Efficient Sampling of Saddle Points with the Minimum-Mode Following Method

The problem of sampling low lying, first-order saddle points on a high dimensional surface is discussed and a method presented for improving the sampling efficiency. The discussion is in the context of an energy surface for a system of atoms and thermally activated transitions in solids treated within the harmonic approximation to transition state theory. Given a local minimum as an initial state and a small, initial displacement, the minimum-mode following method is used to climb up to a saddle point. The goal is to sample as many of the low lying saddle points as possible when such climbs are repeated from different initial displacements. Various choices for the distribution of initial displacements are discussed and a comparison made between (1) displacements along eigenmodes at the minimum, (2) purely random displacements with a maximum cutoff, and (3) Gaussian distribution of displacements. The last choice is found to give best overall results in two test problems studied, a heptamer island on a surface and a grain boundary in a metal. A method referred to as “skipping-path method” is presented to reduce redundant calculations when a climb heads towards a saddle point that has already been identified. The method is found to reduce the computational effort of finding new saddle points to as little as a third, especially when a thorough sampling is performed.