Bézier curve string method for the study of rare events in complex chemical systems

We present a new string method for finding the most probable transition pathway and optimal reaction coordinate in complex chemical systems. Our approach evolves an analytic parametric curve, known as a Bézier curve, to the most probable transition path between metastable regions in configuration space. In addition, we demonstrate that the geometric properties of the Bézier curve can be used to construct the optimal reaction coordinate near the most probable reaction path, and can further be used to devise a ranking vector capable of identifying precisely which collective variables are most important for governing the transition between metastable states. We discuss the algorithmic details of the Bézier curve string method, analyze its stability, accuracy and efficiency, and illustrate its capabilities using model potential energy functions. In particular, we use the degree elevation property of Bézier curves to develop an algorithm that adaptively learns the degree polynomial necessary to accurately represent the most probable transition path. Subsequently, we apply our method to the isomerization of alanine dipeptide, and demonstrate that the reaction coordinate obtained from the Bézier curve string method is in excellent agreement with the optimal reaction coordinate constructed from an aimless shooting and maximum likelihood procedure. Finally, we apply our method to a large complex system and study the homogenous nucleation of benzene from the melt. In these two examples, we illustrate that the ranking vector correctly identifies which collective variables govern these chemical transitions.