phys. Rev. A 82, 062503 (2010).
  34. P. Pyykkö, The physics behind chemistry and the periodic table, Chem. Rev. 112, 371 (2012).
  35. I. I. Tupitsyn and E. V. Berseneva, A single-particle nonlocal potential for taking into account quantum-electrodynamic corrections in calculations of the electronic structure of atoms, Opt. Spectrosc. 114, 682 (2013).
  36. V. M. Shabaev, I. I. Tupitsyn, and V. A. Yerokhin, Model operator approach to the Lamb shift calculations in relativistic many-electron atoms, Phys. Rev. A 88, 012513 (2013).
  37. J. S. M. Ginges and J. C. Berengut, Atomic many-body effects and Lamb shifts in alkali metals, Phys. Rev. A 93, 052509 (2016).
  38. L. V. Skripnikov, Approaching meV level for transition energies in the radium monofluoride molecule RaF and radium cation Ra+ by including quantum-electrodynamics effects, J. Chem. Phys. 154, 201101 (2021).
  39. A. V. Malyshev, D. A. Glazov, V. M. Shabaev, I. I. Tupitsyn, V. A. Yerokhin, and V. A. Zaytsev, Model-QED operator for superheavy elements, Phys. Rev. A 106, 012806 (2022).
  40. V. M. Shabaev, I. I. Tupitsyn, and V. A. Yerokhin, QEDMOD: Fortran program for calculating the model Lamb-shift operator, Comput. Phys. Commun. 189, 175 (2015); 223, 69 (2018).
  41. L. F. Pašteka, E. Eliav, A. Borschevsky, U. Kaldor, and P. Schwerdtfeger, Relativistic coupled cluster calculations with variational quantum electrodynamics resolve the discrepancy between experiment and theory concerning the electron affinity and ionization potential of gold, Phys. Rev. Lett. 118, 023002 (2017).
  42. J. Machado, C. I. Szabo, J. P. Santos, P. Amaro, M. Guerra, A. Gumberidze, G. Bian, J. M. Isac, and P. Indelicato, High-precision measurements of n=2→n=1 transition energies and level widths in He- and Be-like argon ions, Phys. Rev. A 97, 032517 (2018).
  43. R. Si, X. L. Guo, T. Brage, C. Y. Chen, R. Hutton, and C. Froese Fischer, Breit and QED effects on the 3d9D3/22→D5/22 transition energy in Co-like ions, Phys. Rev. A 98, 012504 (2018).
  44. A. Müller, E. Lindroth, S. Bari, A. Borovik, Jr., P.-M. Hillenbrand, K. Holste, P. Indelicato, A. L. D. Kilcoyne, S. Klumpp, M. Martins, J. Viefhaus, P. Wilhelm, and S. Schippers, Photoionization of metastable heliumlike C4+(1s2sS13) ions: Precision study of intermediate doubly excited states, Phys. Rev. A 98, 033416 (2018).
  45. A. Zaitsevskii, N. S. Mosyagin, A. V. Oleynichenko, and E. Eliav, Generalized relativistic small-core pseudopotentials accounting for quantum electrodynamic effects: Construction and pilot applications, Int. J. Quantum Chem. 123, e27077 (2023).
  46. Y. Guo, L. F. Pašteka, Y. Nagame, T. K. Sato, E. Eliav, M. L. Reitsma, and A. Borschevsky, Relativistic coupled-cluster calculations of the electron affinity and ionization potentials of lawrencium, Phys. Rev. A 110, 022817 (2024).
  47. T. Saue, Does chemistry need more physics? Pure Appl. Chem. 97, 1255 (2025).
  48. K. N. Lyashchenko, O. Y. Andreev, and D. Yu, Photon angular distribution in two-photon electron capture by H-like uranium, Phys. Rev. A 111, 032815 (2025).
  49. R. Silwal, S. A. Blundell, S. C. Sanders, Dipti, H. Staiger, A. Hosier, M. G. Fuller, Yu.  Ralchenko, E. Takacs, Testing relativistic atomic structure and bound-state quantum electrodynamics with Co-like ions, Phys. Rev. A 111, 042821 (2025).
  50. M. Athanasakis-Kaklamanakis, S. G. Wilkins, et al., Electron correlation and relativistic effects in the excited states of radium monofluoride, Nature Commun. 16, 2139 (2025).
