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A numerical method for the Hartree equation of the Helium atom

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Abstract

We describe in this paper a numerical method for computing the normalized pointwise positive solution of the Hartree equation for the Helium atom. The method consists of minimizing the Hartree energy by a decomposition coordination method via an augmented Lagrangian. Some numerical results are presented.

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References

  1. P. Bader,Variational method for the Hartree equation of the helium atom, Proc. Roy Soc. Edinburgh Sect. A 82 (1978), 27–39.

    MathSciNet  MATH  Google Scholar 

  2. N. Bazley, M. Reeken, B. Zwahlen,Global properties of the minimal branch of a class of nonlinear variational problems, Math. Z. 123 (1971), 301–309.

    Article  MATH  MathSciNet  Google Scholar 

  3. N. Bazley, R. Seydel,Existence and bounds for critical energies of the Hartree operator, Chem. Phys. Lett. 24 (1974), 128–132.

    Article  Google Scholar 

  4. N. Bazley, B. Zwahlen,A branch of positive solutions of nonlinear eigenvalue problems, Manuscripta Math. 2 (1970), 365–374.

    Article  MATH  MathSciNet  Google Scholar 

  5. V. Benci, D. Fortunato, F. Zirilli,Exponential decay and regularity properties of the Hartree approximation to the bound state wave functions of the helium atom J. Math. Phys. 17 (1976), 1154–1155.

    Article  MathSciNet  Google Scholar 

  6. A. Bove, G. Da Prato, G. Fano,An existence proof for the Hartree-Fock time-dependent problem with bounded two-body interaction, Comm. Math. Phys. 37 (1974), 183–192.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. M. Chadam, R. T. Glassey,Global existence of solutions to the Cauchy problem for time-dependent Hartree equations, J. Math. Phys. 16 (1975), 1122–1130.

    Article  MATH  MathSciNet  Google Scholar 

  8. L. De Loura,Résolution numérique de l'équation de Hartree (1983), Thèse de troisième cycle à l'Université de Paris 6.

  9. V. Fock,Näherungsmethode zur Lösing des quantenmechanischen Mehrkörperproblems, Z. Phys. 61 (1930), 126–148.

    Article  Google Scholar 

  10. G. Fonte, R. Mignani, G. Schiffrer,Solution of the Hartree-Fock equations, Comm. Math. Phys. 33 (1973), 293–304.

    Article  MathSciNet  Google Scholar 

  11. M. Fortin, R. Glowinski,Augmented Lagrangian methods. Applications to the numerical solution of boundary value problems (1983), North-Holland, Amsterdam.

    MATH  Google Scholar 

  12. R. Glowinski,Numerical methods for nonlinear variational problems, (1984), Springer-Verlag, Berlin.

    MATH  Google Scholar 

  13. R. Glowinski, A. Marrocco,Sur l'approximation, par éléments finis d'ordre un et la résolution, par pénalisation-dualité, d'une classe de problèmes de Dirichlet non linéaires, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal Numér. (9) R-2 (1975), 41–76.

    MathSciNet  Google Scholar 

  14. K. Gustafson, D. Sather,Branching analysis of the Hartree equations, Rend. Mat. (6) 4 (1971), 723–734.

    MathSciNet  Google Scholar 

  15. D. R. Hartree,The wave mechanics of a atom with a non-coulomb central field. Part. I. Theory and methods, Proceedings of the Cambridge Philosophical Society 24 (1928), 89–110 Part. II. Some results and discussions, Proceedings of the Cambridge Philosophical Society 24 (1928), 111–132 Part. III. Term values and intensities in series in optical spectra, Proceedings of the Cambridge Philosophical Society 24 (1928), 426–437 Part. IV. Further results relating to terms of the optical spectrum, Proceedings of the Cambridge Philosophical Society 25 (1929), 310–314.

    Google Scholar 

  16. D. R. Hartree,The calculation of atomic structures (1957),John Wiley & Sons,New York.

    MATH  Google Scholar 

  17. O. A. Ladyzhenskaya,The mathematical theory of viscous incompressible flow (1969), Gordon and Breach, New York.

    MATH  Google Scholar 

  18. E. H. Lieb,Thomas-Fermi and Hartree-Fock theory, Proc. 1974 Inter. Congress Math. 2 (1974), 383–386.

    Google Scholar 

  19. E. H. Lieb, B. Simon,On solutions to the Hartree-Fock problem for atoms and molecules, J. Chem. Phys. 61 (1974), 735–736.

    Article  MathSciNet  Google Scholar 

  20. E. H. Lieb, B. Simon,The Hartree-Fock theory for coulomb systems, Comm. Math. Phys. 53 (1977), 185–194.

    Article  MathSciNet  Google Scholar 

  21. P. L. Lions,Some remarks on Hartree equation, Publications du Laboratoire d'Analyse Numérique de l'Université Pierre et Marie Curie 80018 (1980), 1–19.

  22. M. Reeken,General theorem on bifurcation and its application to the Hartree equation of the helium atom, J. Math. Phys. 11 (1970), 2505–2512.

    Article  MathSciNet  Google Scholar 

  23. J. C. Slater,A note on Hartree's method, Phys. Rev. (2) 35 (1930), 210–211.

    Article  Google Scholar 

  24. C. A. Stuart,Existence theory for the Hartree equation, Arch. Rational Mech. Anal. 51 (1973), 60–69.

    Article  MATH  MathSciNet  Google Scholar 

  25. C. A. Stuart,An exemple in nonlinear Functional Analysis: the Hartree equation, J. Math. Anal. Appl. 49 (1975), 725–733.

    Article  MATH  MathSciNet  Google Scholar 

  26. J. Wolkowisky,Existence of solutions of the Hartree equations for N electrons. An application of the Schauder-Tychonoff theorem, Indiana Univ. Math. J. 22 (1972), 551–568.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Université Pierre et Marie Curie, L. A. 189, Paris, France

    L. De Loura

  2. Dep. Matemática, Instituto Superior Técnico, Lisboa, Portugal

    L. De Loura

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  1. L. De Loura
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De Loura, L. A numerical method for the Hartree equation of the Helium atom. Calcolo 23, 185–207 (1986). https://doi.org/10.1007/BF02576528

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  • DOI: https://doi.org/10.1007/BF02576528

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