A method for finding the ridge between saddle points applied to rare event rate estimates

A method is presented for finding the ridge between first order saddle points on a multidimensional surface. For atomic scale systems, such saddle points on the energy surface correspond to atomic rearrangement mechanisms. Information about the ridge can be used to test the validity of the harmonic approximation to transition state theory, in particular to verify that second order saddle points—maxima along the ridge—are high enough compared to the first order saddle points. New minima along the ridge can also be identified during the path optimisation, thereby revealing additional transition mechanisms. The method is based on a string of discretisation points along a path between the first order saddle points and using an iterative optimisation which requires only the force acting on the atoms. At each iteration during the optimisation, the force is inverted along an unstable eigenmode perpendicular to the path. The method is applied to Al adatom diffusion on the Al(100) surface to find the ridge between 2-, 3- and 4-atom concerted displacements and hop mechanisms. A correction to the harmonic approximation of transition state theory was estimated by direct evaluation of the configuration integral along the ridge.