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A Posteriori Error Estimate and Adaptivity for QM/MM Models of Crystalline Defects

Yangshuai Wang, James R. Kermode, Christoph Ortner, Lei Zhang

TL;DR

A robust adaptive QM/MM method for practical material defect simulation, based on a developed residual-based error estimator originating from [54], is proposed, and the robustness of the adaptive algorithm on numerical simulations for various complex crystalline defects is validated.

Abstract

Hybrid quantum/molecular mechanics (QM/MM) models play a pivotal role in molecular simulations. These models provide a balance between accuracy, surpassing pure MM models, and computational efficiency, offering advantages over pure QM models. Adaptive approaches have been developed to further improve this balance by allowing on-the-fly selection of the QM and MM subsystems as necessary. We propose a novel and robust adaptive QM/MM method for practical material defect simulations. To ensure mathematical consistency with the QM reference model, we employ machine-learning interatomic potentials (MLIPs) as the MM models. Our adaptive QM/MM method utilizes a residual-based error estimator that provides both upper and lower bounds for the approximation error, thus indicating its reliability and efficiency. Furthermore, we introduce a novel adaptive algorithm capable of anisotropically updating the QM/MM partitions. This update is based on the proposed residual-based error estimator and involves solving a free interface motion problem, which is efficiently achieved using the fast marching method. We demonstrate the robustness of our approach via numerical tests on a wide range of crystalline defects.

A Posteriori Error Estimate and Adaptivity for QM/MM Models of Crystalline Defects

TL;DR

A robust adaptive QM/MM method for practical material defect simulation, based on a developed residual-based error estimator originating from [54], is proposed, and the robustness of the adaptive algorithm on numerical simulations for various complex crystalline defects is validated.

Abstract

Hybrid quantum/molecular mechanics (QM/MM) models play a pivotal role in molecular simulations. These models provide a balance between accuracy, surpassing pure MM models, and computational efficiency, offering advantages over pure QM models. Adaptive approaches have been developed to further improve this balance by allowing on-the-fly selection of the QM and MM subsystems as necessary. We propose a novel and robust adaptive QM/MM method for practical material defect simulations. To ensure mathematical consistency with the QM reference model, we employ machine-learning interatomic potentials (MLIPs) as the MM models. Our adaptive QM/MM method utilizes a residual-based error estimator that provides both upper and lower bounds for the approximation error, thus indicating its reliability and efficiency. Furthermore, we introduce a novel adaptive algorithm capable of anisotropically updating the QM/MM partitions. This update is based on the proposed residual-based error estimator and involves solving a free interface motion problem, which is efficiently achieved using the fast marching method. We demonstrate the robustness of our approach via numerical tests on a wide range of crystalline defects.
Paper Structure (30 sections, 60 equations, 16 figures, 2 algorithms)

This paper contains 30 sections, 60 equations, 16 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. QM/MM Coupling for Crystalline Defects
  3. Variational model for crystalline defects
  4. Displacement space
  5. The QM site potential
  6. Equilibration of crystalline defects
  7. QM/MM Coupling
  8. Domain decomposition
  9. Specification of MM model
  10. The QM/MM hybrid model
  11. A posteriori error estimates
  12. An idealised error estimator
  13. A practical error estimator
  14. Truncation
  15. Approximated residual force
  16. ...and 15 more sections

Figures (16)

  • Figure 1: Decomposition of (001)[100] edge dislocation in W into QM, MM, buffer (BUF), and far-field (FF) regions.
  • Figure 2: The distributions of the a posteriori error estimator (left) defined on $\Lambda_{\rm est}$ and the speed (right) defined on $\mathcal{N}_{\rm est}$.
  • Figure 3: The illustration of the "Mark & Refine" step plotted on $\mathcal{N}_{\rm est}$: current QM interface (red solid line), new QM interface (red dashed line) and new buffer interface (purple dashed line). Colors represent the normalization of error estimator (speed) defined on $\mathcal{N}_{\rm est}$.
  • Figure 4: QM/MM errors and error estimators plotted against $N_{\rm QM}$ in the adaptive QM/MM Algorithm \ref{['alg:main']}. Different $T_{\rm new}$ employed in Algorithm \ref{['alg:fmm']} can lead to qualitatively different behaviour in the adaptive computations.
  • Figure 5: Domain decomposition for in-plane crack in W.
  • ...and 11 more figures

Theorems & Definitions (3)

  • proof
  • proof
  • proof