Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T09:19:01.267Z Has data issue: false hasContentIssue false

Variational method for the Hartree equation of the helium atom

Published online by Cambridge University Press:  14 November 2011

Peter Bader
Affiliation:
Mathematics Research Center, University of Wisconsin-Madison, Madison, U.S.A.

Synopsis

It is shown that in the Hartree approximation the energy functional of the helium atom reaches its minimum and that the corresponding minimizing function is a solution of the Hartree equation.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Agmon, S.. Lectures on elliptic boundary value problems (New York: Van Nostrand, 1965).Google Scholar
2Bader, P.. Méthode variationnelle pour l'équation de Hartree (Ecole Polytechnique Fédérale de Lausanne Ph.D. Thesis, 1976).Google Scholar
3Bazley, N.. Lower bounds for eigenvalues with application to the helium atom. Phys. Rev. 120 (1960), 144149.CrossRefGoogle Scholar
4Bazley, N. and Zwahlen, B.. A branch of positive solutions of nonlinear eigenvalue problems. Manuscripta Math. 2 (1970), 365374.CrossRefGoogle Scholar
5Bazley, N., Reeken, M. and Zwahlen, B.. Global properties of the minimal branch of a class of nonlinear variational problems. Math. Z. 123 (1971), 301309.CrossRefGoogle Scholar
6Bazley, N. and Seydel, R.. Existence and bounds for critical energies of the Hartree operator. Chem. Phys. Lett. 24 (1974), 128132.Google Scholar
7Behling, R., Bongers, A. and Küpper, T.. Upper and lower bounds to critical values of the Hartree operator. Intemat. J. Quantum Chem. 10 (1976), 985992.CrossRefGoogle Scholar
8Fock, V.. Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems. Z. Phys. 61 (1930), 126148.Google Scholar
9Gustafson, K.. Stability inequalities for semimonotonically perturbed nonhomogeneous boundary problems. SIAM J. Appl. Math. 15 (1967), 368391.CrossRefGoogle Scholar
See also: A priori bounds with applications to integro-dijferential boundary problems (Univ. of Maryland Ph.D. Thesis, 1965).Google Scholar
10Gustafson, K. and Sather, D.. A branching analysis of the Hartree equation. Rend. Mat. 4 (1971), 723734.Google Scholar
11Hartree, D. R.. The wave mechanics of an atom with a non-Coulomb central field, I, II, III, Proc. Cambridge Philos. Soc. 24 (1928), 89-110, 111-132, 426437.CrossRefGoogle Scholar
12Kato, T.. Perturbation theory for linear operators (Berlin: Springer, 1966).Google Scholar
13Ladyzhenskaya, O. A.. The mathematical theory of viscous incompressible flow, 2nd edn (New York: Gordon and Breach, 1969).Google Scholar
14Lieb, E. H.. Thomas-Fermi and Hartree-Fock theory. Proc. Intemat. Congr. Math., Vancouver 2 (1974), 383.Google Scholar
15Lieb, E. H.. Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Studies in Appl. Math. 57 (1977), 93105.CrossRefGoogle Scholar
16Lieb, E. H. and Simon, B.. On solutions to the Hartree-Fock problem for atom^ and molecules. J. Chem. Phys. 61 (1974), 735736.Google Scholar
17Lieb, E. H. and Simon, B.. The Hartree-Fock theory for Coulomb systems. Comm. Math. Phys. 53 (1977), 185194.Google Scholar
18Reeken, M.. General theorem on bifurcation and its application to the Hartree equation of the helium atom. J. Math. Phys. 11 (1970), 25052512.CrossRefGoogle Scholar
19Stuart, C. A.. Existence theory for the Hartree equation. Arch. Rational Mech. Anal. 51 (1973), 6069.CrossRefGoogle Scholar
20Wolkowisky, J. H.. Existence of solutions of the Hartree equations for N electrons. An application of the Schauder-Tychonoff theorem. Indiana Univ. Math. J. 22 (1972), 551568.CrossRefGoogle Scholar
21Yosida, K.. Functional analysis, 2nd edn (Berlin: Springer, 1968).Google Scholar