Shrinking Dimer Dynamics and Its Applications to Saddle Point Search

Saddle point search on an energy surface has broad applications in fields like materials science, physics, chemistry, and biology. In this paper, we present the shrinking dimer dynamics (SDD), a dynamic system which can be applied to locate a transition state on an energy surface corresponding to an index-1 saddle point where the Hessian has a negative eigenvalue. By searching for the saddle point and the associated unstable direction simultaneously in a single dynamic system defined in an extended space, we show that unstable index-1 saddle points of the energy become linearly stable steady equilibria of the SDD which makes the SDD a robust approach for the computation of saddle points. The time discretization of the SDD is connected to various iterative algorithms, including the popular dimer method used in many practical applications. Our study of these discretization schemes lays a rigorous mathematical foundation for the corresponding iterative saddle point search algorithms. Both linear local asymptotic stability analysis and optimal error reduction (convergence) rates are presented and further confirmed by numerical experiments. Global convergence and nonlinear asymptotic stability are also illustrated for some simpler systems. Applications of the SDD in both finiteand infinite-dimensional energy spaces are discussed.