sequences from an initial path. The end minima that must be linked in such calculations may be separated by relatively large distances, and a detailed algorithm was described for building up a connected path using successive transition state searches. The performance of the DNEB/L-BFGS approach is sufficiently good that we have changed this connection strategy in our OPTIM program. In particular, the DNEB/L-BFGS method can often provide good candidates for more than one transition state at a time, and may even produce all the necessary transition states on a long path. However, it is still generally necessary to consider multiple searches between different minima in order to connect a pair of endpoints. In particular, we would like to use the minimum number of NEB images possible for reasons of efficiency, but automate the procedure so that it eventually succeeds or gives up after an appropriate effort for any pair of minima that may arise in a discrete path sampling run. These calculations may involve the construction of many thousands of discrete paths. As in previous work we converge the NEB transition state candidates using eigenvector-following techniques and then use L-BFGS energy minimisation to calculate approximate steepest-descent paths. These paths usually lead to local minima, which we also converge tightly. The combination of NEB and hybrid eigenvector-following techniques [22,23] is similar to using NEB with a `climbing image' as described in Reference henkelmanuj00. The initial parameters assigned to each DNEB run are the number of images and the number of iterations, which we specify by image and iteration densities. The iteration density is the maximum number of iterations per image, while the image density is the maximum number of images per unit distance. The distance in question is the Euclidean separation of the endpoints, which provides a crude estimation of the integrated path length. This approach is based on the idea that knowing the integrated path length, which means knowing the answer before we start, we could have initiated each DNEB run with the same number of images per unit of distance along the path. In general it is also impossible to provide a lower bound on the number of images necessary to fully resolve the path, since this would require prior knowledge of the number of intervening stationary points. Our experience suggests that a good strategy is to employ as small an image and iteration density as possible at the start of a run, and only increase these parameters for connections that fail.
All NEB images, 
, for which 
are considered for further EF refinement. The resulting distinct transition states are stored in a database and the corresponding energy minimised paths were used to identify the minima that they connect. New minima are also stored in a database, while for known minima new connections are recorded. Consecutive DNEB runs aim to build up a connected path by progressively filling in connections between the endpoints or intermediate minima to which they are connected. This is an advantageous strategy because the linear interpolation guesses usually become better as the separation decreases, and therefore fewer optimisation steps are required. Working with sections of a long path one at a time is beneficial because it allows the algorithm to increase the resolution only where it is needed. Our experience is that this approach is generally significantly faster than trying to characterise the whole of a complex path with a single chain of images.
When an overall path is built up using successive DNEB searches we must select the two endpoints for each new search from the database of known minima. It is possible to base this choice on the order in which the transition states were found, which is basically the strategy used in our previous work [133,8]. We have found that this approach is not flexible or general enough to overcome difficulties that arise in situations when irrelevant transition states are present in the database. A better strategy is to connect minima based upon their Euclidean separation. For this purpose it is convenient to classify all the minima into those already connected to the starting endpoint (the S set),the final endpoint (the F set), and the remaining minima, which are not connected to either endpoint (the U set). The endpoints for the next DNEB search are then chosen as the two that are separated by the shortest distance, where one belongs to S or F, and the other belongs to a different set. The distance between these endpoints is then minimised with respect to overall rotation and translation, and an initial guess for the image positions is obtained using linear interpolation. Further details of the implementation of this algorithm and the OPTIM program are available online [138].