Summary
Molecular Hartree-Fock equations for iteration-variation calculations in momentum space based on trial functions expressed as linear combinations of spherical Gaussian functions are obtained. They are applied to the hydrogen molecule to test their validity. The significant improvements brought by a first iteration are accessed through an asymptotic analysis.
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References
Abramovitz, M., Stegun, I.A. (1968): Handbook of mathematical functions. Dover, New York
Ahlrichs, R., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Morgan III, J.D. (1981): Bounds on the decay of electron densities with screening. Phys. Rev.A23, 2106–2117
Berthier, G., Defranceschi, M., Delhalle, J. (1989): Electronic structure theory in momentum space. In: M. Defranceschi, J. Delhalle, eds., Numerical Determination of the Electronic Structure of Atoms, Diatomic and Polyatomic Molecules, pp. 209–238. NATO-ASI, vol. C271. Kluwer, Dordrecht
Berthier, G., Defranceschi, M., Navaza, J., Suard, M., Tsoucaris, G. (1985): Direct momentum space calculations for molecules. J. Mol. Struct. (Theochem.)120, 343–350
Blum, E.K. (1972): Numerical Analysis and Computation. Addison-Wesley, New York
Cohen-Tannoudji, C., Diu, B., Laloe, F. (1982): Mecanique quantique, I. Hermann, Paris
Defranceschi, M., Suard, M., Berthier, G. (1984): Numerical solution of Hartree-Fock equations for a polyatomic molecule: linearH 3 in momentum space. Internat. J. Quantum Chem.25, 863–867
Defranceschi, M., Delhalle, J. (1986): Numerical solution of the Hartree-Fock equations for quasi-one-dimensional systems: prototypical calculations on the (−H−) x chain. Phys. Rev.B34, 5862–5873
Defranceschi, M., Sarrazin, M. (1989): Numerical integration over a spherical shell. Comput. Phys. Com.52, 409–414
Delhalle, J., Fripiat, J.G., Defranceschi, M. (1987): Momentum space as an alternative to improve ab initio calculations using atomic gaussian basis sets. Preliminary investigation on hydrogen atom. Annales Soc. Scient. Brux.101, 9–21
Delhalle, J., Fripiat, J.G., Defranceschi, M. (1990): Improving the one-electron states of ab initio GTO calculations in momentum space. Tests on two-electron systems: H−, He and Li+. Bull. Soc. Chim. Belg.99, 135–145
De Windt, L., Delhalle, J., Fripiat, J.G., Defranceschi, M. (1992): Improving the one-electron states of ab initio LCAO-GTO calculations in momentum space. Tests on the Be and B+ atoms. J. Mol. Struct (Theochem)254, 145–159
Fonte, G. (1981): Convergence of the Rayleig-Ritz Method in Self-Consistent-Field calculations. Theoret. Chim. Acta59, 533–549
Fraga, S., Ransil, B.J. (1961): Studies in molecular structure. VI. J. Chem. Phys.35, 1967–1977
Fripiat, J.G., Delhalle, J., Defranceschi, M. (1989): A momentum space approach to improve ab initio Hartree-Fock results based on the LCAO-GTF approximation: In: M. Defranceschi, J. Delhalle, eds., Numerical Determination of the Electronic Structure of Atoms, Diatomic and Polyatomic Molecules, pp. 263–268. NATO-ASI, vol. C271. Kluwer, Dordrecht
Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T. (1977): Schrödinger inequalities and asymptotic behavior of the electron density of atoms and molecules. Phys. Rev.A16, 1782–1785
Kato, T. (1966): Perturbation theory for linear operators. Springer, Berlin Heidelberg New York
Lassettre, E.N. (1985): Momentum eigenfunctions in the complex momentum plane, V. J. Chem. Phys.82, 827–840
Lebedev, N.N. (1972): Special Functions and their Applications. Dover, New York
McCarthy, I.E., Weigold, E. (1991): Electron momentum spectroscopy for atoms and molecules. Rep. Prog. Phys.54, 789–879
McWeeny, R. (1989): Methods of Molecular Quantum Mechanics, 2nd ed. Academic Press, New York
Morse, P.M., Feshbach, H. (1953): Methods of Theoretical Physics, Part II, chapter 9. Mac Graw-Hill, New York
Navaza, J., Tsoucaris, G. (1981): Molecular wavefunctions in momentum space. Phys. Rev.A24, 683–692
Roman, P. (1965): Advanced Quantum Theory, pp. 3–23. Addison-Wesley, Reading
Roothaan, C.C.J. (1951): New developments in molecular orbital theory. Rev. Mod. Phys.23, 69–89
Spanier, J., Oldham, K.B. (1987): An atlas of functions. Hemisphere, Washington
Svartholm, N.V. (1945): The binding energies of light nuclei. Ph.D. thesis, Stockholm-Lund
Williams, B.G. (1977): The experimental determination of electron momentum densities. Physica Scripta15, 69–79
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Fischer, P., Defranceschi, M. & Delhalle, J. Molecular Hartree-Fock equations for iteration-variation calculations in momentum space. Numer. Math. 63, 67–82 (1992). https://doi.org/10.1007/BF01385848
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DOI: https://doi.org/10.1007/BF01385848