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A numerical comparison between Ehrenfest dynamics and purification of the density matrix method for Born-Oppenheimer dynamics
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
(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 [en]
molecular dynamics, Ehrenfest dynamics, Born-Oppenheimer dynamics
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-195097OAI: oai:DiVA.org:kth-195097DiVA: diva2:1044047
Funder
Swedish e‐Science Research Center
Note

QC 20161103

Available from: 2016-11-01 Created: 2016-11-01 Last updated: 2016-11-03Bibliographically approved
In thesis
1. 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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Citation style
  • apa
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