Skip to main content

Advertisement

SpringerLink
Log in

Structural modifications of two-electron systems under isotropic harmonic confinement

  • Regular Article - Atomic Physics
  • Published:
The European Physical Journal D Aims and scope Submit manuscript

Abstract

Atomic systems placed in external potentials manifest various characteristic features which provide useful knowledge about the surroundings. We have studied the structural properties of the ground state of different two-electron systems under isotropic harmonic confinement (IHC). In particular, we have considered negative hydrogen ion, neutral helium atom and positive singly ionized lithium ion to cover all types of charge states. In addition, we have also studied the system of two electrons inside IHC. The wave function is expanded in Hylleraas basis to incorporate the effect of electron correlation in an explicit manner and Ritz variational calculations are performed to obtain the energy eigenvalues and the relative wave functions. The energy levels become more positive with increasing strength of the confining potential. The results show that for ionic systems, the two-electron energy level crosses the respective one-electron threshold at a certain value of the potential beyond which the two-electron level becomes quasi-bound. In order to get deeper insight into such threshold ionization phenomenon, we have examined the contribution of the correlated energy \(E_\mathrm{corr}\) [sum of radial and angular correlation energy] and radial correlation energy \(E_\mathrm{rad.corr}\) to the total energy for different two-electron systems under IHC. The Hellmann–Feynman theorem and the virial theorem have been verified as a quantitative validation of the accuracy of our results. The one- and two-electron radial densities have also been analyzed to gain a physical insight into the structural changes of the two-electron systems under IHC. Moreover, the expectation values of different radial and angular variables are also reported which are important to estimate different geometrical and spectral properties.

Graphic abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The data are given in the tables and plots. All data are also available from the authors on request. There are no other associated data].

References

  1. R.C. Ashoori, Electrons in artificial atoms. Nature 379(6564), 413–419 (1996)

    Article  ADS  Google Scholar 

  2. N.F. Johnson, Quantum dots: few-body, low-dimensional systems. J. Phys.: Condens. Matter 7(6), 965–989 (1995)

    ADS  Google Scholar 

  3. V.K. Dolmatov, Photoionization of Atoms Encaged in Spherical Fullerenes, Advances in Quantum Chemistry, vol. 58 (Academic Press, Cambridge, 2000), pp. 13–68

    Google Scholar 

  4. J.A. Ludlow, T.-G. Lee, M.S. Pindzola, Double photoionization of atoms and ions confined in charged fullerenes. J. Phys. B: Atomic, Mol. Opt. Phys. 43(23), 235202 (2010)

    Article  ADS  Google Scholar 

  5. T. Nakano, Y. Nozue, Electrons of alkali metals in regular nanospaces of zeolites. Adv. Phys.: X 2(2), 254–280 (2017)

    Google Scholar 

  6. W. Si-Ming, X.-Y. Yang, C. Janiak, Confinement effects in zeolite-confined noble metals. Angew. Chemie Int. Ed. 58(36), 12340–12354 (2019)

    Article  Google Scholar 

  7. A.W. Hauser, A. Volk, P. Thaler, W.E. Ernst, Atomic collisions in suprafluid helium-nanodroplets: timescales for metal-cluster formation derived from he-density functional theory. Phys. Chem. Chem. Phys. 17, 10805–10812 (2015)

    Article  Google Scholar 

  8. A.S. Chatterley, B. Shepperson, H. Stapelfeldt, Three-dimensional molecular alignment inside helium nanodroplets. Phys. Rev. Lett. 119, 073202 (2017)

    Article  ADS  Google Scholar 

  9. M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon and Breach, London, 1969)

    Google Scholar 

  10. P.A. Maksym, T. Chakraborty, Role of electron-electron interactions, quantum dots in a magnetic field. Phys. Rev. Lett. 65, 108–111 (1990)

    Article  ADS  Google Scholar 

  11. J. Cioslowski, K. Pernal, The ground state of harmonium. J. Chem. Phys. 113(19), 8434–8443 (2000)

    Article  ADS  Google Scholar 

  12. D.P. O’Neill, P.M.W. Gill, Wave functions and two-electron probability distributions of the Hooke’s-law atom and helium. Phys. Rev. A 68, 022505 (2003)

    Article  ADS  Google Scholar 

  13. P.-F. Loos, Hooke’s law correlation in two-electron systems. Phys. Rev. A 81, 032510 (2010)

    Article  ADS  Google Scholar 

  14. V.I. Pupyshev, H.E. Montgomery, Spherically symmetric states of hookium in a cavity. Physica Scr. 90(8), 085401 (2015)

    Article  ADS  Google Scholar 

  15. R. J. Eden, V. J. Emery, N. F. Mott. The binding energies of atomic nuclei I. introduction and general method, in Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, vol. 248(1253), pp. 266–281 (1958)

  16. N.R. Kestner, O. Sinanoglu, Study of electron correlation in helium-like systems using an exactly soluble model. Phys. Rev. 128, 2687–2692 (1962)

