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Accurate time propagation for the Schrödinger equation with an explicitly time-dependent Hamiltonian
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Physical and Analytical Chemistry, Quantum Chemistry.
2008 (English)In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 128, 184101:1-11 p.Article in journal (Refereed) Published
Place, publisher, year, edition, pages
2008. Vol. 128, 184101:1-11 p.
National Category
Computational Mathematics Computer Science
Identifiers
URN: urn:nbn:se:uu:diva-16180DOI: 10.1063/1.2916581ISI: 000255983500008OAI: oai:DiVA.org:uu-16180DiVA: diva2:43951
Available from: 2008-05-10 Created: 2008-05-10 Last updated: 2017-12-08Bibliographically approved
In thesis
1. Numerical methods for quantum molecular dynamics
Open this publication in new window or tab >>Numerical methods for quantum molecular dynamics
2009 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

The time-dependent Schrödinger equation models the quantum nature of molecular processes. Numerical simulations of these models help in understanding and predicting the outcome of chemical reactions.

In this thesis, several numerical algorithms for evolving the Schrödinger equation with an explicitly time-dependent Hamiltonian are studied and their performance is compared for the example of a pump-probe and an interference experiment for the rubidium diatom. For the important application of interaction dynamics between a molecule and a time-dependent field, an efficient fourth order Magnus-Lanczos propagator is derived. Error growth in the equation is analyzed by means of a posteriori error estimation theory and the self-adjointness of the Hamiltonian is exploited to yield a low-cost global error estimate for numerical time evolution. Based on this theory, an h,p-adaptive Magnus-Lanczos propagator is developed that is capable to control the global error. Numerical experiments for various model systems (including a three dimensional model and a dissociative configuration) show that the error estimate is effective and the number of time steps needed to meet a certain accuracy is reduced due to adaptivity.

Moreover, the thesis proposes an efficient numerical optimization framework for the design of femtosecond laser pulses with the aim of manipulating chemical reactions. This task can be formulated as an optimal control problem with the electric field of the laser being the control variable. In the algorithm described here, the electric field is Fourier transformed and it is optimized over the Fourier coefficients. Then, the frequency band is narrowed which facilitates the application of a quasi-Newton method. Furthermore, the restrictions on the frequency band make sure that the optimized pulse can be realized by the experimental equipment. A numerical comparison shows that the new method can outperform the Krotov method, which is a standard scheme in this field.

Place, publisher, year, edition, pages
Uppsala University, 2009
Series
Information technology licentiate theses: Licentiate theses from the Department of Information Technology, ISSN 1404-5117 ; 2009-004
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-108366 (URN)
Supervisors
Available from: 2009-10-09 Created: 2009-09-16 Last updated: 2017-08-31Bibliographically approved
2. Efficient and Reliable Simulation of Quantum Molecular Dynamics
Open this publication in new window or tab >>Efficient and Reliable Simulation of Quantum Molecular Dynamics
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The time-dependent Schrödinger equation (TDSE) models the quantum nature of molecular processes.  Numerical simulations based on the TDSE help in understanding and predicting the outcome of chemical reactions. This thesis is dedicated to the derivation and analysis of efficient and reliable simulation tools for the TDSE, with a particular focus on models for the interaction of molecules with time-dependent electromagnetic fields.

Various time propagators are compared for this setting and an efficient fourth-order commutator-free Magnus-Lanczos propagator is derived. For the Lanczos method, several communication-reducing variants are studied for an implementation on clusters of multi-core processors. Global error estimation for the Magnus propagator is devised using a posteriori error estimation theory. In doing so, the self-adjointness of the linear Schrödinger equation is exploited to avoid solving an adjoint equation. Efficiency and effectiveness of the estimate are demonstrated for both bounded and unbounded states. The temporal approximation is combined with adaptive spectral elements in space. Lagrange elements based on Gauss-Lobatto nodes are employed to avoid nondiagonal mass matrices and ill-conditioning at high order. A matrix-free implementation for the evaluation of the spectral element operators is presented. The framework uses hybrid parallelism and enables significant computational speed-up as well as the solution of larger problems compared to traditional implementations relying on sparse matrices.

As an alternative to grid-based methods, radial basis functions in a Galerkin setting are proposed and analyzed. It is found that considerably higher accuracy can be obtained with the same number of basis functions compared to the Fourier method. Another direction of research presented in this thesis is a new algorithm for quantum optimal control: The field is optimized in the frequency domain where the dimensionality of the optimization problem can drastically be reduced. In this way, it becomes feasible to use a quasi-Newton method to solve the problem.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2012. 52 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 970
Keyword
time-dependent Schrödinger equation, quantum optimal control, exponential integrators, spectral elements, radial basis functions, global error control and adaptivity, high-performance computing implementation
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-180251 (URN)978-91-554-8466-8 (ISBN)
Public defence
2012-10-19, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:00 (English)
Opponent
Supervisors
Projects
eSSENCE
Funder
eSSENCE - An eScience Collaboration
Available from: 2012-09-27 Created: 2012-09-01 Last updated: 2013-01-23Bibliographically approved

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Kormann, KatharinaHolmgren, SverkerKarlsson, Hans O.

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