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

Claude Bardos*, Isabelle Catto, Norbert Mauser, Saber Trabelsi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

18 Scopus citations


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 L2 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.

Original languageEnglish (US)
Pages (from-to)273-330
Number of pages58
JournalArchive for Rational Mechanics and Analysis
Issue number1
StatePublished - 2010
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering


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