Abstract
We suggest to use the Newton iteration method for constructing a (locally unique) solution of the atomic and nuclear Hartree-Fock equations for an arbitrary number of particles. Our proposal is based on a theorem by Kantorovič and rests on the following points: 1) the two-body potential must satisfy a boundedness condition; 2) the zero-order approximation, used to start the iteration sequence, must satisfy certain conditions, to be proved numerically. Condition 1) holds, for instance, for all local potentials, defined by a bounded function and for a class of nonlocal potentials; it does not hold for local potentials, behaving as 1/r near the origin.
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References
Hartree, D. R.: Proc. Cambr. Phil. Soc.24, 89 (1927–28); V. Fock, Z. Phys.61, 126 (1930)
Hartree, D. R.: The calculation of atomic structures, New York: J. Wiley 1957
Kerman, A. K.: Proc. of International Conference on properties of nuclear states, ed. by M. Harvey, Presses de l'Université de Montréal, 1969
Moshinsky, M.: Amer. Journ. Phys.36, 52–53 (1968); The harmonic oscillator in modern physics: from atoms to quarks. New York: Gordon & Breach 1969
Reeken, M.: Journ. Math. Phys.11, 2505–2512 (1970)
Catara, F., Di Toro, M., Pace, E., Schiffrer, G.: Nuovo Cim.11A, 733–748 (1972)
Kantorovič, L. V.: Usp. Mat. Nauk3, 89–185 (1948)
Kantorovič, L. V., Akilov, G. P.: Funkcional'nyj analiz v normirovannyh prostranstvah, Moskva: Fizmatgiz 1959; English transl.: Functional analysis in normed spaces, Oxford: Pergamon Press 1964
Schiffrer, G.: Proc. of Symposium on present status and novel developments in the nuclear many-body problem, (Rome, 19–23 sept. 1972), ed. by F. Calogero and C. Ciofi degli Atti, to be published.
Catara, F., Di Toro, M., Lombardo, U.: Lett. Nuovo Cim.4, 849–852 (1972)
Smirnov, V. I.: Kurs vysšej matematiki, tom 5, Moskva: Fizmatgiz 1960; English transl.: A course of higher mathematics, vol. 5, Oxford: Pergamon Press 1964
Lučka, A. Ju.: Teorija i primenenie metoda osrednenija funkcional'nyh popravok, Kiev: AN USSR 1963; English transl.: The method of averaging functional corrections, New York: Academic Press 1965
Rall, L. B.: Computational solution of nonlinear operator equations, New York: J. Wiley 1969
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This work has been supported in part by Istituto Nazionale di Fisica Nucleare (Sezione di Catania) and by Centro Siciliano di Fisica Nucleare e di Struttura della Materia (Catania).
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Fonte, G., Mignani, R. & Schiffrer, G. Solution of the Hartree-Fock equations. Commun.Math. Phys. 33, 293–304 (1973). https://doi.org/10.1007/BF01646742
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DOI: https://doi.org/10.1007/BF01646742