Natural orbital description of the halo nucleus 6He

  • Chrysovalantis Constantinou
  • Mark A. Caprio
  • James P. Vary
  • Pieter Maris


Ab initio calculations of nuclei face the challenge of simultaneously describing the strong short-range internucleon correlations and the long-range properties of weakly bound halo nucleons. Natural orbitals, which diagonalize the one-body density matrix, provide a basis which is better matched to the physical structure of the many-body wave function. We demonstrate that the use of natural orbitals significantly improves convergence for ab initio no-core configuration interaction calculations of the neutron halo nucleus 6He, relative to the traditional oscillator basis.


Neutron halo nucleus 6He Nuclear structure Nuclear theory 



We thank G. Hupin for valuable discussions on the formulation of the nuclear natural orbital problem and M. A. McNanna for carrying out informative preliminary studies in one dimension.


  1. 1.
    S.C. Pieper, R.B. Wiringa, J. Carlson, Quantum Monte Carlo calculations of excited states in A = 6–8 nuclei. Phys. Rev. C 70, 054325 (2004). CrossRefGoogle Scholar
  2. 2.
    T. Neff, H. Feldmeier, Cluster structures within fermionic molecular dynamics. Nucl. Phys. A 738, 357–361 (2004). CrossRefGoogle Scholar
  3. 3.
    G. Hagen, D.J. Dean, M. Hjorth-Jensen et al., Benchmark calculations for \(^3\)H, \(^4\)He, \(^{16}\)O, and \(^{40}\)Ca with ab initio coupled-cluster theory. Phys. Rev. C 76, 044305 (2007). CrossRefGoogle Scholar
  4. 4.
    S. Quaglioni, P. Navrátil, Ab initio many-body calculations of nucleon–nucleus scattering. Phys. Rev. C 79, 044606 (2009). CrossRefGoogle Scholar
  5. 5.
    S. Bacca, N. Barnea, A. Schwenk, Matter and charge radius of \(^6\)He in the hyperspherical-harmonics approach. Phys. Rev. C 86, 034321 (2012). CrossRefGoogle Scholar
  6. 6.
    N. Shimizu, T. Abe, Y. Tsunoda et al., New-generation Monte Carlo shell model for the K computer era. Prog. Theor. Exp. Phys. 2012, 01A205 (2012). CrossRefGoogle Scholar
  7. 7.
    T. Dytrych, K.D. Launey, J.P. Draayer et al., Collective modes in light nuclei from first principles. Phys. Rev. Lett. 111, 252501 (2013). CrossRefGoogle Scholar
  8. 8.
    B.R. Barrett, P. Navrátil, J.P. Vary, Ab initio no core shell model. Prog. Part. Nucl. Phys. 69, 131–181 (2013). CrossRefGoogle Scholar
  9. 9.
    S. Baroni, P. Navrátil, S. Quaglioni, Unified ab initio approach to bound and unbound states: no-core shell model with continuum and its application to \(^{7}\)He. Phys. Rev. C 87, 034326 (2013). CrossRefGoogle Scholar
  10. 10.
    R.B. Wiringa, V.G.J. Stoks, R. Schiavilla, Accurate nucleon–nucleon potential with charge-independence breaking. Phys. Rev. C 51, 38 (1995). CrossRefGoogle Scholar
  11. 11.
    D.R. Entem, R. Machleidt, Accurate charge-dependent nucleon-nucleon potential at fourth order of chiral perturbation theory. Phys. Rev. C 68, 041001(R) (2003). CrossRefGoogle Scholar
  12. 12.
    A.M. Shirokov, J.P. Vary, A.I. Mazur et al., Realistic nuclear Hamiltonian: ab exitu approach. Phys. Lett. B 644, 33–37 (2007). CrossRefGoogle Scholar
  13. 13.
    E. Epelbaum, H.-W. Hammer, U.-G. Meißner, Modern theory of nuclear forces. Rev. Mod. Phys. 81, 1773 (2009). CrossRefGoogle Scholar
  14. 14.
