Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics

Othmar Koch; Christian Lubich

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 2, page 315-331
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Koch, Othmar, and Lubich, Christian. "Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics." ESAIM: Mathematical Modelling and Numerical Analysis 41.2 (2007): 315-331. <http://eudml.org/doc/250059>.

@article{Koch2007,
abstract = { 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. },
author = {Koch, Othmar, Lubich, Christian},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {MCTDH method; wavepacket propagation; nonlinear evolution equation; well-posedness; regularity.},
language = {eng},
month = {6},
number = {2},
pages = {315-331},
publisher = {EDP Sciences},
title = {Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics},
url = {http://eudml.org/doc/250059},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Koch, Othmar
AU - Lubich, Christian
TI - Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/6//
PB - EDP Sciences
VL - 41
IS - 2
SP - 315
EP - 331
AB - 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.
LA - eng
KW - MCTDH method; wavepacket propagation; nonlinear evolution equation; well-posedness; regularity.
UR - http://eudml.org/doc/250059
ER -

References

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