Our principal concern in this chapter is the development of the double-ended NEB approach [29,28,27,30]. The earliest double-ended methods were probably the linear and quadratic synchronous transit algorithms (LST and QST) [125], which are entirely based on interpolation between the two endpoints. In LST the highest energy structure is located along the straight line that links the two endpoints. QST is similar in spirit, but approximates the reaction path using a parabola instead of a straight line. Neither interpolation is likely to provide a good estimate of the path except for very simple reactions, but they may nevertheless be useful to generate initial guesses for more sophisticated double-ended methods.

Another approach is to reduce the distance between reactant and product by some arbitrary value to generate an `intermediate', and seek the minimum energy of this intermediate structure subject to certain constraints, such as fixed distance to an endpoint. This is the basis of the `Saddle' optimisation method [126] and the `Line Then Plane' [127] algorithm, which differ only in the definition of the subspace in which the intermediate is allowed to move. The latter method optimises the intermediate in the hyperplane perpendicular to the interpolation line, while `Saddle' uses hyperspheres. The minimised intermediate then replaces one of the endpoints and the process is repeated.

There are also a number of methods that are based on a `chain-of-states' (CS) approach, where several images of the system are somehow coupled together to create an approximation to the required path. The CS methods mainly differ in the way in which the initial guess to the path is refined. In the `Chain' method [128] the geometry of the highest energy image is relaxed first using only the component of the gradient perpendicular to the line connecting its two neighbours. The process is then repeated for the next-highest energy neighbours. The optimisation is terminated when the gradient becomes tangential to the path. The `Locally Updated Planes' method [129] is similar, but the images are relaxed in the hyperplane perpendicular to the reaction coordinate, rather than along the line defined by the gradient, and all the images are moved simultaneously.

The NEB approach introduced some further refinements to these CS methods [30]. It is based on a discretised representation of the path originally proposed by Elber and Karplus [88], with modifications to eliminate corner-cutting and sliding-down problems [27], and to improve the stability and convergence properties [29]. Maragakis *et al.* applied the NEB method to various physical systems ranging from semiconductor materials to biologically relevant molecules. They report that use of powerful minimisation methods in conjunction with the NEB approach was unsuccessful [107]. These problems were attributed to instabilities with respect to the extra parameters introduced by the springs.

The main result of the present contribution is a modified `doubly nudged' elastic band (DNEB) method, which is stable when combined with the L-BFGS minimiser. In comparing the DNEB approach with other methods we have also analysed quenched velocity Verlet minimisation, and determined the best point at which to remove the kinetic energy. Extensive tests show that the DNEB/L-BFGS combination provides a significant performance improvement over previous implementations. We therefore outline a new strategy to connect distant minima, which is based on successive DNEB searches to provide transition state candidates for refinement by eigenvector-following.