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Reaction mechanisms are an important tool for chemists in the determination of thermodynamic and kinetic properties of chemical reactions.(Hanggi & Borkovec 1990; Heidrich 1995; March 1992; Tolman 1925) The mechanisms are integral in the understanding of detailed molecular or chemical transitions from one equilibrium state (reactant) to another equilibrium state (product). In computational chemistry, the reaction mechanism is often represented as a reaction path on the Born-Oppenheimer potential energy surface (PES) of the system of interest through construction of a potential energy function of the nuclear coordinates.(Bader & Gangi 1975; Lewars 2011; Mezey 1987; Truhlar 2001; Wales 2003) The PES serves as an important theoretical construct to provide a framework to describe the transition between different states in detail. The equilibrium states correspond to local minimum on the PES with zero first order derivatives (gradient) in all directions and all positive eigenvalues of the second order derivative (Hessian) matrix, excluding rotation and translation degrees of freedom. The transition states (TSs), based on the transition state theory (TST),(Doll 2005; Eyring 1935; Laidler & King 1983; Pechukas 1981; Truhlar et al. 1983; Wigner 1938; Yamamoto 1960) are the first order saddle points with zero gradient and only one negative eigenvalue of the Hessian matrix. The equilibrium states are often easy to identify through experimental or computational studies. Understanding the detailed transition process between equilibrium states is of more interest in research, but unfortunately is very difficult to study experimentally. On a given PES, one can imagine that there could exist an infinite number of possible routes connecting two predefined states on that surface. However, not every route has the same weight in elucidating of reaction mechanisms. In the static point of view, the minimum energy path (MEP) is the route that needs the least amount of potential energy for the system to undergo the transition. The MEP connecting two local minima must go through one or more TSs, and is identified as a representative reaction path. In the statistical point of view, the minimum free energy path (MFEP) is the most probable transition path connects two metastable states. The simulation of the systems either through molecular dynamics (MD) or Monte Carlo (MC) sampling on the PES could generate an ensemble of transition paths, from which the MFEP can be identified. Both an MEP and an MFEP can be used to predict important properties, such as a reaction’s kinetic isotope effect.

Reaction Path Optimization and Sampling Methods and Their Applications for Rare Events