    Article  ADS  Google Scholar 

  17. S. Kais, D.R. Herschbach, R.D. Levine, Dimensional scaling as a symmetry operation. J. Chem. Phys. 91(12), 7791–7796 (1989)

    Article  ADS  Google Scholar 

  18. M. Taut, Two electrons in an external oscillator potential: particular analytic solutions of a coulomb correlation problem. Phys. Rev. A 48, 3561–3566 (1993)

    Article  ADS  Google Scholar 

  19. D. Bielińska-Wa̧ż, J. Karwowski, G.H.F. Diercksen, Spectra of confined two-electron atoms. J. Phys. B Atomic Mol. Opt. Phys. 34, 1987–2000 (2001)

    Article  ADS  Google Scholar 

  20. T. Sako, G.H.F. Diercksen, Confined quantum systems: dipole polarizability of the two-electron quantum dot, the hydrogen negative ion and the helium atom. J. Phys. B: At. Mol. Opt. Phys. 36, 3743–3759 (2003)

    Article  ADS  Google Scholar 

  21. D. Bielińska-Wa̧ż, G.H.F. Diercksen, M. Klobukowski, Quantum chemistry of confined systems: structure and vibronic spectra of a confined hydrogen molecule. Chem. Phys. Lett. 349, 215–21 (2001)

    Article  ADS  Google Scholar 

  22. S. Mandal, P.K. Mukherjee, G.H.F. Diercksen, Two electrons in a harmonic potential: an approximate analytical solution. J. Phys. B Atomic Mol. Opt. Phys. 36(22), 4483–4494 (2003)

    Article  ADS  Google Scholar 

  23. A. Robles-Navarro, P. Fuentealba, F. Muñoz, C. Cárdenas, Electronic structure of first and second row atoms under harmonic confinement. Int. J. Quantum Chem. 120(7), e26132 (2020)

    Article  Google Scholar 

  24. L.S.F. Olavo, A.M. Maniero, C.R. de Carvalho, F.V. Prudente, G. Jalbert, Choice of atomic basis set for the study of two electrons in a harmonic anisotropic quantum dot using a configuration interaction approach. J. Phys. B Atomic Mol. Opt. Phys. 49(14), 145004 (2016)

    Article  ADS  Google Scholar 

  25. H.E. Caicedo-Ortiz, H.O. Castañeda Fernández, E. Santiago-Cortès, D.A. Mantilla-Sandoval, Energy levels in a single-electron quantum dot with hydrostatic pressure. Acta Phys. Pol. A 134(2), 570–573 (2018)

    Article  ADS  Google Scholar 

  26. F.S. Nammas, Solvable model of the thermal persistent current at low temperatures of two-electron parabolic GaAs quantum dot. J. Low Temp. Phys. 200(1), 76–89 (2020)

    Article  ADS  Google Scholar 

  27. T. Chakraborty, Q. Dots, A Survey of the Properties of Artificial Atoms (Elsevier, Amsterdam, 1999)

    Google Scholar 

  28. E. Tiesinga, C.J. Williams, F.H. Mies, P.S. Julienne, Interacting atoms under strong quantum confinement. Phys. Rev. A 61, 063416 (2000)

    Article  ADS  Google Scholar 

  29. P.-O. Löwdin, Correlation problem in many-electron quantum mechanics I. review of different approaches and discussion of some current ideas. Adv. Chem. Phys. 2, 207–322 (1958)

    Google Scholar 

  30. A. Samanta, S.K. Ghosh, Correlation in an exactly solvable two-particle quantum system. Phys. Rev. A 42, 1178–1183 (1990)

    Article  ADS  Google Scholar 

  31. P. Ziesche, V.H. Smith, M. Hô, S.P. Rudin, P. Gersdorf, M. Taut, The He isoelectronic series and the Hooke’s law model: correlation measures and modifications of Collins’ conjecture. J. Chem. Phys. 110(13), 6135–6142 (1999)

    Article  ADS  Google Scholar 

  32. B. Szafran, J. Adamowski, S. Bednarek, Electron-electron correlation in quantum dots. Physica E 5, 185–195 (1999)

    Article  ADS  Google Scholar 

  33. J. Stiehler, J. Hinze, D. Andrae, M. Reiher, P. Schiffels, J. Neugebauer, Numerical Wavefunction Calculation for Atoms (University of Bielefeld, Bielefeld, 2000)

    Google Scholar 

  34. R.G. Parr, Y. Weitao, Density-Functional Theory of Atoms and Molecules, International Series of Monographs on Chemistry (Oxford University Press, New York, 1994)

    Google Scholar 

  35. P. Kościk, J.K. Saha, Entanglement in helium atom confined in an impenetrable cavity. Eur. Phys. J. D 69, 250 (2015)

    Article  ADS  Google Scholar 

  36. C.-H. Lin, Y.K. Ho, Shannon information entropy in position space for two-electron atomic systems. Chem. Phys. Lett. 633, 261–264 (2015)

