1. Chain-of-States Methods

In chain-of-states methods, the transition pathway is represented as a number of intermediate states, which can be looked as snapshots of configurations as the atomic structure transforms from initial to final state along the transition pathway. After the search converges, the discrete states are chained to each other, usually by interpolating between the states, to obtain the transition pathway and the saddle point. They work well in transitions where there may be more than one saddle point. In situations where there may be multiple transition pathways, the methods will converge to the pathway closest to the initial guess for the transition pathway.

Nudged Elastic Band (NEB)

The Nudged Elastic Band (NEB) method and its extensions are among the most used and well-developed chain of states methods. NEB is an extension of the plain elastic band (PEB) method, where images, which are points in the configuration space corresponding to intermediate states, are connected by springs. In PEB, images move subject to both the true force due to the gradient of potential energy and the spring force. The relaxed band should converge to the MEP. However, when the spring constant is large enough, the perpendicular components of spring forces (with respect to the direction of path) pull images away from saddle points at the sharp turns of the path. This corner cutting makes PEB overestimate the saddle point energy. On the other hand, when the spring constant is small, the parallel components of true forces make images slide down towards the minima. This sliding down reduces the resolution of the region of interest (ROI) near the saddle point. The NEB method is to solve the problems of corner cutting and sliding down by removing the perpendicular component of spring force and the parallel component of true force in the total force on each image. The only effect of the springs now is to keep images evenly distributed within the path. Nevertheless, when the potential energy changes rapidly, the path has kinks where the parallel component of the energy gradient is much larger than the perpendicular one, because the restoring perpendicular components of forces are weak. Another shortcoming of NEB is that the actual saddle point may not be located by one of the images directly. These issues are solved by some further extensions of the NEB method.

References: Jonsson, H., Mills, G., and Jacobsen, K., 1998. Classical and Quantum Dynamics in Condensed Phase Simulations. World Scientific, Hackensack, NJ, Chap. 16, pp. 385-404. See also http://www.hi.is/~hj/paperNEBleri.pdf

The improved tangent NEB (IT-NEB) method reduces the chances of getting kinks by better estimating the tangent direction of the path to approximate MEP at each image. Instead of the central finite difference approximation between the image and its two neighbours as in the original NEB, only one neighbor is used for tangent estimation. The climbing image NEB (CI-NEB) method was further developed so that the image with the highest energy actively climbs up to locate the actual saddle point. An alternative to resolve the issue of kinks is the doubly nudged elastic band (DNEB) method where a manipulated perpendicular component of spring force is introduced back into the total force such that MEP can be restored. To locate the actual saddle points, the eigenvector following (EF) optimization approach can be applied to the result of NEB. To increase the resolution of ROI, adaptive spring constants can be applied such as energy-weighted or gradient-weighted. The original NEB method requires that the initial and final states should be known. A free end NEB (FENEB) method was recently proposed to keep one end of the band free so that it can swing at a fixed energy level, thus the number of images can be reduced while still maintaining enough resolution. Similar to the String method, Spline-based interpolation was also used in NEB.

References: Henkelman, G., and Jonsson, H., 2000. “Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points”, J. Chem. Phys., 113, pp. 9978-9985. Henkelman, G., Uberuaga, B., and Jonsson, H., 2000. “A climbing image nudged elastic band method for finding saddle points and minimum energy paths”, J. Chem. Phys., 113, pp. 9901-9904. Trygubenko, S., and Wales, D., 2004. “A doubly nudged elastic band method for finding transition states”, J. Chem. Phys., 120, 2082-2094. Zhu, T., Li, J., Samanta, A., Kim, H.G., and Suresh,S., 2007. "Interfacial plasticity governs strain rate sensitivity and ductility in nanostructured metals", Proceedings of the National Academy of Sciences of the USA, 104, 3031-3036 Galvan, I., and Field, M., 2008. “Improving the Efficiency of the NEB Reaction Path Finding Algorithm”, J. Comp. Chem., 29, pp. 139-143.

String Methods

The second group of chain-of-states methods is the String methods. The transition path is represented continuously as Splines. Subject to perpendicular forces the curve evolves and converges to the MEP alternately in two steps. In the evolution step the curve is discretized as a string of points and solved by standard ODE solvers. In the reparameterization step, the points are redistributed along the string based on parameterization constraints and Spline interpolations. Compared to NEB methods where the number of images is fixed during searching, the number of points in the String method can be dynamically modified based on needs. The Growing String method is an extension where the points of strings are initially located at the two ends of reactant and product. Then the string grows by inserting points so as to meet at the saddle point upon convergence. The quadratic String method is formulated based on a multi-objective optimization approach. Based on the local quadratic approximation of the PES, the quasi-Newton technique is applied to search the MEP.

References: E, W., Ren, W., and Vanden-Eijnden, E., 2002. “String method for the study of rare events”, Phys. Rev. B, 66, pp. 052301(1)-052301(4). Ren, W., 2003. “Higher order string method for finding minimum energy path”, Comm. Math. Sci., 1, pp. 377-384. E, W., Ren, W., and Vanden-Eijnden, E., 2007. “A Simplified and improved string method for computing the minimum energy paths in barrier-crossing events”, J. Chem. Phys., 126, pp. 164103(1) -164103(8). Peters, B., Heyden, A., Bell, A., and Chakraborty, A., 2004. “A growing string method for determining transition states: Comparison to the nudged elastic band and string methods”, J. Chem. Phys., 120, 7877-7886. Burger, S., and Yang, W., 2006. “Quadratic string method for determining the minimum-energy path based on multiobjective optimization”, J. Chem. Phys., 124, pp. 054109(1)- 054109(12).

2. Other Methods

In the conjugate peak refinement (CPR) method, saddle points and the approximated MEP are found by searching the maximum of one direction and the minima of all other conjugate directions iteratively, because exactly one eigenvalue of the Hessian matrix at the saddle points is negative. The accelerated Langevin dynamics (ALD) method is a stochastic transition path sampling method. Paths are sampled by solving the modified Langevin equation with negative friction coefficients to accelerate the transition. The MEP is the average of all sampled paths. The Hamilton-Jacobi (HJ) method generates the MEP by solving a Hamilton-Jacobi type equation with special cost functions.

References: Fischer, S., and Karplus, M., 1992. “Conjugate Peak Refinement: an algorithm for finding reaction paths and accurate transition states with many degrees of freedom”, Chem. Phys. Lett., 194, 252-261. Chen, L., Ying, C., and Ala-Nissila, T., 2002. “Finding transition paths and rate coefficients through accelerated Langevin dynamics”, Phys. Rev. E, 65, pp. 042101(1)-042101(4). Dey, B., and Ayers, P., 2006. “A Hamilton–Jacobi type equation for computing minimum potential energy paths”, Mol. Phys., 104, pp. 541–558.