Explore Transition States on Energy Surface: Numerical Analysis and Algorithmic Developments

Many important physical or chemical events such as phase transition, protein folding or macromolecular conformational changes are typical examples of transition processes between metastable states (local minima) on potential energy surface in very high dimension. At low temperatures, these transitions rarely occur, but once they occur, they happen in the way of crossing the relevant transition states by surmounting significant energy barriers. These transition states supply the key chemical or physical information like the energy barriers and the transition rates which can be calculated from transition state theory. When the system is extremely complex to have a very rugged potential energy surface, the system may show a smoother free energy surface obtained by some dimension reduction procedure. Transition states on this free energy surface, rather than those on the original potential energy landscape, will o?er clearer insights of the underlying transition processes for these complex systems. Mathematically, these transition states are index-1 saddle points on the energy surface in R3N where N is the number of atoms. Such high dimensionality challenges numerical search for these index-1 saddle points. Standard Newton-Raphson methods for root-finding may su?er from extraordinary cost of computation. The natural relaxation of the system, such as the conventional molecular dynamics simulation, is helpful in finding local metastable states, but not possible in locating saddle points. PI proposed a new dynamics for the system, gentlest ascent dynamics (GAD), so that the relaxation of this GAD will converge to index-1 saddle points. This new dynamics is motivated by the methodology of min-mode-following, where only the minimum mode of the Hessian is required. In this project, our main goal is to propose a novel algorithmic framework for the searching of index-1 saddle points that are capable of accelerating the existing numerical methods from linear convergence to quadratic or super-linear convergence while maintaing the same computational cost. Our framework is an iterative minimisation scheme with quadratic iterative rate and is also sufficiently general to rediscover and analyse many existing algorithms from traditional viewpoint of numerical analysis. Optimal numerical implementation for this framework will be investigated in this project and applied to both potential energy landscape (without dimension reduction) and the free energy surface (with dimension reduction). We also address some important issues related to real applications such as initialization.