Journal of Mathematical Chemistry

, Volume 50, Issue 9, pp 2397–2407 | Cite as

Lower bound to the ground-state expectation value of a positive unbounded operator using related bounded operators

  • M. G. Marmorino
  • Abdullah Almayouf
  • Tyler Krause
  • Doan Le
Original Paper


Error bars around observables of quantum mechanical systems are extremely lacking; in most cases only an upper bound to the energy is practical. We present a new lower bound to the expectation value of an operator that is most similar to the lower bound of Weinhold. While Weinhold’s bound has flexibility by incorporating expectation values (some of which may not exist) of different moments of the operator to be bounded, the flexibility of our lower bound relies on the form of a similar, but bounded, operator. Like Weinhold’s bound, ours is limited to non-negative operators and the ground-state of the system. Our lower bound is shown to have properties which allow it to converge to the true expectation value of the ground state, but a practical application to the Helium atom shows that Weinhold’s bound is superior in this case.


Lower bound Expectation value Helium 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Mazziotti A., Parr R.G.: J. Chem. Phys. 52, 1605 (1970)CrossRefGoogle Scholar
  2. 2.
    Temple G.: Proc. R. Soc. Lond. Ser. A 119, 276 (1928)CrossRefGoogle Scholar
  3. 3.
    Bazley N.W.: Phys. Rev. 120, 144 (1960)CrossRefGoogle Scholar
  4. 4.
    Bazley N.W., Fox D.W.: J. Math. Phys. 7, 413 (1966)CrossRefGoogle Scholar
  5. 5.
    Jennings P., Wilson E.B. Jr: J. Chem. Phys. 45, 1847 (1966)CrossRefGoogle Scholar
  6. 6.
    Weinhold F.: Adv. Quant. Chem. 6, 299 (1972)CrossRefGoogle Scholar
  7. 7.
    Eckart C.: Phys. Rev. 36, 878 (1930)CrossRefGoogle Scholar
  8. 8.
    Marmorino M.G., Cassella K.: Int. J. Quantum Chem. 111, 3588 (2011)Google Scholar
  9. 9.
    Weinhold F.: J. Math. Phys. 11, 2127 (1970)CrossRefGoogle Scholar
  10. 10.
    Banerjee A., Sen K.D., Garza J., Vargas R.: J. Chem. Phys. 116, 4054 (2002)CrossRefGoogle Scholar
  11. 11.
    Sen K.D.: J. Chem. Phys. 122, 1943241 (2005)CrossRefGoogle Scholar
  12. 12.
    Wolfram Research, Inc., Mathematica, Version 6.0 (Wolfram Research, Inc., Champaign, IL, 2007)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • M. G. Marmorino
    • 1
  • Abdullah Almayouf
    • 1
  • Tyler Krause
    • 1
  • Doan Le
    • 2
  1. 1.Department of Chemistry and BiochemistryIndiana University South BendSouth BendUSA
  2. 2.Department of Mathematical SciencesIndiana University South BendSouth BendUSA

Personalised recommendations