This paper gives an error analysis of the multi-configuration time-dependent Hartree (MCTDH) method for the approximation of multi-particle time-dependent Schrödinger equations. The MCTDH method approximates the multivariate wave function by a linear combination of products of univariate functions and replaces the high-dimensional linear Schrödinger equation by a coupled system of ordinary differential equations and low-dimensional nonlinear partial differential equations. The main result of this paper yields an L2 error bound of the MCTDH approximation in terms of a best-approximation error bound in a stronger norm and of lower bounds of singular values of matrix unfoldings of the coefficient tensor. This result permits us to establish convergence of the MCTDH method to the exact wave function under appropriate conditions on the approximability of the wave function, and it points to reasons for possible failure in other cases.
@article{M2AN_2010__44_4_759_0,
author = {Conte, Dajana and Lubich, Christian},
title = {An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {44},
year = {2010},
pages = {759-780},
doi = {10.1051/m2an/2010018},
mrnumber = {2683582},
zbl = {1192.81125},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2010__44_4_759_0}
}
Conte, Dajana; Lubich, Christian. An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010) pp. 759-780. doi : 10.1051/m2an/2010018. http://gdmltest.u-ga.fr/item/M2AN_2010__44_4_759_0/
[1] , , and , The multiconfiguration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wavepackets. Phys. Rep. 324 (2000) 1-105.
[2] , The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions. Arch. Ration. Mech. Anal. 169 (2003) 35-71. | Zbl 1035.81069
[3] and , Matrix Analysis. Cambridge Univ. Press, UK (1985). | Zbl 0704.15002
[4] , Structured rank-(R1, ..., Rd) decomposition of function-related tensors in . Comput. Meth. Appl. Math. 6 (2006) 194-220. | Zbl 1120.65052
[5] and , Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics. ESAIM: M2AN 41 (2007) 315-331. | Numdam | Zbl 1135.81380
[6] and , Dynamical low-rank approximation. SIAM J. Matrix Anal. Appl. 29 (2007) 434-454. | Zbl 1145.65031
[7] and , Dynamical tensor approximation. Preprint (2009). | Zbl pre05859901
[8] and , Tensor decompositions and applications. SIAM Rev. 51 (2009) 455-500. | Zbl 1173.65029
[9] , Solutions of the multiconfiguration equations in quantum chemistry. Arch. Ration. Mech. Anal. 171 (2004) 83-114. | Zbl 1063.81102
[10] , On variational approximations in quantum molecular dynamics. Math. Comp. 74 (2005) 765-779. | Zbl 1059.81188
[11] , From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis. Europ. Math. Soc., Zurich, Switzerland (2008). | Zbl 1160.81001
[12] , and , Eds., Multidimensional Quantum Dynamics: MCTDH Theory and Applications. Wiley, New York, USA (2009).
[13] , and , The multi-configurational time-dependent Hartree approach. Chem. Phys. Lett. 165 (1990) 73-78.
[14] and , Quantum molecular dynamics: propagating wavepackets and density operators using the multi-configuration time-dependent Hartree (MCTDH) method. Theo. Chem. Acc. 109 (2003) 251-267.
[15] , , and , Molecular dynamics of pyrazine after excitation to the S2 electronic state using a realistic 24-mode model Hamiltonian. J. Chem. Phys. 110 (1999) 936-946.
[16] , Visual Quantum Mechanics. Springer, New York, USA (2000). | Zbl 1056.81001
[17] and , Multilayer formulation of the multiconfiguration time-dependent Hartree theory. J. Chem. Phys. 119 (2003) 1289-1299.