Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence of the Discrete Minimum Energy Path

Xuanyu Liu, Huajie Chen, Christoph Ortner

TL;DR

This work studies the convergence of discrete minimum energy paths (MEPs) obtained via the nudged elastic band (NEB) method. It formulates the continuous MEP problem with arc-length parameterization and analyzes its discretization, establishing an optimal convergence rate: the discrete MEP error satisfies $\|\bar{\varphi}-\bar{\phi}_M\| \le C M^{-1}$ in a discrete $C^1$-type norm, while the energy barrier error obeys $|\Delta E(\bar{\varphi})-\Delta_h E(\bar{\phi}_h)| \le C h^2$, with $h=1/M$. The proof hinges on a linearized stability analysis of the discrete MEP operator and a careful consistency/stability/inverse-function argument, under natural assumptions (A) and (B) about the energy landscape. Numerical experiments on toy, Muller, and Lennard-Jones-type systems corroborate the theoretical rates and illustrate the influence of the assumptions on observed convergence. The results provide a sharp, quantitative error bound for MEP discretization, informing image-count choices and offering a framework potentially extendable to other MEP methods such as the string method.

Abstract

The minimum energy path (MEP) describes the mechanism of reaction, and the energy barrier along the path can be used to calculate the reaction rate in thermal systems. The nudged elastic band (NEB) method is one of the most commonly used schemes to compute MEPs numerically. It approximates an MEP by a discrete set of configuration images, where the discretization size determines both computational cost and accuracy of the simulations. In this paper, we consider a discrete MEP to be a stationary state of the NEB method and prove an optimal convergence rate of the discrete MEP with respect to the number of images. Numerical simulations for the transitions of some several proto-typical model systems are performed to support the theory.

Convergence of the Discrete Minimum Energy Path

TL;DR

This work studies the convergence of discrete minimum energy paths (MEPs) obtained via the nudged elastic band (NEB) method. It formulates the continuous MEP problem with arc-length parameterization and analyzes its discretization, establishing an optimal convergence rate: the discrete MEP error satisfies in a discrete -type norm, while the energy barrier error obeys , with . The proof hinges on a linearized stability analysis of the discrete MEP operator and a careful consistency/stability/inverse-function argument, under natural assumptions (A) and (B) about the energy landscape. Numerical experiments on toy, Muller, and Lennard-Jones-type systems corroborate the theoretical rates and illustrate the influence of the assumptions on observed convergence. The results provide a sharp, quantitative error bound for MEP discretization, informing image-count choices and offering a framework potentially extendable to other MEP methods such as the string method.

Abstract

The minimum energy path (MEP) describes the mechanism of reaction, and the energy barrier along the path can be used to calculate the reaction rate in thermal systems. The nudged elastic band (NEB) method is one of the most commonly used schemes to compute MEPs numerically. It approximates an MEP by a discrete set of configuration images, where the discretization size determines both computational cost and accuracy of the simulations. In this paper, we consider a discrete MEP to be a stationary state of the NEB method and prove an optimal convergence rate of the discrete MEP with respect to the number of images. Numerical simulations for the transitions of some several proto-typical model systems are performed to support the theory.
Paper Structure (11 sections, 6 theorems, 103 equations, 12 figures)

This paper contains 11 sections, 6 theorems, 103 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Minimum energy path
  4. The discrete MEP
  5. Convergence of the discrete MEP
  6. Numerical experiments
  7. Conclusions
  8. Proofs: Convergence of the discrete MEP
  9. The abstract MEP and its discretization
  10. Stability of the discrete MEP
  11. Convergence analysis

Key Result

Theorem 2.1

Let $\bar{\varphi}\in C^3( [0,1];\mathbb{R}^{N} )$ be a solution of mep satisfying Assumptions (A) and (B). Then, for sufficiently small $h$ (equivalently, sufficiently large $M$), there exists a solution $\bar{\phi}_h$ of dMEP:NEB such that where $C_{\rm p}$ and $C_{\rm e}$ are positive constants depending only on $E$ and $\bar{\varphi}$.

Figures (12)

  • Figure 2.1: (Example 1) The contour lines of the energy landscape for $E$, with two minimizers and the MEP (red line).
  • Figure 2.2: (Example 1) The convergence of the discrete MEP ($\sigma_A$ is simple and lowest eigenvalue).
  • Figure 2.3: (Example 1) The contour lines of the energy landscape for $E_1$, with two minimizers and the MEP (red line).
  • Figure 2.4: (Example 1) The convergence of the discrete MEP ($\sigma_A$ is a degenerated eigenvalue).
  • Figure 2.5: (Example 1) The contour lines of the energy landscape for $E_2$, with two minimizers and the MEP (red line).
  • ...and 7 more figures

Theorems & Definitions (14)

  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 4.1
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 4 more