In practice we represent the imaginary-time trajectories using a ring-polymer discretization, such that the instanton can be optimized using a saddle-point search in the extended space of the Cartesian coordinates of the ring-polymer beads and the τ variable simultaneously. Unlike standard transition-state searches, however, an instanton in the inverted regime constitutes a high-index saddle point. Because the required index is known precisely in terms of the number of degrees of freedom and of the number of ring-polymer beads used to discretize the product state, efficient eigenvector-following algorithms can be employed. Once the optimal tunneling pathway has been located, the thermal rate constant predicted by the semiclassical instanton (SCI) approximation to Fermi's golden rule can be calculated as k