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Free-basis-set method to describe the helium atom confined by a spherical box with finite and infinite potentials

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Abstract

The finite difference method based on a non-regular mesh is used to solve Hartree–Fock and Kohn–Sham equations for the helium atom confined by finite and infinite potentials. The reliability of this approach is shown when this is contrasted with the Roothaan’s approach, which depends on a basis set and therefore its exponents must be optimized for each confinement imposed over the helium atom. The comparison between our numerical approach and the Roothaan’s approach was done by using total and orbitals energies from the Hartree–Fock method where there are several sources of comparison. By the side of the Kohn–Sham method there are a few published results and consequently the results reported here can be used as benchmark for future comparisons. The electron density, through the Shannon’s entropy, was also used for the comparison between our approach and other reports. This entropy shows that the helium atom confined by an infinite potential can be described almost with any approach, Hartree–Fock or Kohn–Sham give almost same results. This conclusion cannot be applied for finite potential since Hartree–Fock and Kohn–Sham methods present large differences between themselves. This study represents the first step to develop a numerical code free of basis sets to obtain the electronic structure of many-electron atoms.

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Data Availibility Statement

This manuscript has no associated data or the data will not be deposited [Authors’ comment: All data generated or analysed during this study are included in this published article.]

References

  1. K.D. Sen (ed.), Electronic Structure of Quantum Confined Atoms and Molecules (Springer, 2014)

  2. E. Ley-Koo, Recent progress in confined atoms and molecules: superintegrability and symmetry breakings. Rev. Mex. Fis. 64, 326–363 (2018)

    Article  MathSciNet  Google Scholar 

  3. A. Michels, J. De Boer, A. Bijl, Remarks concerning molecular interaction and their influence on the polarisability. Physica 4, 981–994 (1937)

    Article  ADS  Google Scholar 

  4. S.R. de Groot, C.A. ten Seldam, On the energy levels of a model of the compressed hydrogen atom. Physica 12, 669–682 (1946)

    Article  ADS  Google Scholar 

  5. E. Ley-Koo, S. Rubinstein, The hydrogen atom within spherical boxes with penetrable walls. J. Chem. Phys. 71, 351–357 (1979)

    Article  ADS  Google Scholar 

  6. E.V. Ludeña, SCF calculations for hydrogen in a spherical box. J. Chem. Phys. 66, 468–470 (1977)

    Article  ADS  Google Scholar 

  7. N. Aquino, Accurate energy eigenvalues for enclosed hydrogen-atom within spherical impenetrable boxes. Int. J. Quantum Chem. 54, 107–115 (1995)

    Article  Google Scholar 

  8. E.V. Ludeña, SCF Hartree–Fock calculations of ground-state wavefunctions of compressed atoms. J. Chem. Phys. 69, 1770–1775 (1978)

    Article  ADS  Google Scholar 

  9. M. van Faassen, Atoms in boxes: from confined atoms to electron-atom scattering. J. Chem. Phys. 131, 1 (2009)

    Google Scholar 

  10. J.P. Connerade, V. Dolmatov, Controlling orbital collapse from inside and outside a transition element. J. Phys. B: At. Mol. Opt. Phys. 31, 3557–3564 (1998)

    Article  ADS  Google Scholar 

  11. J. Garza, R. Vargas, A. Vela, Numerical self-consistent-field method to solve the Kohn–Sham equations in confined many-electron atoms. Phys. Rev. E 58, 3949–3954 (1998)

    Article  ADS  Google Scholar 

  12. A. Sarsa, C. Le Sech, Variational Monte Carlo method with Dirichlet boundary conditions: application to the study of confined systems by impenetrable surfaces with different symmetries. J. Chem. Theory Comput. 7, 2786–2794 (2011)

    Article  Google Scholar 

  13. J. Garza, J.-M. Hernández-Pérez, J.-Z. Ramírez, R. Vargas, Basis set effects on the Hartree–Fock description of confined many-electron atoms. J. Phys. B: At. Mol. Opt. Phys. 45, 015002 (2012)

    Article  ADS  Google Scholar 

  14. E. García-Hernández, C. Díaz-García, R. Vargas, J. Garza, Four-index integral transformation in many-body perturbation theory and electron propagator to second order on GPUS for confined atoms. AIP Conf. Proc. 1558, 1528 (2013)

    Article  ADS  Google Scholar 

  15. M. Rodriguez-Bautista, C. Díaz-García, A.M. Navarrete-López, R. Vargas, J. Garza, Roothaan’s approach to solve the Hartree–Fock equations for atoms confined by soft walls: basis set with correct asymptotic behavior. J. Chem. Phys. 143, 34103 (2015)

