A Stability Boundary Based Method for Finding Saddle Points on Potential Energy Surfaces

The task of finding saddle points on potential energy surfaces plays a crucial role in understanding the dynamics of a micromolecule as well as in studying the folding pathways of macromolecules like proteins. The problem of finding the saddle points on a high dimensional potential energy surface is transformed into the problem of finding decomposition points of its corresponding nonlinear dynamical system. This paper introduces a new method based on TRUST-TECH (TRansformation Under STability reTained Equilibria CHaracterization) to compute saddle points on potential energy surfaces using stability boundaries. Our method explores the dynamic and geometric characteristics of stability boundaries of a nonlinear dynamical system. A novel trajectory adjustment procedure is used to trace the stability boundary. Our method was successful in finding the saddle points on different potential energy surfaces of various dimensions. A simplified version of the algorithm has also been used to find the saddle points of symmetric systems with the help of some analytical knowledge. The main advantages and effectiveness of the method are clearly illustrated with some examples. Promising results of our method are shown on various problems with varied degrees of freedom.

[1]  Gg Balint-Kurti,et al.  Lecture notes in Chemistry , 2000 .

[2]  D. Heidrich,et al.  Properties of Chemically Interesting Potential Energy Surfaces , 1991 .

[3]  M. Karplus,et al.  Simulation of activation free energies in molecular systems , 1996 .

[4]  D. Heidrich,et al.  Searching for saddle points of potential energy surfaces by following a reduced gradient , 1998 .

[5]  J. Polanyi,et al.  Location of Energy Barriers. I. Effect on the Dynamics of Reactions A + BC , 1969 .

[6]  Hsiao-Dong Chiang,et al.  Finding saddle points using stability boundaries , 2005, SAC '05.

[7]  H. Chiang,et al.  Quasi-stability regions of nonlinear dynamical systems: theory , 1996 .

[8]  Eamonn F. Healy,et al.  Location of transition states in reaction mechanisms , 1984 .

[9]  T. Helgaker.  Transition-state optimizations by trust-region image minimization , 1991 .

[10]  A. Gordus,et al.  Single-molecule transition-state analysis of RNA folding , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[11]  H. Chiang,et al.  A systematic search method for obtaining multiple local optimal solutions of nonlinear programming problems , 1996 .

[12]  F. A. Seiler,et al.  Numerical Recipes in C: The Art of Scientific Computing , 1989 .

[13]  M. Karplus,et al.  Conjugate peak refinement: an algorithm for finding reaction paths and accurate transition states in systems with many degrees of freedom , 1992 .

[14]  R. Miron,et al.  The Step and Slide method for finding saddle points on multidimensional potential surfaces , 2001 .

[15]  G. Henkelman,et al.  A climbing image nudged elastic band method for finding saddle points and minimum energy paths , 2000 .

[16]  R. Jernigan,et al.  Equilibrium states of rigid bodies with multiple interaction sites: Application to protein helices , 1997 .

[17]  K. Dill,et al.  Phi values in protein-folding kinetics have energetic and structural components. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[18]  William H. Press,et al.  The Art of Scientific Computing Second Edition , 1998 .

[19]  A. I. Panin,et al.  Search for stationary points of arbitrary index by augmented Hessian method , 1995 .

[20]  G. Barkema,et al.  Identification of Relaxation and Diffusion Mechanisms in Amorphous Silicon , 1998, cond-mat/9804317.

[21]  K. Müller,et al.  Location of saddle points and minimum energy paths by a constrained simplex optimization procedure , 1979 .

[22]  H. Jónsson,et al.  Nudged elastic band method for finding minimum energy paths of transitions , 1998 .

[23]  Felix F. Wu,et al.  Stability regions of nonlinear autonomous dynamical systems , 1988 .

[24]  B. Eckhardt.  Irregular scattering , 1988 .

[25]  J. Ben Rosen,et al.  Protein structure prediction and potential energy landscape analysis using continuous global minimization , 1997, RECOMB '97.

[26]  P. Schleyer,et al.  Internal conrotation and disrotation in H2BCH2BH2 and diborylmethane 1,3 H exchange , 1997 .

[27]  R. Elber,et al.  Reaction path study of conformational transitions in flexible systems: Applications to peptides , 1990 .

[28]  E. Carter,et al.  Ridge method for finding saddle points on potential energy surfaces , 1993 .

[29]  G. Henkelman,et al.  Methods for Finding Saddle Points and Minimum Energy Paths , 2002 .

[30]  Barkema,et al.  Event-Based Relaxation of Continuous Disordered Systems. , 1996, Physical review letters.

[31]  J. Baker.  An algorithm for the location of transition states , 1986 .

[32]  Hsiao-Dong Chiang,et al.  TRUST-TECH-Based Expectation Maximization for Learning Finite Mixture Models , 2008, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[33]  J. Doye,et al.  Saddle Points and Dynamics of Lennard-Jones Clusters, Solids and Supercooled Liquids , 2001, cond-mat/0108310.

[34]  Hsiao-Dong Chiang,et al.  A dynamical trajectory-based methodology for systematically computing multiple optimal solutions of general nonlinear programming problems , 2004, IEEE Transactions on Automatic Control.

[35]  G. Henkelman,et al.  A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives , 1999 .