    M. Freer, The clustered nucleus–cluster structures in stable and unstable nuclei. Rep. Prog. Phys. 70, 2149–2210 (2007). CrossRefGoogle Scholar
  15. 15.
    B. Jonson, Light dripline nuclei. Phys. Rep. 389, 1–59 (2004). CrossRefGoogle Scholar
  16. 16.
    I. Tanihata, H. Savajols, R. Kanungo, Recent experimental progress in nuclear halo structure studies. Prog. Part. Nucl. Phys. 68, 215–313 (2013). CrossRefGoogle Scholar
  17. 17.
    P.-O. Löwdin, Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction. Phys. Rev. 97, 1474 (1955). MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    H. Shull, P.-O. Löwdin, Natural spin orbitals for helium. J. Chem. Phys. 23, 1565 (1955). CrossRefGoogle Scholar
  19. 19.
    P.-O. Löwdin, H. Shull, Natural orbitals in the quantum theory of two-electron systems. Phys. Rev. 101, 1730 (1956). CrossRefMATHGoogle Scholar
  20. 20.
    E.R. Davidson, Properties and uses of natural orbitals. Rev. Mod. Phys. 44, 451 (1972). CrossRefGoogle Scholar
  21. 21.
    C. Mahaux, R. Sartor, Single-particle motion in nuclei. Adv. Nucl. Phys. 20, 1–223 (1991). Google Scholar
  22. 22.
    M.V. Stoitsov, A.N. Antonov, S.S. Dimitrova, Natural orbital representation and short-range correlations in nuclei. Phys. Rev. C 48, 74 (1993). CrossRefGoogle Scholar
  23. 23.
    T. Helgaker, P. Jørgensen, J. Olsen, Molecular Electron-Structure Theory (Wiley, Chichester, 2000)Google Scholar
  24. 24.
    A.E. McCoy, M.A. Caprio, Algebraic evaluation of matrix elements in the Laguerre function basis. J. Math. Phys. 57, 021708 (2016). MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    M. Moshinsky, Y.F. Smirnov, The Harmonic Oscillator in Modern Physics (Harwood Academic Publishers, Amsterdam, 1996)MATHGoogle Scholar
  26. 26.
    J. Suhonen, From Nucleons to Nucleus (Springer, Berlin, 2007)CrossRefMATHGoogle Scholar
  27. 27.
    R.J. Furnstahl, G. Hagen, T. Papenbrock, Corrections to nuclear energies and radii in finite oscillator spaces. Phys. Rev. C 86, 031301(R) (2012). CrossRefGoogle Scholar
  28. 28.
    S.N. More, A. Ekström, R.J. Furnstahl et al., Universal properties of infrared oscillator basis extrapolations. Phys. Rev. C 87, 044326 (2013). CrossRefGoogle Scholar
  29. 29.
    P. Maris, J.P. Vary, A.M. Shirokov, Ab initio no-core full configuration calculations of light nuclei. Phys. Rev. C 79, 014308 (2009). CrossRefGoogle Scholar
  30. 30.
    P. Maris, J.P. Vary, Ab initio nuclear structure calculations of p-shell nuclei with JISP16. Int. J. Mod. Phys. E 22, 1330016 (2013). CrossRefGoogle Scholar
  31. 31.
    M.A. Caprio, P. Maris, J.P. Vary et al., Collective rotation from ab initio theory. Int. J. Mod. Phys. E 24, 1541002 (2015). CrossRefGoogle Scholar
  32. 32.
    L. Schäfer, H.A. Weidenmüller, Self-consistency in the application of Brueckner’s method to doubly-closed-shell nuclei. Nucl. Phys. A 174, 1–25 (1971). CrossRefGoogle Scholar
  33. 33.
    M.A. Caprio, P. Maris, J.P. Vary, Coulomb–Sturmian basis for the nuclear many-body problem. Phys. Rev. C 86, 034312 (2012). CrossRefGoogle Scholar
  34. 34.
    G. Hagen, M. Hjorth-Jensen, N. Michel, Gamow shell model and realistic nucleon–nucleon interactions. Phys. Rev. C 73, 064307 (2006). CrossRefGoogle Scholar
  35. 35.