    Article  Google Scholar 

  16. J. Garza, R. Vargas, Comparative study between the Hartree–Fock and Kohn–Sham models for the lowest singlet and triplet states of the confined helium atom. Adv. Quantum Chem. 57, 241–254 (2009)

    Article  ADS  Google Scholar 

  17. T.D. Young, R. Vargas, J. Garza, A Hartree–Fock study of the confined helium atom: local and global basis set approaches. Phys. Lett. A 380, 712–717 (2016)

    Article  ADS  Google Scholar 

  18. R. Cabrera-Trujillo, S.A. Cruz, Confinement approach to pressure effects on the dipole and the generalized oscillator strength of atomic hydrogen. Phys. Rev. A 87, 012502 (2013)

    Article  ADS  Google Scholar 

  19. R. Cabrera-Trujillo, Sum rules and the role of pressure on the excitation spectrum of a confined hydrogen atom by a spherical cavity. J. Phys. B: At. Mol. Opt. Phys. 50, 155006 (2017)

    Article  ADS  Google Scholar 

  20. C. Martínez-Flores, R. Cabrera-Trujillo, High pressure effects on the excitation spectra and dipole properties of Li, Be\(^+\), and B\(^{2+}\) atoms under confinement. Matt. Rad. Ext. 5, 024401 (2020)

    Google Scholar 

  21. M.-A. Martínez-Sánchez, R. Vargas, J. Garza, in Asymptotic Behavior: An Overview. Chapter Asymptotic Behavior for the Hydrogen Atom Confined by Different Potentials. (NOVA Science Publishers, New York, 2020), pp. 101–132

  22. J.L. Marin, S.A. Cruz, Use of the direct variational method for the study of one- and two-electron atomic systems confined by spherical penetrable boxes. J. Phys. B At. Mol. Opt. Phys. 25, 4365 (1992)

    Article  ADS  Google Scholar 

  23. M. Rodriguez-Bautista, R. Vargas, N. Aquino, J. Garza, Electron-density delocalization in many-electron atoms confined by penetrable walls: a Hartree–Fock study. Int. J. Quantum Chem. 118, e25571 (2018)

    Article  Google Scholar 

  24. F.-A. Duarte-Alcaráz, M.-A. Martínez-Sánchez, M. Rivera-Almazo, R. Vargas, R.-A. Rosas-Burgos, J. Garza, Testing one-parameter hybrid exchange functionals in confined atomic systems. J. Phys. B: At. Mol. Opt. Phys. 52, 135002 (2019)

  25. M.-A. Martínez-Sánchez, M. Rodriguez-Bautista, R. Vargas, J. Garza, Solution of the Kohn–Sham equations for many-electron atoms confined by penetrable walls. Theor. Chem. Acc. 135, 207 (2016)

    Article  Google Scholar 

  26. S. Majumdar, A.K. Roy, Shannon entropy in confined He-like ions within a density functional formalism. Quantum Rep. 2, 189–207 (2020)

    Article  Google Scholar 

  27. C. Aslangul, R. Constanciel, R. Daudel, P. Kottis, Aspects of the localizability of electrons in atoms and molecules: Loge theory and related methods. Adv. Quantum Chem. 6, 93–141 (1972)

    Article  ADS  Google Scholar 

  28. K.D. Sen, F. De Proft, A. Borgoo, P. Geerlings, N-derivative of Shannon entropy of shape function for atoms. Chem. Phys. Lett. 410, 70–76 (2005)

    Article  ADS  Google Scholar 

  29. K.Ch. Chatzisavvas, Ch.C. Moustakidis, C.P. Panos, Information entropy, information distances, and complexity in atoms. J. Chem. Phys. 123, 174111 (2005)

  30. K.D. Sen, Characteristic features of Shannon information entropy of confined atoms. J. Chem. Phys. 123, 074110 (2005)

    Article  ADS  Google Scholar 

  31. N. Aquino, A. Flores-Riveros, J.F. Rivas-Silva, Shannon and Fisher entropies for a hydrogen atom under soft spherical confinement. Phys. Lett. A 377, 2062–2068 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. 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 

  33. S.-B. Liu, Information-theoretic approach in density functional reactivity theory. Acta. Phys. Chim. Sin. 32, 98 (2016)

    Article  Google Scholar 

  34. L.R. Zan, L.G. Jiao, J. Ma, Y.K. Ho, Information-theoretic measures of hydrogen-like ions in weakly coupled debye plasmas. Phys. Plasmas 24, 122101 (2017)

    Article  ADS  Google Scholar 

  35. J.-H. Ou, Y.K. Ho, Shannon information entropy in position space for the ground and singly excited states of helium with finite confinements. Atoms 5, 66 (2017)

    Article  Google Scholar 

  36. L.G. Jiao, L.R. Zan, Y.Z. Zhang, Y.K. Ho, Benchmark values of Shannon entropy for spherically confined hydrogen atom. Int. J. Quantum Chem. 117, e25375 (2017)

