Inspired by the success of ring-polymer molecular dynamics (RPMD), we derive a transition-state-theory version (RPTST) with an optimal dividing surface which is, in general, conical. This explains why centroid-based quantum TSTs are inaccurate for asymmetric barriers at low temperatures. The geometry of the ring-polymer transition state describes a finite-difference approximation to the semi-classical instanton trajectory. Based on this, a new practical method for locating multidimensional instantons is proposed, by computing saddle points on the ring-polymer surface, and a method for computing the reaction rate constant is derived based on the ImF premise. We show that a free-energy version of the ImF instanton theory is related to RPTST and thereby provide an explanation for why RPMD produces accurate results in the deep-tunnelling regime. We also present a ring-polymer instanton derivation of a theory for calculating tunnelling splittings leading to a practical method to...
