Energy landscapes: calculating pathways and rates

The stationary points of a potential energy surface provide a convenient framework for coarse-graining calculations of thermodynamics and kinetics. Thermodynamic properties can be extracted from a database of local minima using the superposition approach, where the total partition function is written as a sum over the contributions from each minimum. To analyse kinetics, we must also consider the transition states that link individual local minima, and evaluate rate constants for the corresponding elementary rearrangements. For small molecules the assignment of separate thermodynamic quantities, such as free energies, to individual isomers, and the notion of isomerisation rates between these structures, is usually straightforward. However, for larger systems the experimental states of interest generally correspond to sets of local minima with some common feature, such as a particular structural motif. This review focuses upon the discrete path sampling approach to obtaining phenomenological two-state rate constants between ensembles of local minima that are distinguished by suitable order parameters. Examples are discussed for atomic and molecular clusters, and for two peptides.