Local Search Methods

The automated surface walking algorithm is one of the original saddle point search methods. With local quadratic approximations of PES, the search is based on eigenvectors of the Hessian matrix, which is updated iteratively similar to the quasi-Newton technique. It also involves the scaling of one of the active coordinates in order to make the Hessian eigenvalues lie in a required range. In the partitioned rational function optimization method, PES is approximated by rationalized quadratic surfaces, and the search is conducted by partitioning the minimax problem into two separate maximization and minimization subproblems.

References: Simons, J., Joergensen, P., Taylor, H., and Ozment, J. (1983) Walking on Potential Energy Surfaces. J. Phys. Chem., 87: 2745- 2753 Banerjee, A., Adams, N., Simons, J., and Shepard, R. (1985) Search for Stationary Points on Surfaces. J. Phys. Chem., 89: 52-57

More recently, the ridge method searches the saddle point by traveling down along the ridge with a pair of images. The direction of the connecting line between the two images can be constrained to make sure the pair is kept on the ridge. Similarly, the dimer method searches the saddle point based on a pair of images. However, the small distance between the two images is fixed. Starting from one basin, the dimer moves uphill in the translation step. In the rotation step, it rotates towards the lowest curvature mode of PES by minimizing the dimer energy through the conjugate gradient approach. Later in the improved dimer method, the curvature is calculated differently with reduced numbers of gradient calculations to improve the performance and robustness under numerical noises. In the synchronous transit method, the transition state is initially estimated by minimizing the interpolated inter-atomic distances. Then the saddle point estimate is further refined with the conjugate gradient optimization. In the reduced gradient following method and the reduced potential energy surface model method, stationary points are intersections of zero-gradient curves and surfaces respectively with distinguished coordinates removed. Saddle point search is within the subspace of these zero-gradient curves or surfaces.

References: Ionova, I.V. and Carter, E.A. (1993) Ridge method for finding saddle points on potential energy surfaces. J. Chem. Phys., 98: 6377-6386 Henkelman, G. and Jónsson, H. (1999) A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives. J. Chem. Phys., 111(15): 7010-7022 Heyden, A., Bell, A.T., and Keil, F.J. (2005) Efficient methods for finding transition states in chemical reactions: Comparison of improved dimer method and partitioned rational function optimization method. J. Chem. Phys., 123: 224101-224114 Halgren, T.A. and Lipscomb, W.N. (1977) The synchronous-transit method for determining reaction pathways and locating molecular transition states. Chemical Physics Letters, 49(2): 225-232 Govind, N., Petersen, M., Fitzgerald, G., King-Smith, D., and Andzelm, J. (2003) A generalized synchronous transit method for transition state location. Computational Materials Science, 28: 250–258 Quapp, W., Hirsch, M., Imig, O., and Heidrich, D. (1998) Searching for Saddle Points of a Potential Energy Surface by Following a Reduced Gradient. J Comput Chem, 19: 1087-1100 Hirsch, M. and Quapp, W. (2002) Improved RGF Method to Find Saddle Points. J. Comput. Chem., 23: 887-894 Anglada, J.M., Besalú, E., Bofill, J.M., and Crehuet, R. (2001) On the Quadratic Reaction Path Evaluated in a Reduced Potential Energy Surface Model and the Problem to locate Transition States. J. Comput. Chem., 22: 387-406 Rothman, M.J. and Lohr, L.L. (1980) Analysis of an energy minimization method for locating transition states on potential energy hypersurfaces. Chem. Phys. Lett., 70(2): 405-409 Scharfenberg, P. (1982) An improved method for the evaluation of transition states. J Comput Chem, 3: 277-282

Global Search Methods

Different from local search, the global search methods ensure the saddle point with the maximum energy is located if the search converges to one. The DHS method searches the saddle point by iteratively reducing the distance between reactant and product images while minimizing the energy subject to the equal distance constraint at each step. The Activation-Relaxation Technique searches saddle points in two steps. In the activation step, one image jumps from a local minimum towards a saddle point according to a controlled force. In the relaxation step, it moves from the saddle point to another minimum. Thus it can traverse many saddle points without the knowledge of final product. In the Step and Slide method, the energy levels of two images from the initial and final states are gradually increased. Then the distance between the two is minimized while both images are kept in the same isoenergy surfaces. The interval Newton’s method finds all stationary points by solving the equation of vanishing gradient.

References: Dewar, M.J.S., Healy, E.F., and Stewart, J.J.P. (1984) Location of Transition States in Reaction Mechanisms. J. Chem. Soc., Faraday Trans., 2(80): 227 Mousseau, N. and Barkema, G.T. (1998) Traveling through potential energy landscapes of disordered materials: The activation-relaxation technique. Physical Review E, 57(2): 2419-2424 Miron, R.A. and Fichthorn, K.A. (2001) The Step and Slide method for finding saddle points on multidimensional potential surfaces. J. Chem. Physics, 115(19): 8742-8747 Lin, Y. and Stadtherr, M.A. (2004) Locating stationary points of Sorbate-Zeolite potential energy surfaces using interval analysis. J. Chem. Phys., 121(20): 10159-10166