Basin constrained κ-dimer method for saddle point finding

Within the harmonic approximation to transition state theory, the rate of escape from a reactant is calculated from local information at saddle points on the boundary of the state. The dimer minimum-mode following method can be used to find such saddle points. But as we show, dimer searches that are initiated from a reactant state of interest can converge to saddles that are not on the boundary of the reactant state. These disconnected saddles are not directly useful for calculating the escape rate. Additionally, the ratio of disconnected saddles can be large, especially when the dimer searches are initiated far from the reactant minimum. The reason that the method finds disconnected saddles is a result of the fact that the dimer method tracks local ridges, defined as the set of points where the force is perpendicular to the negative curvature mode, and not the true ridge, defined as the boundary of the set of points which minimize to the reactant. The local ridges tend to deviate from the true ridge away from saddle points. Furthermore, the local ridge can be discontinuous and have holes which allow the dimer to cross the true ridge and escape the initial state. To solve this problem, we employ an alternative definition of a local ridge based upon the minimum directional curvature of the isopotential hyperplane, κ, which provides additional local information to tune the dimer dynamics. We find that hyperplanes of κ = 0 pass through all saddle points but rarely intersect with the true ridge elsewhere. By restraining the dimer within the κ < 0 region, the probability of converging to disconnected saddles is significantly reduced and the efficiency of finding connected saddles is increased.