Methods to locate Saddle Points in Complex Landscapes

TL;DR

The paper introduces simple, efficient algorithms for locating saddle points in complex energy landscapes without prior knowledge or second derivative calculations, using only potential values and gradients.

Contribution

It presents a novel approach that uses two nearby points and weak noise to find saddle points, including those not reachable by traditional slowest ascent methods.

Findings

Successfully finds correct saddle points on Muller-Brown surface.

Effectively locates low-lying saddle points with high probability.

Computational effort scales linearly with degrees of freedom.

Abstract

We present a class of simple algorithms that allows 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 configuration space to locate the path of slowest ascent. By introducing a weak noise term, the algorithm is able to find even low-lying saddle points that are not reachable by means of a slowest ascent path. Since the algorithm makes only use of the value of the potential and its gradient, the computational effort to find saddles is linear in the number of degrees of freedom, if the potential is short-ranged. We test the performance of the algorithm for two potential energy landscapes. For the M\"uller-Brown surface we find that the algorithm always finds the correct saddle point. For the modified M\"uller-Brown surface, which has a saddle point that is not reachable by means of a slowest ascent path, the algorithm is still able to find this saddle point with high probability.