    Article  Google Scholar 

  37. M. Alipour, Z. Badooei, Toward electron correlation and electronic properties from the perspective of information functional theory. J. Phys. Chem. A 122, 6424–6437 (2018)

    Article  Google Scholar 

  38. G.A. Sekh, A. Saha, B. Talukdar, Shannon entropies and Fisher information of K-shell electrons of neutral atoms. Phys. Lett. A 382, 315–320 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. N. Mukherjee, A.K. Roy, Information-entropic measures in free and confined hydrogen atom. Int. J. Quantum Chem. 118, e25596 (2018)

    Article  Google Scholar 

  40. J.-H. Ou, Y.K. Ho, Shannon, Rényi, Tsallis entropies and Onicescu Information energy for low-lying singly excited states of helium. Atoms 7, 66 (2019)

    Article  ADS  Google Scholar 

  41. S. Majumdar, N. Mukherjee, A.K. Roy, Information entropy and complexity measure in generalized Kratzer potential. Chem. Phys. Lett. 716, 257–264 (2019)

    Article  ADS  Google Scholar 

  42. M.-A. Martínez-Sánchez, R. Vargas, J. Garza, Shannon entropy for the hydrogen atom confined by four different potentials. Quantum Rep. 1, 208–218 (2019)

  43. J.-H. Ou, Y.K. Ho, Benchmark calculations of Rényi, Tsallis entropies, and Onicescu information energy for ground state helium using correlated Hylleraas wave functions. Int. J. Quantum Chem. 119, e25928 (2019)

  44. S.J.C. Salazar, H.G. Laguna, V. Prasad, R.P. Sagar, Shannon-information entropy sum in the confined hydrogenic atom. Int. J. Quantum Chem. 120, e26188 (2020)

    Article  Google Scholar 

  45. I. Nasser, A. Abdel-Hady, Fisher information and Shannon entropy calculations for two-electron systems. C. J. Phys. 98, 784–789 (2020)

    Article  ADS  Google Scholar 

  46. I. Nasser, M. Zeama, A. Abdel-Hady, Rényi, Fisher, Shannon, and their electron correlation tools for two-electron series. Phys. Scr. 95, 095401 (2020)

    Article  ADS  Google Scholar 

  47. S. Saha, J. Jose, Shannon entropy as a predictor of avoided crossing in confined atoms. Int. J. Quantum Chem. e26374 (2020)

  48. C.R. Estañón, N. Aquino, D. Puertas-Centeno, J.S. Dehesa, Two-dimensional confined hydrogen: an entropy and complexity approach. Int. J. Quantum Chem. 120, e26192 (2020)

    Article  Google Scholar 

  49. S. Hazra, A comparative study on information theoretic approach for atomic and molecular systems. Comput. Theor. Chem. 1179, 112801 (2020)

    Article  Google Scholar 

  50. C.E. Shannon, A mathematical theory of communication. Bell Syst. Technol. 27, 379–423 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  51. C.F. Matta, L. Massa, A.V. Gubskaya, E. Knoll, Can one take the logarithm or the sine of a dimensioned quantity or a unit? Dimensional analysis involving transcendental functions. J. Chem. Edu. 88, 67–70 (2011)

    Article  Google Scholar 

  52. C.F. Matta, M. Sichinga, P.W. Ayers, Information theoretic properties from the quantum theory of atoms in molecules. Chem. Phys. Lett. 514, 379–383 (2011)

    Article  ADS  Google Scholar 

  53. N. Flores-Gallegos, A new approach of Shannon’s entropy in atoms. Chem. Phys. Lett. 650, 57–59 (2016)

    Article  ADS  Google Scholar 

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Acknowledgements

C.M-.F., M.-A.M-.S. and J. G. thank CONACYT for the postdoctoral fellowship, Ph.D. scholarship (574390) and the project FC-2016/2412.

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Authors and Affiliations

  1. Departamento de Química, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Col. Vicentina, Iztapalapa, C. P. 09340, México D.F., Mexico

    César Martínez-Flores, Michael-Adán Martínez-Sánchez, Rubicelia Vargas & Jorge Garza

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  1. César Martínez-Flores
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  2. Michael-Adán Martínez-Sánchez
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  3. Rubicelia Vargas
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  4. Jorge Garza
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All the authors were involved in the preparation of the manuscript. All the authors have read and approved the final manuscript.

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Correspondence to Jorge Garza.

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Martínez-Flores, C., Martínez-Sánchez, MA., Vargas, R. et al. Free-basis-set method to describe the helium atom confined by a spherical box with finite and infinite potentials. Eur. Phys. J. D 75, 100 (2021). https://doi.org/10.1140/epjd/s10053-021-00110-x

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