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On variational approximations in quantum molecular dynamics
Author(s):
Christian
Lubich.
Journal:
Math. Comp.
74
(2005),
765-779.
MSC (2000):
Primary 65M15, 81Q05;
Secondary 35Q40
Posted:
May 25, 2004
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Abstract:
The Dirac-Frenkel-McLachlan variational principle is the basic tool for obtaining computationally accessible approximations in quantum molecular dynamics. It determines equations of motion for an approximate time-dependent wave function on an approximation manifold of reduced dimension. This paper gives a near-optimality result for variational approximations. It bounds the error in terms of the distance of the exact wave function to the approximation manifold and identifies the parameters that control the deviation of the variational approximation from the best approximation on the manifold.
References:
-
- 1.
- M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1987.
- 2.
- M. Baer, G.D. Billing, eds., The Role of Degenerate States in Chemistry, Advances in Chemical Physics, Vol. 124, Wiley, 2002.
- 3.
- M.H. Beck, A. Jäckle, G.A. Worth, H.-D. Meyer, The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propagating wavepackets, Phys. Reports 324 (2000), 1-105.
- 4.
- G.D. Billing, The Quantum Classical Theory, Oxford Univ. Press, 2003.
- 5.
- P.G. Ciarlet, Introduction to the finite element method, in Handbook of Numerical Analysis, Vol. II (P.G. Ciarlet and J.L. Lions, eds.), North-Holland, Amsterdam, 1991, 17-352. MR 92f:65001
- 6.
- P.A.M. Dirac, Note on exchange phenomena in the Thomas atom, Proc. Cambridge Phil. Soc. 26 (1930), 376-385.
- 7.
- J. Frenkel, Wave Mechanics, Advanced General Theory, Clarendon Press, Oxford, 1934.
- 8.
- E.J. Heller, Time dependent variational approach to semiclassical dynamics, J. Chem. Phys. 64 (1976), 63-73.MR 56:4540
- 9.
- T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, 1980.MR 53:11389
- 10.
- T. Klett, Das Multikonfigurations-Hartree-Verfahren in der Quantendynamik, Diplomarbeit, Fakultät für Mathematik und Physik, Univ. Tübingen, 2004.
- 11.
- A. Martinez, An Introduction to Semiclassical and Microlocal Analysis, Springer, New York, 2002. MR 2003b:35010
- 12.
- A.D. McLachlan, A variational solution of the time-dependent Schrodinger equation, Mol. Phys. 8 (1964), 39-44.MR 31:4358
- 13.
- A. Messiah, Quantum Mechanics, Dover Publ., 1999 (reprint of the two-volume edition published by Wiley, 1961-1962). MR 23:B2826; MR 26:4643
- 14.
- A. Raab, On the Dirac-Frenkel/McLachlan variational principle, Chem. Phys. Lett. 319 (2000), 674-678.
- 15.
- S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, Lecture Notes in Mathematics 1821, Springer, Berlin, 2003.
- 16.
- H. Yserentant, On the regularity of the Schrödinger equation in Hilbert spaces of mixed derivatives, Numer. Math., 2004 (to appear).
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Additional Information:
Christian
Lubich
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
Email:
lubich@na.uni-tuebingen.de
DOI:
10.1090/S0025-5718-04-01685-0
PII:
S 0025-5718(04)01685-0
Keywords:
Quantum dynamics,
Dirac-Frenkel-McLachlan variational principle,
time-dependent Hartree and Hartree-Fock methods,
optimality,
error bounds
Received by editor(s):
September 8, 2003
Posted:
May 25, 2004
Copyright of article:
Copyright
2004,
American Mathematical Society
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