    Article  ADS  Google Scholar 

  37. J.A. Nelder, R. Mead, A simplex method for function minimization. Comput. J. 7, 308–313 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  38. J.K. Saha, T.K. Mukherjee, Doubly excited bound and resonance \(({^{3}P}^{e})\) states of helium. Phys. Rev. A 80, 022513 (2009)

    Article  Google Scholar 

  39. H. Nakamura, N. Hatano, S. Garmon, T. Petrosky, Quasibound states in the continuum in a two channel quantum wire with an Adatom. Phys. Rev. Lett. 99, 210404 (2007)

    Article  ADS  Google Scholar 

  40. Z.-Q. Fu, K.-K. Bai, Y.-N. Ren, J.-J. Zhou, L. He, Coulomb interaction in quasibound states of graphene quantum dots. Phys. Rev. B 101, 235310 (2020)

    Article  ADS  Google Scholar 

  41. C.-D. Han, H.-Y. Xu, Y.-C. Lai, Electrical confinement in a spectrum of two-dimensional Dirac materials with classically integrable, mixed, and chaotic dynamics. Phys. Rev. Res. 2, 013116 (2020)

    Article  Google Scholar 

  42. S. Bhattacharyya, J.K. Saha, T.K. Mukherjee, Nonrelativistic structure calculations of two-electron ions in a strongly coupled plasma environment. Phys. Rev. A 91, 042515 (2015)

    Article  ADS  Google Scholar 

  43. J. Garza, R. Vargas, N. Aquino, K.D. Sen, DFT reactivity indices in confined many-electron atoms. J. Chem. Sci. 117, 379–386 (2005)

    Article  Google Scholar 

  44. E. García-Hernández, C. Díaz-García, R. Vargas, J. Garza, Implementation of the electron propagator to second order on GPUs to estimate the ionization potentials of confined atoms. J. Phys. B: At. Mol. Opt. Phys. 47, 185007 (2014)

    Article  ADS  Google Scholar 

  45. S.-K. Son, R. Thiele, Z. Jurek, B. Ziaja, R. Santra, Quantum-mechanical calculation of ionization-potential lowering in dense plasmas. Phys. Rev. X 4, 031004 (2014)

    Google Scholar 

  46. R. Jha, S. Giri, P.K. Chattaraj, Does confinement alter the ionization energy and electron affinity of atoms? Eur Phys J D 75, 88 (2021)

    Article  ADS  Google Scholar 

  47. J. Zeng, Y. Li, Y. Hou, C. Gao, J. Yuan, Ionization potential depression and ionization balance in dense carbon plasma under solar and stellar interior conditions. Astronomy Astrophys. 644, A92 (2020)

    Article  ADS  Google Scholar 

  48. C.L. Wilson, H.E. Montgomery, K.D. Sen, D.C. Thompson, Electron correlation energy in confined two-electron systems. Phys. Lett. A 374, 4415–4419 (2010)

    Article  MATH  ADS  Google Scholar 

  49. T. Koga, Radial correlation limits of helium and heliumlike atoms. Z Physik D Atoms Molecules Clust 37, 301 (1996)

    Article  ADS  Google Scholar 

  50. F. Arias, E. de Saavedra, F.J.G. Buendia, Single-particle and electron-pair densities at the origin in the ground state of helium-like ions. J Phys B: Atomic, Mol Opt Phys 27(21), 5131–5137 (1994)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The financial assistance provided through Grant Number 23(Sanc.)/ST/P/S&T/16G-26/2017 by DHESTBT, Govt. of West Bengal, India, is gratefully acknowledged by AH and SB. JKS acknowledges partial financial support from DHESTBT, Govt. of West Bengal, India, under Grant Number 249(Sanc.)/ST/P/S&T/16G-26/2017. KDS thanks Indian National Science Academy, New Delhi, for award of a senior scientist grant.

Author information

Author notes

  1. Ashoke Hazra and Santanu Mondal have contributed equally to this work.

Authors and Affiliations

  1. Department of Physics, Acharya Prafulla Chandra College, New Barrackpore, Kolkata, 700131, India

    Ashoke Hazra & Sukhamoy Bhattacharyya

  2. Department of Physics, Aliah University, Newtown, Kolkata, 700156, India

    Santanu Mondal & Jayanta K. Saha

  3. School of Chemistry, University of Hyderabad, Hyderabad, 500046, India

    K. D. Sen

Authors
  1. Ashoke Hazra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Santanu Mondal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sukhamoy Bhattacharyya
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jayanta K. Saha
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. K. D. Sen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Sukhamoy Bhattacharyya or Jayanta K. Saha.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hazra, A., Mondal, S., Bhattacharyya, S. et al. Structural modifications of two-electron systems under isotropic harmonic confinement. Eur. Phys. J. D 75, 186 (2021). https://doi.org/10.1140/epjd/s10053-021-00196-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjd/s10053-021-00196-3

Search

Navigation