Abstract
We present a framework for solving the Schrödinger equation modeling the interaction of a dissociative quantum system with a laser field. A perfectly matched layer (PML) is used to handle non-reflecting boundaries and the Schrödinger equation is discretized with high-order finite differences in space and an h, p-adaptive Magnus–Arnoldi propagator in time. We use a posteriori error estimation theory to control the global error of the numerical discretization. The parameters of the PML are chosen to meet the same error tolerance. We apply our framework to the IBr molecule, for which numerical experiments show that the total error can be controlled efficiently. Moreover, we provide numerical evidence that the Magnus–Arnoldi solver outperforms the implicit Crank–Nicolson scheme by far.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Blanes,S., Casas, F., Ros, J.: Improved high order integrators based on the Magnus expansion, BIT Numer. Math. 40, 434–450 (2000)
Cao,Y., Petzold, L.: A posteriori error estimate and global error control for ordinary differential equations by the adjoint method. SIAM J. Sci. Comput. 26, 359–374 (2004)
Guo, H.: The effect of nonadiabatic coupling in the predissociation dynamics of IBr. J. Chem. Phys. 99, 1685–1692 (1993)
Gustafsson, M.: A PDE solver framework optimized for clusters of multicore processors. Master’s thesis, UPTEC Report F09 004, Uppsala University (2009)
Hagstrom, T.: New results on absorbing layers and radiation boundary conditions. In: M. Ainsworth, P. Davies, D. Duncan, P. Martin, B. Rynne (Eds.), Topics in Computational Wave Propagation, vol. 31 of Lecture Notes in Computational Science and Engineering, pp. 1–42. Springer, New York (2003)
Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19, 1552–1574 (1998)
Karlsson, H.O.: Accurate resonances and effective absorption of flux using smooth exterior scaling. J. Chem. Phys. 109, 9366–9371 (1998)
Kormann, K., Holmgren, S., Karlsson, H.O.: Accurate time propagation for the Schrödinger equation with an explicitly time-dependent Hamiltonian. J. Chem. Phys. 128, 184101 (2008)
Kormann, K., Holmgren, S., Karlsson, H.O.: Global error control of the time-propagation for the Schrödinger equation with a time-dependent Hamiltonian. Technical Report 2009-021, Uppsala University (2009)
Lubich, C.: From quantum to classical molecular dynamics: Reduced models and numerical analysis. Eur. Math. Soc., Zürich 9, 147–179 (2011)
Nissen, A., Kreiss, G.: An optimized perfectly matched layer for the Schrödinger equation. Commun. Comput. Phys.
Tannor, D.J.: Introduction to Quantum Mechanics: A Time-Dependent Perspective. University Science Books, Sausalito (2007)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kormann, K., Nissen, A. (2010). Error Control for Simulations of a Dissociative Quantum System. In: Kreiss, G., Lötstedt, P., Målqvist, A., Neytcheva, M. (eds) Numerical Mathematics and Advanced Applications 2009. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11795-4_56
Download citation
DOI: https://doi.org/10.1007/978-3-642-11795-4_56
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11794-7
Online ISBN: 978-3-642-11795-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)