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Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics

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Abstract

The Bogoliubov–Dirac–Fock (BDF) model is the mean-field approximation of no-photon quantum electrodynamics. The present paper is devoted to the study of the minimization of the BDF energy functional under a charge constraint. An associated minimizer, if it exists, will usually represent the ground state of a system of N electrons interacting with the Dirac sea, in an external electrostatic field generated by one or several fixed nuclei. We prove that such a minimizer exists when a binding (HVZ-type) condition holds. We also derive, study and interpret the equation satisfied by such a minimizer. Finally, we provide two regimes in which the binding condition is fulfilled, obtaining the existence of a minimizer in these cases. The first is the weak coupling regime for which the coupling constant α is small whereas αZ and the particle number N are fixed. The second is the non-relativistic regime in which the speed of light tends to infinity (or equivalently α tends to zero) and Z, N are fixed. We also prove that the electronic solution converges in the non-relativistic limit towards a Hartree–Fock ground state.

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Authors and Affiliations

  1. Department of Mathematics, University of Alabama at Birmingham, Campbell Hall, 1300 University Boulevard, Birmingham, AL, 35294, USA

    Christian Hainzl

  2. CNRS and Département de Mathématiques (CNRS UMR 8088), Université de Cergy-Pontoise, 2, avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex, France

    Mathieu Lewin

  3. CEREMADE (CNRS UMR 7534), Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775, Paris Cedex 16, France

    Éric Séré

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  1. Christian Hainzl
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  2. Mathieu Lewin
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  3. Éric Séré
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Correspondence to Éric Séré.

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Communicated by G. Friesecke

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Hainzl, C., Lewin, M. & Séré, É. Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics. Arch Rational Mech Anal 192, 453–499 (2009). https://doi.org/10.1007/s00205-008-0144-2

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