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

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 the Hessian information, the SPM evolves a single pair of spring-coupled particles on an energy surface. By 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. The SPM fundamentally differs from traditional surface walking methods that rely on the eigenvectors of the Hessian, which may deviate from the MEP tangent and potentially lead to convergence failure or undesired saddle points. The efficiency of the SPM for finding saddle points is verified by ample examples, including Lennard-Jones clusters, Morse clusters, water clusters, and the Landau energy functional involving quasicrystal phase transitions.