Hostname: page-component-5c6d5d7d68-wp2c8 Total loading time: 0 Render date: 2024-08-13T10:52:52.624Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  16 June 2007

Othmar Koch  and
Christian Lubich
Othmar Koch
Affiliation:
Universität Tübingen, Mathematisches Institut, Auf der Morgenstelle 10, 72076 Tübingen, Germany.
Christian Lubich
Affiliation:
Universität Tübingen, Mathematisches Institut, Auf der Morgenstelle 10, 72076 Tübingen, Germany.
Get access

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.

Keywords

MCTDH methodwavepacket propagationnonlinear evolution equationwell-posednessregularity.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 2: Special issue on Molecular Modelling , March 2007 , pp. 315 - 331
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

H.W. Alt, Lineare Funktionalanalysis. Springer Verlag, Berlin-Heidelberg-New York, 3rd edition (1999).
M. Baer and G.D. Billing Eds., The Role of Degenerate States in Chemistry, Advances in Chemical Physics 124, Wiley (2002).
Beck, M.H. and Meyer, H.-D., An efficient and robust integration scheme for the equations of the multiconfiguration time-dependent Hartree (MCTDH) method. Z. Phys. D 42 (1997) 113–129. CrossRef
Beck, M.H., Jäckle, A., Worth, G.A. and Meyer, H.-D., The multiconfiguration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wavepackets. Phys. Rep. 324 (2000) 1–105. CrossRef
Bove, A., Da Prato, G. and Fano, G., An existence proof for the Hartree-Fock time-dependent problem with bounded two-body interaction. Comm. Math. Phys. 37 (1974) 183–191. CrossRef
Bove, A., Da Prato, G. and Fano, G., On the Hartree-Fock time-dependent problem. Comm. Math. Phys. 49 (1976) 25–33. CrossRef
Burghardt, I., Meyer, H.-D. and Cederbaum, L.S., Approaches to the approximate treatment of complex molecular systems by the multiconfiguration time-dependent Hartree method. J. Chem. Phys. 111 (1999) 2927–2939. CrossRef
Caillat, J., Zanghellini, J., Kitzler, M., Kreuzer, W., Koch, O. and Scrinzi, A., Correlated multielectron systems in strong laser pulses – an MCTDHF approach. Phys. Rev. A 71 (2005) 012712. CrossRef
Chadam, J.M. and Glassey, R.T., Global existence of solutions to the Cauchy problem for time-dependent Hartree equations. J. Math. Phys. 16 (1975) 1122–1130. CrossRef
Dirac, P.A.M., Note on exchange phenomena in the Thomas atom. Proc. Cambridge Phil. Soc. 26 (1930) 376–385. CrossRef
W. Domcke, D.R. Yarkony and H. Köppel Eds., Conical Intersections. Electronic Structure, Dynamics & Spectroscopy. World Scientific, Singapore, 2004.
Faou, E. and Lubich, C., Poisson, A integrator for Gaussian wavepacket dynamics. Comput. Visual. Sci. 9 (2005) 45–55. CrossRef
J. Frenkel, Wave Mechanics, Advanced General Theory. Clarendon Press, Oxford (1934).
Friesecke, G., The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions. Arch. Ration. Mech. Anal. 169 (2003) 35–71. CrossRef
Koch, O., Kreuzer, W. and Scrinzi, A., Approximation of the time-dependent electronic Schrödinger equation by MCTDHF. Appl. Math. Comput. 173 (2006) 960–976.
Lewin, M., Solutions of the multiconfiguration equations in quantum chemistry. Arch. Ration. Mech. Anal. 171 (2004) 83–114. CrossRef
Lubich, C., A variational splitting integrator for quantum molecular dynamics. Appl. Numer. Math. 48 (2004) 355–368. CrossRef
Lubich, C., On variational approximations in quantum molecular dynamics. Math. Comp. 74 (2005) 765–779. CrossRef
McLachlan, A.D., A variational solution of the time-dependent Schrödinger equation. Mol. Phys. 8 (1964) 39–44. CrossRef
Meyer, H.-D. and Worth, G.A., Quantum molecular dynamics: propagating wavepackets and density operators using the multi-configuration time-dependent Hartree (MCTDH) method. Theo. Chem. Acc. 109 (2003) 251–267. CrossRef
Meyer, H.-D., Manthe, U. and Cederbaum, L.S., The multi-configurational time-dependent Hartree approach. Chem. Phys. Lett. 165 (1990) 73–78.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).
Raab, A., Worth, G.A., Meyer, H.-D. and Cederbaum, L.S., Molecular dynamics of pyrazine after excitation to the S 2 electronic state using a realistic 24-mode model Hamiltonian. J. Chem. Phys. 110 (1999) 936–946. CrossRef
Rowe, D.J., Ryman, A. and Rosensteel, G., Many-body quantum mechanics as a symplectic dynamical system. Phys. Rev. A 22 (1980) 2362–2373. CrossRef
Wang, H. and Thoss, M., Multilayer formulation of the multiconfiguration time-dependent Hartree theory. J. Chem. Phys. 119 (2003) 1289–1299. CrossRef