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Modèles mathématiques de la chimie quantique atomique & dynamique quantique et spectre multifractal

Abstract : All electrons in heavy atoms and ions, and especially those close to the nucleus, feels important relativistic effects. Therefore, it is necessay to take into account these effects for a precise description of the energy levels. Relativistic quantum mechanics dates back to P.A.M. Dirac in 1928. In his fundamental works, Dirac theoretically described the existence of antiparticles, who were experimentally discovered few years later. Indeed, the quantum Hamiltonien he obtained for describing the energies of the hydrogen atom makes sense if one reinterpret the negative energies as the total energy of an infinite sea of virtual electrons. A hole in the negative energy spectrum correspond to the creation of an anti-particle: the positron. Later on, in 1938, for studying atoms with many electrons, Swirles proposed a model yielding the so-called Dirac-Fock equations. This approach is self-consistent, and the resulting equations are nonlinear. It allows numerical studies in good agreement with experimental datas. However, the physical motivation for this approach, based on nonrelativistic models, remains unsatisfactory since it does not take into account Dirac's interpretation and the existence of positrons. Moreover, the link between the Dirac-Fock equations and the theoretical approach given by the full quantum electrodynamics remains elusive. In particular, the definition of the right Hilbert space describing the space of electronic states remains open. The present work is an attempt to give rigorous justifications to this problem. We start by the construction of a family of functionals depending on the choice of the one-electron space. We will work in the Hartree-Fock approximation. We will then study the stability property, as well as the existence of minima for these functionals, with or without a constraint on the total charge of the system. Then we will present some results concerning the connections between the « Dirac-Fock equations » approach and the « QED in Hartree-Fock approximation » approach. The results deal separately with the open shell case and the closed shell case. In the latter case, we show that each approach yields the same results.
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Habilitation à diriger des recherches
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Contributor : Jean-Marie Barbaroux <>
Submitted on : Tuesday, October 4, 2005 - 10:59:19 AM
Last modification on : Monday, March 25, 2019 - 4:52:05 PM
Long-term archiving on: : Friday, September 14, 2012 - 2:45:17 PM

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Jean-Marie Barbaroux. Modèles mathématiques de la chimie quantique atomique & dynamique quantique et spectre multifractal. Mathématiques [math]. Université de Nantes, 2005. ⟨tel-00010385⟩

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