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

被引:19
|
作者
Bardos, Claude [1 ]
Catto, Isabelle [2 ,3 ]
Mauser, Norbert [4 ]
Trabelsi, Saber [1 ]
机构
[1] Univ Paris 07, Lab JL Lions, F-75013 Paris, France
[2] CNRS, UMR 7534, F-75016 Paris, France
[3] Univ Paris 09, CEREMADE, F-75016 Paris, France
[4] Univ Wien, Wolfgang Pauli Inst, Fak Math, A-1090 Vienna, Austria
关键词
MULTICONFIGURATION EQUATIONS; CAUCHY-PROBLEM; EXISTENCE; DYNAMICS; APPROXIMATION; MOLECULES; ATOMS;
D O I
10.1007/s00205-010-0308-8
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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 Schrodinger equation based on (time-dependent) linear combinations of (time-dependent) Slater determinants. The "one-electron" wave-functions satisfy nonlinear Schrodinger-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 role 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.
引用
收藏
页码:273 / 330
页数:58
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