Ring-polymer instanton theory

Instanton theory provides a simple description of a quantum tunnelling process in terms of an optimal tunnelling pathway. The theory is rigorously based on quantum mechanics principles and is derived from a semiclassical approximation to the path-integral formulation. In multidimensional systems, the optimal tunnelling pathway is generally different from the minimum-energy pathway and is seen to cut the corner' around the transition state. A ring-polymer formulation of instanton theory leads to a practical computational method for applying the theory to describe, simulate and predict quantum tunnelling effects in complex molecular systems. It can be used to compute either the rate of a tunnelling process leading to a chemical reaction or the tunnelling splitting pattern of a molecular cluster. In this review, we introduce a unification of the theory's derivation and discuss recent improvements to the numerical implementation.