A spring pair method of finding saddle points using the minimum energy path as a compass

TL;DR

The paper introduces the spring pair method (SPM), a Hessian-free approach that efficiently finds index-1 saddle points by evolving a spring-coupled particle pair guided by the minimum energy path, improving reliability over traditional methods.

Contribution

The spring pair method (SPM) is a novel, Hessian-free technique that accurately locates saddle points by aligning with the minimum energy path using a coupled particle system.

Findings

SPM effectively finds saddle points in high-dimensional systems.

SPM outperforms traditional surface walking methods in reliability.

Validated on Lennard-Jones clusters and quasicrystal phase transitions.

Abstract

Finding index-1 saddle points is crucial for understanding phase transitions. In this work, we propose a simple yet efficient approach, the spring pair method (SPM), to accurately locate saddle points. Without requiring Hessian information, SPM evolves a single pair of spring-coupled particles on the potential energy surface. By cleverly designing complementary drifting and climbing dynamics based on gradient decomposition, the spring pair converges onto the minimum energy path (MEP) and spontaneously aligns its orientation with the MEP tangent, providing a reliable ascent direction for efficient convergence to saddle points. SPM fundamentally differs from traditional surface walking methods, which rely on the eigenvectors of Hessian that may deviate from the MEP tangent, potentially leading to convergence failure or undesired saddle points. The efficiency of SPM for finding saddle points is verified by ample examples, including high-dimensional Lennard-Jones cluster rearrangement and the Landau energy functional involving quasicrystal phase transitions.