  51. B.-B. Li, L. Wu, D.-H. Zhang, C.-Z. Dong, and J. Jiang, The energy levels and hyperfine structures constants for n=2 states in boron-like ions, J. Quant. Spectrosc. Radiat. Transfer 345, 109573 (2025).
  52. H. Lu, B. Li, M. Yang, L. Dong, Y. Wang, M. Li, L. Wu, J. Li, J. Jiang, C. Dong, and D. Zhang, Isotope shifts due to the 1s22sS1/22→1s22pP1/2,3/22 transitions of Li-like Th87+ ions, Chin. Phys. B 34, 073203 (2025).
  53. V. M. Shabaev, I. I. Tupitsyn, M. Y. Kaygorodov, Y. S. Kozhedub, A. V. Malyshev, and D. V. Mironova, QED corrections to the P1/22−P3/22 fine structure in fluorinelike ions: Model Lamb-shift-operator approach, Phys. Rev. A 101, 052502 (2020).
  54. W. H. Furry, On bound states and scattering in positron theory, Phys. Rev. 81, 115 (1951).
  55. W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140, A1133 (1965).
  56. J. C. Slater, A simplification of the Hartree-Fock method, Phys. Rev. 81, 385 (1951).
  57. J. Sapirstein and K. T. Cheng, S-matrix calculations of energy levels of sodiumlike ions, Phys. Rev. A 91, 062508 (2015).
  58. R. Latter, Atomic energy levels for the Thomas-Fermi and Thomas-Fermi-Dirac potential, Phys. Rev. 99, 510 (1955).
  59. V. A. Yerokhin and A. V. Maiorova, Calculations of QED effects with the Dirac Green function, Symmetry 12, 800 (2020).
  60. A. V. Malyshev, D. A. Glazov, A. V. Volotka, I. I. Tupitsyn, V. M. Shabaev, G. Plunien, and Th. Stöhlker, Ground-state ionization energies of boronlike ions, Phys. Rev. A 96, 022512 (2017).
  61. V. A. Yerokhin, P. Indelicato, and V. M. Shabaev, Evaluation of the two-loop self-energy correction to the ground state energy of H-like ions to all orders in Zα, Eur. Phys. J. D 25, 203 (2003).
  62. V. A. Yerokhin, Z. Harman, and C. H. Keitel, Two-loop electron self-energy with accelerated partial-wave expansion, Phys. Rev. A 111, 042820 (2025).
  63. V. M. Shabaev, Two-time Green's function method in quantum electrodynamics of high-Z few-electron atoms, Phys. Rep. 356, 119 (2002).
  64. V. M. Shabaev, Schrodinger-like equation for the relativistic few-electron atom, J. Phys. B: At. Mol. Opt. Phys. 26, 4703 (1993).
  65. V. M. Shabaev, Quantum electrodynamics effects in atoms and molecules, in Comprehensive Computational Chemistry (1st ed.), edited by M. Yáñez, R. J. Boyd, pp. 94–128 (Elsevier, Oxford, 2024).
  66. I. I. Tupitsyn, V. M. Shabaev, J. R. Crespo López-Urrutia, I. Draganić, R. Soria Orts, and J. Ullrich, Relativistic calculations of isotope shifts in highly charged ions, Phys. Rev. A 68, 022511 (2003).
  67. I. I. Tupitsyn, N. A. Zubova, V. M. Shabaev, G. Plunien, and Th. Stöhlker, Relativistic calculations of x-ray transition energies and isotope shifts in heavy atoms, Phys. Rev. A 98, 022517 (2018).
  68. V. F. Bratzev, G. B. Deyneka, and I. I. Tupitsyn, Bull. Acad. Sci. USSR, Phys. Ser. 41, 173 (1977) [Izv. Acad. Nauk SSSR, Ser. Fiz. 41, 2655 (1977)].
  69. J. Sapirstein, K. T. Cheng, and M. H. Chen, Potential independence of the solution to the relativistic many-body problem and the role of negative-energy states in heliumlike ions, Phys. Rev. A 59, 259 (1999).
  70. A. Kramida, Y. Ralchenko, J. Reader, and NIST ASD Team, NIST Atomic Spectra Database (ver. 5.12) [Online], National Institute of Standards and Technology, Gaithersburg, MD, 2026, https://physics.nist.gov/asd.
  71. J. Sapirstein and K. T. Cheng, S-matrix calculations of energy levels of the lithium isoelectronic sequence, Phys. Rev. A 83, 012504 (2011).

Outline

Information

Phys. Rev. A 114, 012820– Published 28 July, 2026
  • Received 26 March 2026
  • Accepted 30 June 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation