One of the two most important results of this chapter is probably the doubly nudged elastic band formulation, in which a portion of the spring gradient perpendicular to the path is retained. With this modification we found that L-BFGS minimisation of the images is stable, thus providing a significant improvement in efficiency. Constraints such as elimination of overall rotation and translation are not required, and the DNEB/L-BFGS method has proved to be reliable for relatively complicated cooperative rearrangements in a number of clusters.
In comparing the performance of the L-BFGS and quenched velocity Verlet (QVV) methods for optimising chains of images we have also investigated a number of alternative QVV schemes. We found that the best approach is to quench the velocity after the coordinate update, so that the velocity response to the new gradient lags one step behind the coordinate updates. However, this slow-response QVV (SQVV) method does not appear to be competitive with L-BFGS.
We have revised our previous scheme  for constructing connections between distant minima using multiple transition state searches. Previously we have used an NEB/L-BFGS framework for this purpose, with eigenvector-following refinement of transition state candidates and characterisation of the connected minima using energy minimised approximations to the steepest-descent paths [133,8]. When the DNEB/L-BFGS approach is used we have found that it is better to spend more effort in the DNEB phase of the calculation, since a number of good transition state guesses can often be obtained even when the number of images is relatively small. In favourable cases a complete path linking the required endpoints may be obtained in one cycle. Of course, this was always the objective of the NEB approach [29,28,27,30], but we have not been able to achieve such results reliably for complex paths without the current modifications.
When a number of transition states are involved we still find it more efficient to build up the overall path in stages, choosing endpoints that become progressively closer in space. This procedure has been entirely automated within the OPTIM program, which can routinely locate complete paths for highly cooperative multi-step rearrangements, such as those connecting different morphologies of the LJ and LJ clusters.
For complex rearrangements the number of elementary steps involved may be rather large, and new methods for constructing an initial path are needed. This path does not need to be the shortest, or the fastest, but it does need to be fully connected. The second most important result of this chapter is probably a connection procedure based upon Dijkstra's shortest path algorithm, which enables us to select the most promising paths that include missing connections for subsequent double-ended searches. We have found that this approach enables initial paths containing more than a hundred steps to be calculated automatically for a variety of systems. Results have been presented for trpzip peptides here and for buckminsterfullerene, the GB1 hairpin and the villin headpiece subdomain elsewhere . These paths will be employed to seed future discrete path sampling calculations. This approach greatly reduces the computational demands of the method, and has allowed us to tackle more complicated problems. The new algorithm has also been implemented within our OPTIM program, and a public domain version is available for download from the Internet .