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Error Estimation and Adaptive Methods for Molecular Dynamics
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
2014 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This  thesis consists of two  papers that  concern error estimates for the Born-Oppenheimer molecular dynamics, and adaptive algorithms for the Car-Parrinello and Ehrenfest molecular dynamics. In Paper I, we study error estimates for Born-Oppenheimer molecular dynamics with  nearly crossing potential  surfaces. The paper first proves an error estimate showing that  the difference of the values of observables for the time- independent Schrödinger equation, with matrix valued potentials, and the values of observables for ab initio Born-Oppenheimer molecular dynamics, of the ground state, depends on the probability  to be in excited states and the electron/nuclei mass ratio.  Then we present a numerical method to determine the probability to be in excited states, based on Ehrenfest molecular dynamics, and stability analysis of a perturbed eigenvalue problem. In Paper II, we present an approach, motivated by the Landau-Zener probability estimation, to systematically choose the artificial  electron mass parameter appearing in the Car-Parrinello  and Ehrenfest molecular dynamics methods to achieve both  good accuracy in approximating  the Born-Oppenheimer molecular dynamics solution, and high computational  efficiency. This makes the Car- Parrinello  and Ehrenfest molecular dynamics methods dependent  only on the problem data.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. , 12 p.
Series
TRITA-MAT-A, 2014:13
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-156035ISBN: 978-91-7595-361-8 (print)OAI: oai:DiVA.org:kth-156035DiVA: diva2:764200
Presentation
2014-11-25, Room 3733, Lindstedtsvägen 25, KTH, Stockholm, 10:15 (English)
Opponent
Supervisors
Note

QC 20141118

Available from: 2014-11-18 Created: 2014-11-18 Last updated: 2014-11-18Bibliographically approved
List of papers
1. Computational error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces
Open this publication in new window or tab >>Computational error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces
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2015 (English)In: Applied Mathematics Research eXpress, ISSN 1687-1200, E-ISSN 1687-1197, no 2, 329-417 p.Article in journal (Refereed) Published
Abstract [en]

The difference of the values of observables for the time-independent Schrödinger equation, with matrix-valued potentials, and the values of observables for ab initio Born-Oppenheimer molecular dynamics, of the ground state, depends on the probability to be in excited states, and the electron/nuclei mass ratio. The paper first proves an error estimate (depending on the electron/nuclei mass ratio and the probability to be in excited states) for this difference of microcanonical observables, assuming that molecular dynamics space-time averages converge, with a rate related to the maximal Lyapunov exponent. The error estimate is uniform in the number of particles and the analysis does not assume a uniform lower bound on the spectral gap of the electron operator and consequently the probability to be in excited states can be large. A numerical method to determine the probability to be in excited states is then presented, based on Ehrenfest molecular dynamics, and stability analysis of a perturbed eigenvalue problem.

Place, publisher, year, edition, pages
Oxford University Press, 2015
Keyword
HIGH-ORDER CORRECTIONS, NUMERICAL-ANALYSIS, QUANTUM, PROPAGATION, ERGODICITY, OPERATORS, APPROXIMATION, SYSTEMS
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-156031 (URN)10.1093/amrx/abv007 (DOI)000366820400007 ()2-s2.0-84941214775 (Scopus ID)
Note

Updated from Manuscript to Article. QC 20151012. QC 20160121

Available from: 2014-11-18 Created: 2014-11-18 Last updated: 2017-12-05Bibliographically approved
2. An Adaptive Mass Algorithm for Car-Parrinello and Ehrenfest ab initio molecular dynamics
Open this publication in new window or tab >>An Adaptive Mass Algorithm for Car-Parrinello and Ehrenfest ab initio molecular dynamics
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Ehrenfest and Car-Parrinello molecular dynamics are computational alternatives to approximate Born-Oppenheimer molecular dynamics without solving the electron eigenvalue problem at each time-step. A non-trivial issue is to choose the artificial electron mass parameter appearing in the Car-Parrinello method to achieve  both good accuracy and high computational efficiency. In this paper, we propose an algorithm, motivated by the Landau-Zener probability, to systematically choose an artificial mass dynamically, which makes the Car-Parrinello and Ehrenfest molecular dynamics methods dependent only on the problem data. Numerical experiments for simple model problems show that the time-dependent adaptive artificial mass parameter improves the efficiency of the Car-Parrinello and Ehrenfest molecular dynamics.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-156033 (URN)
Note

QC 20141118

Available from: 2014-11-18 Created: 2014-11-18 Last updated: 2016-11-04Bibliographically approved

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Licentiate Thesis(513 kB)172 downloads
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Citation style
  • apa
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Output format
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