Exploiting symmetry to speed up symmetric nudge elastic band calculations

Many transitions of interest such as diffusion mechanisms in crystalline structures (vacancy, defect and polaron migration), rotations of ligands on molecules, and even reactions at surfaces (surface migration) there is a reflection relating the initial and final geometries of the reaction. Finding minimum energy paths (MEPs) of thermally activated transitions in such systems with band methods like the nudge elastic band (NEB, which for more than 20 years has been the most popular method), can be speed up if the symmetry is exploited. Here we present the formal properties required from a system to guarantee a reflection symmetric MEP. We present an implementation to prepare and effectively speed up nudge elastic band calculations through symmetry considerations. The implementation contains methods to identify symmetry relations between two given atomic geometries that can be used for many other purposes. The method can also be combined with many other implementations intended for speeding up nudge elastic band calculations. The implementation is validated through examples, showing that the computational time for a NEB calculation can be reduced by a factor between 2 and 10, depending on the considered system.