Figure 3: The consequences of distortion symmetry and balanced forces for NEB calculations.

(a–d) Superimposed images along oxygen (red atom) diffusion paths on graphene (grey carbon atoms connected by grey bonds). In a and b, an initial linear path is assumed for the diffusion of a single oxygen atom from right (λ=−1) to left (λ=+1), across a C₆ graphene ring. The symmetry of the path in a and b is mm2; the symmetry traps the path and prevents convergence to a minimum-energy pathway (MEP). To break the mm2 symmetry, we perturb this linear path as mm2→2 and mm2→1, respectively, as shown schematically exaggerated as green curves in a and b and indicated by the text in the inset. (c,d) The final paths after NEB relaxation starting from the perturbed paths of a and b, respectively, as indicated by red vertical arrows. The paths c and d have distortion symmetries of 2 and m, respectively. The 2 symmetry continues to trap the transition state (just like mm2 did for the linear path), whereas the initial path with trivial symmetry can correctly converge to a MEP with m symmetry. (e) The calculated energies of the images and the interpolation provided by QE’s NEB module⁴⁵. (f) Results for an example two-dimensional potential energy surface inspired by the above problem, using a simple NEB implementation. The plots are smoothed and rescaled histograms showing the frequency of NEB convergence at a given number of iterations in this example system for 100,000 randomly generated initial paths each with m or with trivial symmetry of 1. The two curves are rescaled to the same maximum height. Symmetrizing using the correct symmetry, m (red curve) reduced the number of NEB iterations needed in 98.97% of test cases. The average reduction was ∼2.3 × as compared with conventional symmetry (blue curve). This demonstrates that distortion symmetry can reduce the number of NEB iterations necessary for convergence.