Methods to locate saddle points in complex landscapes

We present a class of simple algorithms that allows us to find the reaction path in systems with a complex potential energy landscape. The approach does not need any knowledge on the product state and does not require the calculation of any second derivatives. The underlying idea is to use two nearby points in the configuration space to locate the path of the slowest ascent. By introducing a weak noise term, the algorithm is able to find even low-lying saddle points that are not directly reachable by means of the slowest ascent path. Since the algorithm only makes use of the value of the potential and its gradient, the computational effort to find saddle points is linear in the number of degrees of freedom if the potential is short-ranged. We test the performance of the algorithm for three potential energy landscapes. For the Müller-Brown surface, we find that the algorithm always finds the correct saddle point. For the modified Müller-Brown surface, which has a saddle point that is not reachable by means of the slowest ascent path, the algorithm is still able to find this saddle point with high probability. For the case of a three-dimensional Lennard-Jones cluster, the algorithm is able to find the lowest energy barrier with high probability, showing that the method is also efficient in landscapes with many dimensions.