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Setting and Analysis of the Multi-configuration Time-dependent Hartree–Fock Equations

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Abstract

In this paper, we formulate and analyze the multi-configuration time-dependent Hartree–Fock (MCTDHF) equations for molecular systems with pairwise interaction. This set of coupled nonlinear PDEs and ODEs is an approximation of the N-particle time-dependent Schrödinger equation based on (time-dependent) linear combinations of (time-dependent) Slater determinants. The “one-electron” wave-functions satisfy nonlinear Schrödinger-type equations coupled to a linear system of ordinary differential equations for the expansion coefficients. The invertibility of the one-body density matrix (full-rank hypothesis) plays a crucial rôle in the analysis. Under the full-rank assumption a fiber bundle structure emerges and produces unitary equivalence between different useful representations of the MCTDHF approximation. For a large class of interactions (including Coulomb potential), we establish existence and uniqueness of maximal solutions to the Cauchy problem in the energy space as long as the density matrix is not singular. A sufficient condition in terms of the energy of the initial data ensuring the global-in-time invertibility is provided (first result in this direction). Regularizing the density matrix violates energy conservation. However, global well-posedness for this system in L 2 is obtained with Strichartz estimates. Eventually, solutions to this regularized system are shown to converge to the original one on the time interval when the density matrix is invertible.

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Authors and Affiliations

  1. Laboratoire J.-L. Lions, Université Paris 7, 175 rue du Chevaleret, 75013, Paris, France

    Claude Bardos & Saber Trabelsi

  2. Wolfgang Pauli Institute, Nordbergstrasse 15, 1090, Vienna, Austria

    Claude Bardos & Saber Trabelsi

  3. CNRS, UMR 7534, 75016, Paris, France

    Isabelle Catto

  4. Université Paris-Dauphine, CEREMADE, Place du Mal. de Lattre de Tassigny, 75016, Paris, France

    Isabelle Catto

  5. Wolfgang Pauli Institute, c/o Fak. f.Mathematik, Univ. Wien, Nordbergstrasse 15, 1090, Vienna, Austria

    Norbert Mauser

Authors
  1. Claude Bardos
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  2. Isabelle Catto
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  3. Norbert Mauser
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  4. Saber Trabelsi
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Correspondence to Claude Bardos.

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Communicated by G. Friesecke

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Bardos, C., Catto, I., Mauser, N. et al. Setting and Analysis of the Multi-configuration Time-dependent Hartree–Fock Equations. Arch Rational Mech Anal 198, 273–330 (2010). https://doi.org/10.1007/s00205-010-0308-8

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  • DOI: https://doi.org/10.1007/s00205-010-0308-8

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