An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics

Dajana Conte; Christian Lubich

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 4, page 759-780
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Conte, Dajana, and Lubich, Christian. "An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics." ESAIM: Mathematical Modelling and Numerical Analysis 44.4 (2010): 759-780. <http://eudml.org/doc/250714>.

@article{Conte2010,
abstract = { 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. },
author = {Conte, Dajana, Lubich, Christian},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {MCTDH method; multi-dimensional quantum dynamics; low-rank approximation; low-rank approximation},
language = {eng},
month = {6},
number = {4},
pages = {759-780},
publisher = {EDP Sciences},
title = {An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics},
url = {http://eudml.org/doc/250714},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Conte, Dajana
AU - Lubich, Christian
TI - An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/6//
PB - EDP Sciences
VL - 44
IS - 4
SP - 759
EP - 780
AB - 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.
LA - eng
KW - MCTDH method; multi-dimensional quantum dynamics; low-rank approximation; low-rank approximation
UR - http://eudml.org/doc/250714
ER -

References

top
  1. M.H. Beck, A. Jäckle, G.A. Worth and H.-D. Meyer, The multiconfiguration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wavepackets. Phys. Rep.324 (2000) 1–105.  
  2. G. Friesecke, The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions. Arch. Ration. Mech. Anal.169 (2003) 35–71.  Zbl1035.81069
  3. R.A. Horn and C.R. Johnson, Matrix Analysis. Cambridge Univ. Press, UK (1985).  Zbl0576.15001
  4. B.N. Khoromskij, Structured rank-(R1, ..., Rd) decomposition of function-related tensors in d . Comput. Meth. Appl. Math.6 (2006) 194–220.  Zbl1120.65052
  5. O. Koch and C. Lubich, Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics. ESAIM: M2AN41 (2007) 315–331.  Zbl1135.81380
  6. O. Koch and C. Lubich, Dynamical low-rank approximation. SIAM J. Matrix Anal. Appl.29 (2007) 434–454.  Zbl1145.65031
  7. O. Koch and C. Lubich, Dynamical tensor approximation. Preprint (2009).  Zbl1214.15017
  8. T.G. Kolda and B.W. Bader, Tensor decompositions and applications. SIAM Rev.51 (2009) 455–500.  Zbl1173.65029
  9. M. Lewin, Solutions of the multiconfiguration equations in quantum chemistry. Arch. Ration. Mech. Anal.171 (2004) 83–114.  Zbl1063.81102
  10. C. Lubich, On variational approximations in quantum molecular dynamics. Math. Comp.74 (2005) 765–779.  Zbl1059.81188
  11. C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis. Europ. Math. Soc., Zurich, Switzerland (2008).  Zbl1160.81001
  12. H.-D. Meyer, F. Gatti and G.A. Worth, Eds., Multidimensional Quantum Dynamics: MCTDH Theory and Applications. Wiley, New York, USA (2009).  
  13. H.-D. Meyer, U. Manthe and L.S. Cederbaum, The multi-configurational time-dependent Hartree approach. Chem. Phys. Lett.165 (1990) 73–78.  
  14. H.-D. Meyer and G.A. Worth, 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. A. Raab, G.A. Worth, H.-D. Meyer and L.S. Cederbaum, 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. B. Thaller, Visual Quantum Mechanics. Springer, New York, USA (2000).  Zbl1056.81001
  17. H. Wang and M. Thoss, Multilayer formulation of the multiconfiguration time-dependent Hartree theory. J. Chem. Phys.119 (2003) 1289–1299.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.