Advertisement

Positivity

, Volume 12, Issue 4, pp 653–666 | Cite as

Non-existence of a minimizer to the magnetic Hartree-Fock functional

  • Mattias Enstedt
  • Michael Melgaard
Article

Abstract

In the presence of an external magnetic field, we prove absence of a ground state within the Hartree-Fock theory of atoms and molecules. The result is established for a wide class of magnetic fields when the number of electrons is greater than or equal to 2Z + K, where Z is the total charge of K nuclei. Positivity properties are instrumental in the proof of this bound for the maximal ionization.

Keywords

Magnetic Hartree-Fock equations Ionization Positivity 

Mathematics Subject Classification (2000)

Primary 81V45 Secondary 35Q40, 47G20, 47N50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. Baumgartner, On the degree of ionization in the TFW theory, Lett. Math. Phys., 7(5) (1983), 439–441.Google Scholar
  2. 2.
    D. E. Edmunds, W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford University Press, New York (1987).Google Scholar
  3. 3.
    M. Enstedt, M. Melgaard, Existence of solution to Hartree–Fock equations with decreasing magnetic fields (27 pages, 2007), Nonlinear Anal Theory Methods Appl., (2008 in press).Google Scholar
  4. 4.
    V. Fock, Näherungsmethode zur lösung des quantenmechanischen Mehrkörperproblems, Z. Physik, 61 (1930), 126–148.Google Scholar
  5. 5.
    G. Fonte, R. Mignani, G. Schiffrer, Solution of the Hartree–Fock equations. Commun. Math. Phys., 33 (1973), 293–304.Google Scholar
  6. 6.
    K. Gustafson, D. Sather, A branching analysis of the Hartree equation. Rend. Math. 4(6) (1971), 723–734.Google Scholar
  7. 7.
    D. R. Hartree, The wave mechanics of an atom with the non-Coulomb central field, I. Theory and methods, Proc. Camb. Philos. Soc. 24 (1928), 89–132.Google Scholar
  8. 8.
    W. J. Hehre, L. Radom, L., P.v.R. Schleyer, J. A. Pople, Ab initio molecular orbital theory, Wiley, Chichester (1986).Google Scholar
  9. 9.
    H. Leinfelder, C. G. Simader, Schrödinger operators with singular magnetic vector potentials, Math. Z., 176(1) (1981), 1–19.Google Scholar
  10. 10.
    E.H. Lieb, Thomas–Fermi and Hartree-Fock theory, In: Proceedings of the International Congress of Mathematicians (Vancouver, 1974), Vol. 2, pp. 383–386. Canad. Math. Congress, Montreal (1975).Google Scholar
  11. 11.
    E. H. Lieb, Thomas–Fermi and related theories of atoms and molecules, Rev. Mod. Phys., 53 (1981), 603-641 (Errata 54 (1982), 311).Google Scholar
  12. 12.
    E. H. Lieb, Bound on the maximum negative ionization of atoms and molecules, Phys. Rev. A, 29 (1984), 3018–3028.Google Scholar
  13. 13.
    E. H. Lieb, E. H., M. Loss, Analysis. American Mathematical Society, Providence (1997)Google Scholar
  14. 14.
    E. H. Lieb, B. Simon, On solutions to the Hartree–Fock problem for atoms and molecules. J. Chem. Phys., 61 (1974), 735–736.Google Scholar
  15. 15.
    E. H. Lieb, B. Simon, The Hartree–Fock theory for Coulomb systems. Commun. Math. Phys., 53(3) (1977), 185–194.Google Scholar
  16. 16.
    P.-L. Lions, Solutions of Hartree–Fock equations for Coulomb systems. Commun. Math. Phys., 109(1), (1987), 33–97.Google Scholar
  17. 17.
    R. McWeeny, Methods of Molecular Quantum Mechanics, 2nd edn. Academic Press (1992).Google Scholar
  18. 18.
    M. Reed, B Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-adjointness, Academic Press, New York (1975).Google Scholar
  19. 19.
    M. Reeken, General theorem on bifurcation and its application to the Hartree equation of the helium atom. J. Math. Phys. 11 (1970), 2505–2512.Google Scholar
  20. 20.
    M. Reeken, Existence of solutions to the Hartree-Fock equations. In: Eigenvalues of non-linear problems (Centro Internaz. Mat. Estivo C.I. M. E.), III Ciclo, Varenna, 1974, pp. 197–209. Edizioni Cremonese, Rome (1974).Google Scholar
  21. 21.
    J. C. Slater, Note on Hartree’s method, Phys. Rev. 35 (1930), 210–211.Google Scholar
  22. 22.
    J. P. Solovej, The size of atoms in Hartree–Fock theory. In: L. Hörmander et al (eds), Partial differential equations and mathematical physics, Prog. Nonlinear Differ. Equ. Appl. The Danish-Swedish analysis seminar, Proceedings, Birkhäuser, Vol. 21, pp. 321–332 (1995).Google Scholar
  23. 23.
    J. P. Solovej, The ionization conjecture in Hartree–Fock theory. Ann. of Math. 158(2) (2003), 509–576.Google Scholar
  24. 24.
    C. A. Stuart, Existence theory for the Hartree equation. Arch. Ration. Mech. Anal. 51 (1973), 60–69.Google Scholar
  25. 25.
    A. Szabo, N. S. Ostlund, Modern quantum chemistry: an introduction to advanced electronic structure theory, MacMillan, Basingstoke (1982).Google Scholar
  26. 26.
    J. H. Wolkowisky, Existence of solutions of the Hartree equations for N electrons. An application of the Schauder–Tychonoff theorem. Indiana Univ. Math. J. 22 (1972/73), 551–568.Google Scholar
  27. 27.
    E. Zeidler, Applied functional analysis. Main principles and their applications. Applied Mathematical Sciences, Vol. 109. Springer-Verlag, New York (1995).Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden

Personalised recommendations