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Upper and lower bounds to atomic radial position moments

  • M. G. MarmorinoEmail author
Original Paper
  • 15 Downloads

Abstract

A procedure is developed to generate rigorous rough upper bounds to radial moments [r2n] with integer n ≥ − 2 for the ground and excited states of atoms and molecules. These rough upper bounds to [r2n] enable the calculation of accurate upper and lower bounds to the lesser moment [rn] through existing and new formulas that utilize a lower bound to the square overlap of a trial function and true wave function (in this case through an energy lower bound via the Eckart formula). Error bars to [r−1], [r], [r2], and [r4] are calculated for the ground state of the non-relativistic infinite nuclear mass helium atom to yield expectation values within 0.037%, 0.018%, 0.039%, and 0.24% respectively, of the true values. As a byproduct of this investigation, a new formula for the error bar to observables is derived which is a slight improvement upon similar error bars. Also Lieb’s limit on the number of electrons that an atom can bind is reproduced.

Keywords

Upper bounds Lower bounds Position moments Radial moments Helium atom 

Mathematics Subject Classification

35P15 49J40 

Notes

References

  1. 1.
    A. Lüchow, H. Kleindienst, Int. J. Quantum Chem. 51, 211 (1994)CrossRefGoogle Scholar
  2. 2.
    F.W. King, J. Phys. Chem. 102, 8053 (1995)CrossRefGoogle Scholar
  3. 3.
    I. Porras, D.M. Feldmann, F.W. King, Int. J. Quantum Chem. 71, 455 (1999)CrossRefGoogle Scholar
  4. 4.
    N.W. Bazley, D.W. Fox, J. Math. Phys. 7, 413 (1966)CrossRefGoogle Scholar
  5. 5.
    F. Weinhold, J. Math. Phys. 11(7), 2127 (1970)CrossRefGoogle Scholar
  6. 6.
    A. Mazziotti, R.G. Parr, J. Chem. Phys. 52, 1605 (1970)CrossRefGoogle Scholar
  7. 7.
    J. Killingbeck, Phys. Lett. 65A, 84 (1978)Google Scholar
  8. 8.
    J.P. Killingbeck, J. Phys. A Math. Gen. 34, 8309 (2001)CrossRefGoogle Scholar
  9. 9.
    C.S. Lai, J. Phys. A Math. Gen. 16, L181 (1983)CrossRefGoogle Scholar
  10. 10.
    C.R. Handy, Phys. Rev. A 46, 1663 (1992)CrossRefGoogle Scholar
  11. 11.
    D. Bessis, C.R. Handy, Numer. Algorithms 3, 1 (1992)CrossRefGoogle Scholar
  12. 12.
    F.M. Fernandez, J.F. Ogilvie, R.H. Tipping, Phys. Lett. A 116, 407 (1986)CrossRefGoogle Scholar
  13. 13.
    A. Czarnecki, Y. Liang, Phys. Rev. A 91, 012514 (2015)CrossRefGoogle Scholar
  14. 14.
    P. Jennings, E.B. Wilson Jr., J. Chem. Phys. 45, 1847 (1966)CrossRefGoogle Scholar
  15. 15.
    F. Weinhold, Adv. Quantum Chem. 6, 299 (1972)CrossRefGoogle Scholar
  16. 16.
    R. Blau, A.R.P. Rau, L. Spruch, Phys. Rev. A 8, 119 (1973)CrossRefGoogle Scholar
  17. 17.
    M.G. Marmorino, J. Math. Chem. 104, 880 (2005)Google Scholar
  18. 18.
    E.H. Lieb, Phys. Rev. A 29, 3018 (1983)CrossRefGoogle Scholar
  19. 19.
    A.G. Donchev, S.A. Kalachev, N.N. Koesnikov, V.I. Tarasov, Phys. Part. Nucl. Lett. 4(1), 39 (2007)CrossRefGoogle Scholar
  20. 20.
    W.H. Miller, J. Chem. Phys. 50, 4305 (1969)CrossRefGoogle Scholar
  21. 21.
    E. Pollak, J. Chem. Theory Comput. 15(7), 4079 (2019)CrossRefGoogle Scholar
  22. 22.
    E.A. Hylleraas, Adv. Quantum Chem. 1, 1–33 (1964)CrossRefGoogle Scholar
  23. 23.
    C. Eckart, Phys. Rev. 36, 878 (1930)CrossRefGoogle Scholar
  24. 24.
    N.W. Bazley, Phys. Rev. 120, 144 (1960)CrossRefGoogle Scholar
  25. 25.
    G. Temple, Proc. R. Soc. A (Lond.) 119, 276 (1928)CrossRefGoogle Scholar
  26. 26.
    R. Blau, Phys. Rev. A 14, 890 (1976)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Chemistry and BiochemistryIndiana University South BendSouth BendUSA

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