We discuss the multi-configuration time-dependent Hartree (MCTDH) method for the approximation of the time-dependent Schrödinger equation in quantum molecular dynamics. This method approximates the high-dimensional nuclear wave function by a linear combination of products of functions depending only on a single degree of freedom. The equations of motion, obtained via the Dirac-Frenkel time-dependent variational principle, consist of a coupled system of low-dimensional nonlinear partial differential equations and ordinary differential equations. We show that, with a smooth and bounded potential, the MCTDH equations are well-posed and retain high-order Sobolev regularity globally in time, that is, as long as the density matrices appearing in the method formulation remain invertible. In particular, the solutions are regular enough to ensure local quasi-optimality of the approximation and to admit an efficient numerical treatment.
Mots-clés : MCTDH method, wavepacket propagation, nonlinear evolution equation, well-posedness, regularity
@article{M2AN_2007__41_2_315_0,
author = {Koch, Othmar and Lubich, Christian},
title = {Regularity of the multi-configuration time-dependent {Hartree} approximation in quantum molecular dynamics},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {315--331},
publisher = {EDP-Sciences},
volume = {41},
number = {2},
year = {2007},
doi = {10.1051/m2an:2007020},
mrnumber = {2339631},
zbl = {1135.81380},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2007020/}
}
TY - JOUR AU - Koch, Othmar AU - Lubich, Christian TI - Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 315 EP - 331 VL - 41 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2007020/ DO - 10.1051/m2an:2007020 LA - en ID - M2AN_2007__41_2_315_0 ER -
%0 Journal Article %A Koch, Othmar %A Lubich, Christian %T Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 315-331 %V 41 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2007020/ %R 10.1051/m2an:2007020 %G en %F M2AN_2007__41_2_315_0
Koch, Othmar; Lubich, Christian. Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics. ESAIM: Modélisation mathématique et analyse numérique, Special issue on Molecular Modelling, Tome 41 (2007) no. 2, pp. 315-331. doi : 10.1051/m2an:2007020. http://www.numdam.org/articles/10.1051/m2an:2007020/
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