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Computational error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
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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. no 2, 329-417 p.
Keyword [en]
HIGH-ORDER CORRECTIONS, NUMERICAL-ANALYSIS, QUANTUM, PROPAGATION, ERGODICITY, OPERATORS, APPROXIMATION, SYSTEMS
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-156031DOI: 10.1093/amrx/abv007ISI: 000366820400007Scopus ID: 2-s2.0-84941214775OAI: oai:DiVA.org:kth-156031DiVA: diva2:764166
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
In thesis
1. Error Estimation and Adaptive Methods for Molecular Dynamics
Open this publication in new window or tab >>Error Estimation and Adaptive Methods for Molecular Dynamics
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:nbn:se:kth:diva-156035 (URN)978-91-7595-361-8 (ISBN)
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
2. Numerical Methods for Molecular Dynamics with Nearly Crossing Potential Surfaces
Open this publication in new window or tab >>Numerical Methods for Molecular Dynamics with Nearly Crossing Potential Surfaces
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four 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 the 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 the ab initio Born-Oppenheimer molecular dynamics of the ground state depends on the probability to be in the excited states and the nuclei/electron mass ratio. Then we present a numerical method to determine the probability to be in the excited states, based on the Ehrenfest molecular dynamics, and stability analysis of a perturbed eigenvalue problem.

In Paper II, we present an approach, motivated by the so called Landau-Zener probability estimation, to systematically choose the artificial electron mass parameters appearing in the Car-Parrinello and Ehrenfest molecular dynamics methods to approximate the Born-Oppenheimer molecular dynamics solutions.

In Paper III, we extend the work presented in Paper II for a set of more general problems with more than two electron states. A main conclusion of Paper III is that it is necessary to resolve the near avoided conical intersections between all electron eigenvalue gaps, including gaps between the occupied states.

In Paper IV, we numerically compare, using simple model problems, the Ehrenfest molecular dynamics using the adaptive mass algorithm proposed in Paper II and III and the Born-Oppenheimer molecular dynamics based on the so called purification of the density matrix method concluding that the Born-Oppenheimer molecular dynamics based on purification of density matrix method performed better in terms of computational efficiency.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2016
Series
TRITA-MAT-A, 2016:09
Keyword
Numerical Methods, Molecular Dynamics, Nearly Crossing Potential Surfaces, Error Estimation, Adaptive Algorithm
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-195098 (URN)978-91-7729-157-2 (ISBN)
Public defence
2016-12-09, D2, Lindstedtsvägen 5, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Swedish e‐Science Research Center
Note

QC 20161102

Available from: 2016-11-02 Created: 2016-11-01 Last updated: 2016-11-07Bibliographically approved

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