Optimization-based Shrinking Dimer Method for Finding Transition States
Update: 2017-03-02 14:47:27      Author: yangjuan@csrc.ac.cn

Finding transition states on a complex energy landscape, that are given by index-one saddle points of the energy,is a challenging problem that arises in various scientific fields like physics, chemistry, biology and materials science[1]. It has attracted much attention in many interesting applications. A recent review on numerical algorithms for transition states can be found in [?].

In [3], Zhang and Du proposed the shrinking dimer dynamics (SDD) to compute index-1 saddle points, which may be viewed as the continuous limit of the original dimer method popularized by [4]. Inspired by the SDD, we propose a novel optimization-based shrinking dimer (OSD) method.

In the new OSD algorithm, 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 [5] 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. Various numerical examples are tested for both OSD and SDD methods, including some standard lower dimensional test examples, a cluster of seven particles,

and nucleation in phase transformations. The results demonstrate that the OSD method with the proposed Barzilai-Borwein step size is very effective and provides a more efficient implementation than the original SDD algorithm. Our new algorithms offer potential to significantly advance the state-of-the-art in transition state search.




Figure 1: Comparison of SDD and OSD for stingray and Eckhardt energies and a 7-atom island in an FCC crystal.

Lei Zhang (PKU)  Qiang Du  Zhenzhen Zheng (Postdoc, CSRC),SIAM J. Sci. Compu., 38, A528A544, 2016.


[1] D.Wales, Energy landscapes: applications to clusters, biomolecules and glasses, Cambridge Univ Press, 2003.

[2] L.Zhang, W.-Q.Ren, A.Samanta and Q.Du, Recent Developments in Computational Modeling of Nucleation in Phase Transformations, npj Computational Materials, 2 (2016), 16003

[3] J.Y.Zhang and Q.Du, Shrinking dimer dynamics and its applications to saddle point search, SIAM J. Numer. Anal., 50 (2012), 1899-1921.

[4] G.Henkelman and H.Jonsson, A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives, J. Chem. Phys., 111 (1999), 7010.

[5] J.Barzilai and J.Borwein, Two point step size gradient methods, IMA J Num Anal, 8 (1988), 141-148.

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