    M.A. Caprio, P. Maris, J.P. Vary, Halo nuclei \(^{6}\)He and \(^{8}\)He with the Coulomb–Sturmian basis. Phys. Rev. C 90, 034305 (2014). CrossRefGoogle Scholar
  36. 36.
    D.R. Tilley, C.M. Cheves, J.L. Godwin et al., Energy levels of light nuclei \(A=5, 6, 7\). Nucl. Phys. A 708, 3–163 (2002). CrossRefGoogle Scholar
  37. 37.
    J.L. Friar, J. Martorell, D.W.L. Sprung, Nuclear sizes and the isotope shift. Phys. Rev. A 56, 4579 (1997). CrossRefGoogle Scholar
  38. 38.
    L.-B. Wang, P. Mueller, K. Bailey et al., Laser spectroscopic determination of the \(^{6}\)He nuclear charge radius. Phys. Rev. Lett. 93, 142501 (2004). CrossRefGoogle Scholar
  39. 39.
    M. Brodeur, T. Brunner, C. Champagne et al., First direct mass measurement of the two-neutron halo nucleus \(^6\)He and improved mass for the four-neutron halo \(^8\)He. Phys. Rev. Lett. 108, 052504 (2012). CrossRefGoogle Scholar
  40. 40.
    Z.-T. Lu, P. Mueller, G.W.F. Drake et al., Colloquium: laser probing of neutron-rich nuclei in light atoms. Rev. Mod. Phys. 85, 1383 (2013). CrossRefGoogle Scholar
  41. 41.
    P. Maris, M. Sosonkina, J.P. Vary, Scaling of ab-initio nuclear physics calculations on multicore computer architectures. Proc. Comput. Sci. 1, 97–106 (2010). CrossRefGoogle Scholar
  42. 42.
    H.M. Aktulga, C. Yang, E.G. Ng et al., Improving the scalability of a symmetric iterative eigensolver for multi-core platforms. Concurr. Comput. Pract. Exp. 26, 2631–2651 (2013). CrossRefGoogle Scholar
  43. 43.
    S.K. Bogner, R.J. Furnstahl, P. Maris et al., Convergence in the no-core shell model with low-momentum two-nucleon interactions. Nucl. Phys. A 801, 21–42 (2008). CrossRefGoogle Scholar
  44. 44.
    S.A. Coon, M.I. Avetian, M.K.G. Kruse et al., Convergence properties of ab initio calculations of light nuclei in a harmonic oscillator basis. Phys. Rev. C 86, 054002 (2012). CrossRefGoogle Scholar
  45. 45.
    R.J. Furnstahl, S.N. More, T. Papenbrock, Systematic expansion for infrared oscillator basis extrapolations. Phys. Rev. C 89, 044301 (2014). CrossRefGoogle Scholar
  46. 46.
    R.J. Furnstahl, G. Hagen, T. Papenbrock et al., Infrared extrapolations for atomic nuclei. J. Phys. G 42, 034032 (2015). CrossRefGoogle Scholar
  47. 47.
    K.A. Wendt, C. Forssén, T. Papenbrock et al., Infrared length scale and extrapolations for the no-core shell model. Phys. Rev. C 91, 061301(R) (2015). CrossRefGoogle Scholar
  48. 48.
    J.P. Vary, P. Maris, E. Ng et al., Ab initio nuclear structure—the large sparse matrix eigenvalue problem. J. Phys. Conf. Ser. 180, 012083 (2009). CrossRefGoogle Scholar
  49. 49.
    C.F. Bender, E.R. Davidson, A natural orbital based energy calculation for helium hydride and lithium hydride. J. Phys. Chem. 70, 2675–2685 (1966). CrossRefGoogle Scholar

Copyright information

© Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Chinese Nuclear Society, Science Press China and Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  • Chrysovalantis Constantinou
    • 1
    • 2
  • Mark A. Caprio
    • 2
  • James P. Vary
    • 3
  • Pieter Maris
    • 3
  1. 1.Center for Theoretical Physics, Sloane Physics LaboratoryYale UniversityNew HavenUSA
  2. 2.Department of PhysicsUniversity of Notre DameNotre DameUSA
  3. 3.Department of Physics and AstronomyIowa State UniversityAmesUSA

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