## The Doubly Nudged Elastic Band (DNEB) Method

The main problems with finding transition states and pathways using the nudged elastic band (NEB) method are associated with accuracy and efficiency. We have developed a framework for finding transition states that is based on the variable tradeoff between the two, and showed that in most cases the new approach is more than an order of magnitude faster. A short summary of the doubly nudged elastic band approach (DNEB) follows:

- * a version of the limited memory quasi-Newton method (L-BFGS) is used for minimization;
- * the NEB gradient is modified to include a portion of spring gradient perpendicular to the path to achieve stability during optimization;
- * overall rotation and translation on each image are not removed.

We use DNEB method with a revised connection algorithm that is an efficient way of constructing connections between distant minima using multiple DNEB transition state searches. It automatically optimizes transition state candidates from each DNEB run, constructs steepest-descent pathways and analyses the database of minima and transition states to choose which connections need to be established to produce a connected rearrangement pathway. The main features of our revised connection algorithm are as follows:

- * decision as to which minima to connect is based on the Euclidean distance;
- * DNEB settings (number of images and number of iterations) are controlled by image and iteration densities;
- * distance between the endpoints is minimized and initial resources are assigned based on the endpoint separation;
- * we use Dijkstra algorithm to find the shortest unconnected pathway at every cycle of connection algorithm.

This work was described in detail in:

S. A. Trygubenko and D. J. Wales, 'A Doubly Nudged Elastic Band Method for Finding Transition States' [JCP Online] [arXiv]

J. M. Carr, S. A. Trygubenko and D. J. Wales, 'Finding pathways between distant local minima' [JCP Online] [arXiv]

As of 2020, benchmarks of DNEB, its predecessors and successors are available in a more structured form, thanks to the efforts of the group of Graeme Henkelman, creator of NEB.

## Test cases: rearrangements of 38- and 75-atom Lennard-Jones clusters

The doubly nudged elastic band method and connection algorithm were tested on a number of rearrangements of Lennard-Jones clusters. For this purpose we selected rearrangements between the two lowest-energy minima of the LJ_{38} and LJ_{75} clusters. The potential energy surfaces of these are known to have a double funnel morphology and for both clusters the two lowest energy minima are very distinct structurally.

The endpoint geometries for each cluster as well as output information are available for download. There are two sets of endpoints for each rearrangement, which correspond to different Euclidean separations. Each set is labeled with value of the separation, which is given in the units of σ. The endpoints chosen for these tests are not the ones that give the shortest Euclidean distances, even though the paper states so.

The geometries of the endpoints and the pathways are in the XYZ format (.xyz files) and can be visualized using the XMakemol program. For each pathway the potential energy as a function of the integrated path length is supplied in .eofs files. The energy profiles can be plotted using Gnuplot.

Please note that pathways available here are simply the ones that were found with DNEB/Connect algorithm starting from a band obtained with linear interpolation. These pathways are neither the the shortest nor they are the fastest.