Optimization-based Shrinking Dimer Method for Finding Transition States

We present a new and efficient optimization-based shrinking dimer (OSD) algorithm for finding transition states associated with a given energy landscape. The transition states are given by index-one saddle points of the energy surface. By constructing the rotation and translation steps in the classical dimer method under an optimization framework, we are able to take advantage of more powerful optimization methods to substantially speed up the computation of transition states. Specifically, the Barzilai--Borwein gradient method is proposed as an effective implementation of OSD. We show that the OSD method is the generalized formulation of the original shrinking dimer dynamics (SDD) and enjoys superlinear convergence. We test various numerical examples, including some standard lower dimensional test examples, a cluster of seven particles, seven-atom island on the (111) surface of an FCC crystal, and nucleation in phase transformations. The results demonstrate that the OSD method with the proposed Barzilai--Borwein stepsize is very effective and provides a more efficient implementation than the SDD algorithm and some other dimer-type methods. Our new algorithms offer potential to significantly advance the state-of-the-art in dimer-type methods for transition state search.