Enhanced Climbing Image Nudged Elastic Band method with Hessian Eigenmode Alignment

Accurate determination of transition states is central to an understanding of reaction kinetics. Double-endpoint methods where both initial and final states are specified, such as the climbing image nudged elastic band (CI-NEB), identify the minimum energy path between the two and thereby the saddle point on the energy surface that is relevant for the given transition, thus providing an estimate of the transition state within harmonic transition state theory. Such calculations can, however, incur high computational costs and may suffer stagnation on exceptionally flat or rough energy surfaces. Conversely, methods that only require specification of an initial set of atomic coordinates, such as the minimum mode following (MMF) method, offer efficiency but can converge on saddle points that are not relevant for the transition of interest. Here, we present an adaptive hybrid algorithm that switches between the CI-NEB and the MMF method so as to get faster convergence to the relevant saddle point. The method is benchmarked for the Baker-Chan (BC) saddle point test set using the PET-MAD machine-learned potential as well as 59 transitions of a heptamer island on Pt(111) from the OptBench set. A Bayesian analysis of the performance shows a median reduction of energy and force calculations by 46% [95% CrI: -55%, -37%] relative to CI-NEB for the BC set, while a 28% reduction is found for the transitions of the heptamer island. Calculations of the BC set where a simple switch from the CI-NEB to the MMF method is made when the magnitude of the atomic forces drops below 0.5 eV/AA requires 30% more force calculations than the OCI-NEB algorithm. These results show that an adaptive hybrid method mixing CI-NEB and MMF can be a highly efficient tool for high-throughput automated chemical discovery of atomic rearrangements.

💡 Research Summary

The paper introduces an adaptive hybrid algorithm, termed Off‑path Climbing Image Nudged Elastic Band (OCI‑NEB), that dynamically combines the strengths of the double‑endpoint Climbing Image Nudged Elastic Band (CI‑NEB) method with the single‑point Minimum Mode Following (MMF) approach, specifically the Dimer method. Traditional CI‑NEB requires both reactant and product minima and constructs a discrete chain of images that relaxes toward the minimum energy path (MEP). While robust, CI‑NEB can become computationally expensive, especially on flat or noisy potential energy surfaces where many force evaluations are needed. Conversely, MMF methods start from a single configuration and follow the lowest curvature eigenmode of the Hessian to a first‑order saddle point, offering speed but risking convergence to irrelevant saddles.

OCI‑NEB addresses these limitations by initially performing standard CI‑NEB to bring the climbing image close to the saddle region, then switching to a targeted MMF refinement. The switch is not based on an absolute force threshold (e.g., 0.5 eV/Å) but on a relative trigger: the maximum force on the initial path (F₀) is recorded, and the algorithm activates MMF when the current maximum force falls below λ_rel · F₀ (default λ_rel = 0.5). An optional absolute floor (λ_abs = 0.1 eV/Å) can be used for single‑reaction studies.

During the MMF phase, the algorithm monitors the alignment α = |v̂_min · τ̂| between the Hessian’s minimum eigenmode (v̂_min) and the NEB tangent (τ̂). As long as α exceeds a tolerance (α_tol ≈ 0.9), the Dimer continues; when α drops below this value, MMF stops early, and the climbing image is moved to the configuration that exhibited the most negative curvature during the Dimer run. This alignment‑based early‑termination prevents wasteful Dimer steps and ensures that the refinement stays within the relevant subspace of the reaction path.

Stability is further reinforced by several safeguards: a “stability latch” activates MMF only after the climbing‑image index has remained constant for κ = 5 iterations; the optimizer state is reset if any image moves beyond a prescribed fraction of the total path length; and a “best‑state restoration” mechanism guarantees that, if the Dimer fails (e.g., encounters positive curvature), the algorithm reverts to the best negative‑curvature configuration rather than adopting a deteriorated geometry.

If the Dimer repeatedly fails or yields poor alignment, OCI‑NEB employs an adaptive back‑off strategy. The trigger force for the next MMF activation is reduced by a factor P(α) = B + (1 − B) α^S (default B = 0.4, S = 1.5), effectively lowering the force threshold until the NEB has sufficiently relaxed and re‑aligned the path before another Dimer attempt.

The method is implemented in the eOn 2 software suite and uses the Sequential Image‑Dependent Pair Potential (S‑IDPP) to generate physically reasonable initial paths, avoiding steric clashes before expensive electronic‑structure or machine‑learning interatomic potential (MLIP) evaluations. Spring constants along the NEB are energy‑weighted, concentrating images near high‑energy regions to improve saddle‑point resolution without increasing the number of images.

Performance is benchmarked on two datasets: (1) the Baker‑Chan (BC) saddle‑point test set, and (2) 59 transition events of a heptamer island on Pt(111) taken from the OptBench collection. Both benchmarks employ the PET‑MAD machine‑learned potential for high‑accuracy forces and energies. A Bayesian regression analysis shows that OCI‑NEB reduces the total number of energy and force evaluations by a median of 46 % (95 % credible interval: –55 % to –37 %) for the BC set, and by 28 % for the Pt(111) heptamer transitions. By contrast, a naïve strategy that switches from CI‑NEB to MMF when forces drop below 0.5 eV/Å actually increases the number of force evaluations by ~30 % relative to the pure CI‑NEB approach, highlighting the importance of the relative‑trigger and alignment criteria.

In summary, OCI‑NEB delivers a robust, adaptive framework that leverages the global path‑finding capability of CI‑NEB and the rapid local refinement of MMF, while dynamically controlling the exchange based on physically meaningful metrics (relative force, mode alignment, and adaptive back‑off). The algorithm markedly accelerates transition‑state searches across diverse energy landscapes, making it especially suitable for high‑throughput automated reaction discovery, catalyst surface studies, nanocluster rearrangements, and any application where large numbers of saddle‑point calculations are a computational bottleneck.