A Climbing Multi-String Method to Map Free-Energy Saddles and Minima

Finding stationary points, minima or saddles, on hyperdimensional free energy surface is crucial to gain insights about the possible transition events and to compute the associated transition rates. Though minima can be easily found, the tasks of locating saddles and measuring their energies relative to their associated minima remain challenging, especially in high-dimensional spaces. We propose here a modified climbing string method to locate multiple saddles by initiating multiple strings in one go. For a given minimum, strings propagate by gradient flow in the path space, with one end fixed and other end climbing across the free energy profile to locate a saddle. The convergence of multiple strings to a common saddle is avoided by checking repulsive forces between the climbing ends. The presence of multiple images along the string helps to ensure the direct connectivity between the minimum and saddles and to compute the free energy along the path. Hence, the climbing multi-string method can be used to map the network of directly connected stationary points in the free energy hypersurface. We demonstrate this method to locate saddle points in two-dimensional and four-dimensional free energy surface of alanine dipeptide and alanine tripeptide, respectively.