Abstract
For neutral atoms and molecules and positive ions and radicals, we prove the existence of solutions of the Hartree-Fock equations which minimize the Hartree-Fock energy. We establish some properties of the solutions including exponential falloff.
Research partially supported by U.S. National Science Foundation Grant MCS-75–21684
Research partially supported by U.S. National Science Foundation under Grants MPS-75–11864 and MPS-75–20638. On leave from Departments of Mathematics and Physics, Princeton University, Princeton, NJ 08540, USA
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adams, R.: Sobolev spaces. New York: Academic Press 1976
Bader, P. : Méthode variationelle pour l’équation de Hartree. E.P.F. Lausanne Thesis
Bazley, N., Seydel, R. : Existence and bounds for critical energies of the Hartree operator. Chem. Phys. Letters 24, 128–132 (1974)
Behling, R., Bongers, A., Kuper, T. : Upper and lower bounds to critical values of the Hartree operator. University of Köln (preprint)
Benci, V., Fortunato, D., Zirilli, F.: Exponential decay and regularity properties of the Hartree approximation to the bound state wavefunctions of the helium atom. J. Math. Phys. 17, 1154–1155 (1976)
Bethe, H., Jackiw, R.: Intermediate quantum mechanics. New York: Benjamin 1969
Bove, A., DaPrato, G., Fano, G. : An existence proof for the Hartree-Fock time dependent problem with bounded two-body interaction. Commun, math. Phys. 37, 183–192 (1974)
Chadam, J.M., Glassey, R.T.: Global existence of solutions to the Cauchy problem for time-dependent Hartree equations. J. Math. Phys. 16, 1122–1130 (1975)
Fermi, E.: Un metodo statistico per la determinazione di alcune priorietà dell atome. Rend. Acad. Nat. Lincei 6, 602–607 (1927)
Fock, V. : Näherungsmethode zur Losing des quantenmechanischen Mehrkörperproblems. Z. Phys. 61, 126–148 (1930)
Fonte, G., Mignani, R., Schiffrer, G. : Solution of the Hartree-Fock equations. Commun, math. Phys. 33, 293–304 (1973)
Gustafson, K., Sather, D. : Branching analysis of the Hartree equations. Rend, di Mat. 4, 723–734 (1971)
Handy, N.C, Marron, M.T., Silverstom, H. J. : Long range behavior of Hartree-Fock orbitals. Phys. Rev. 180, 45–47 (1969)
Hartree, D. : The wave mechanics of an atom with a non-coulomb central field. Part I. Theory and methods. Proc. Comb. Phil. Soc. 24, 89–132 (1928)
Jensen, R. : Princeton University Senior Thesis, 1976
Kato, T. : Fundamental properties of Hamiltonian operator of Schrödinger type. Trans. Am. Math. Soc. 70, 195–211 (1951)
Kato, T. : On the eigen functions of many particle systems in quantum mechanics. Comm. Pure Appl. Math. 10, 151–177 (1957)
Kato, T. : Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer 1966
Lieb, E.H.: Thomas-Fermi and Hartree-Fock theory, Proc. 1974 International Congress Mathematicians, Vol. II, pp. 383–386
Lieb, E.H.: The stability of matter. Rev. Mod. Phys. 48, 553–569 (1976)
Lieb, E.H.: Existence and uniqueness of minimizing solutions of Choquard’s non-linear equation. Stud. Appl. Math, (in press)
Lieb, E.H., Simon, B. : On solutions to the Hartree Fock problem for atoms and molecules. J. Chem. Phys. 61, 735–736 (1974)
Lieb, E.H., Simon, B.: The Thomas-Fermi theory of atoms, molecules, and solids. Adv. Math. 23, 22–116(1977)
Reed, M., Simon, B. : Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. New York: Academic Press 1975
Reed, M., Simon, B. : Methods of modern mathematical physics. IV. Analysis of operators. New York: Academic Press 1977
Reeken, M. : General theorem on bifurcation and its application to the Hartree equation of the helium atom. J. Math. Phys. 11, 2505–2512 (1970)
Schaefer III H.F. : The electronic structure of atoms and molecules. Reading: Addison Wesley 1972
Simon, B. : On the infinitude vs. finiteness of the number of bound states of an N-body quantum system. Helv. Phys. Acta 43, 607–630 (1970)
Simon, B. : Pointwise bounds on eigen functions and wave packets in N-body quantum systems. I. Proc. Am. Math. Soc. 42, 395–401 (1974)
Slater, J.C: A note on Hartree’s method. Phys. Rev. 35, 210–211 (1930)
Stein, E. : Singular integrals and differentiability properties of functions. Princeton : University Press 1970
Stuart, C. : Existence theory for the Hartree equation. Arch. Rat. Mech. Anal. 51, 60–69 (1973)
Thomas, L.H.: The calculation of atomic fields. Proc. Comb. Phil. Soc. 23, 542–548 (1927)
Wolkowisky, J.: Existence of solutions of the Hartree equations for N electrons. An application of the Schauder-Tychonoff theorem. Ind. Univ. Math. Journ. 22, 551–558 (1972)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lieb, E.H., Simon, B. (2001). The Hartree-Fock Theory for Coulomb Systems. In: Thirring, W. (eds) The Stability of Matter: From Atoms to Stars. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04360-8_21
Download citation
DOI: https://doi.org/10.1007/978-3-662-04360-8_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-04362-2
Online ISBN: 978-3-662-04360-8
eBook Packages: Springer Book Archive