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Numerical Methods for Molecular Dynamics with Nearly Crossing Potential Surfaces
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). (Numerical Analysis)
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 [en]
Numerical Methods, Molecular Dynamics, Nearly Crossing Potential Surfaces, Error Estimation, Adaptive Algorithm
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-195098ISBN: 978-91-7729-157-2OAI: oai:DiVA.org:kth-195098DiVA: diva2:1044055
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
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 (ScopusID)
Note

Updated from Manuscript to Article. QC 20151012. QC 20160121

Available from: 2014-11-18 Created: 2014-11-18 Last updated: 2016-11-08Bibliographically 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
3. An Adaptive Mass Algorithm for Ehrenfest and Car-Parrinello ab initio Molecular Dynamics: dynamics with several electronic states
Open this publication in new window or tab >>An Adaptive Mass Algorithm for Ehrenfest and Car-Parrinello ab initio Molecular Dynamics: dynamics with several electronic states
(English)Manuscript (preprint) (Other academic)
Abstract [en]

This paper extends previous numerical studies on ficticious adaptive mass algorithms for Car-Parrinello and Ehrenfest dynamics to problems with more than two electron states. The main conclusion in this work is that it is necessary to resolve the near avoided conicial intersections between all electron eigenvalue gaps, also beween occupied states.

Keyword
molecular dynamics, adaptive algorithm
National Category
Computational Mathematics Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-195096 (URN)
Funder
Swedish Research Council, 621-2010-5647Swedish e‐Science Research Center
Note

QC 20161103

Available from: 2016-11-01 Created: 2016-11-01 Last updated: 2016-11-03Bibliographically approved
4. A numerical comparison between Ehrenfest dynamics and purification of the density matrix method for Born-Oppenheimer dynamics
Open this publication in new window or tab >>A numerical comparison between Ehrenfest dynamics and purification of the density matrix method for Born-Oppenheimer dynamics
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we numerically compare the computational efficiency between the Born-Oppenheimer dynamics based on purification of the density matrix, and the Ehrenfest molecular dynamics. In particular we study a set of problems when the ground state and excited state eigenvalues of the electronic energy surfaces come close to each other creating the so called near avoided conical intersections. The numerical results based on a simple model problem show that the Born-Oppenheimer molecular dynamics based on purification of the density matrix performs better compared to the Ehrenfest molecular dynamics. For instance, the Ehrenfest dynamics needs to resolve all spectral gaps, also between occupied states, whereas only the spectral gap between the highest occupied and lowest unoccupiedmolecular orbitals is required to be resolved for the Born-Oppenheimer molecular dynamics based on the purification of the density matrix method.

Keyword
molecular dynamics, Ehrenfest dynamics, Born-Oppenheimer dynamics
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-195097 (URN)
Funder
Swedish e‐Science Research Center
Note

QC 20161103

Available from: 2016-11-01 Created: 2016-11-01 Last updated: 2016-11-03Bibliographically approved

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