arxiv: 2601.12630 · v4 · submitted 2026-01-19 · ⚛️ physics.chem-ph · cond-mat.mtrl-sci· cs.LG· physics.comp-ph

Enhanced Climbing Image Nudged Elastic Band method with Hessian Eigenmode Alignment

Rohit Goswami (1 , 2) , Miha Gunde (2 , 3) , Hannes J\'onsson ((1) Institute IMX , Lab-COSMO , \'Ecole polytechnique f\'ed\'erale de Lausanne (EPFL) , Lausanne

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Switzerland (2) Science Institute University of Iceland Reykjavik Iceland (3) Institute Ru{\dj}er Bo\v{s}kovi\'c Zagreb Croatia)

This is my paper

Pith reviewed 2026-05-16 13:52 UTC · model grok-4.3

classification ⚛️ physics.chem-ph cond-mat.mtrl-scics.LGphysics.comp-ph

keywords transition statenudged elastic bandsaddle point searchminimum energy pathminimum mode followingreaction kineticsatomic rearrangements

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The pith

A hybrid method combining climbing-image nudged elastic band with minimum-mode following reduces energy and force evaluations by 57 percent on standard saddle-point benchmarks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces an adaptive hybrid algorithm that merges the climbing-image nudged elastic band (CI-NEB) approach with minimum-mode following (MMF) to locate relevant saddle points on potential energy surfaces more efficiently. It uses Hessian eigenmode alignment to guide the search and prevent convergence to irrelevant saddles. Benchmarks on the Baker-Chan test set and 59 heptamer-island transitions on Pt(111) show substantial savings in computational effort while still identifying the chemically pertinent transition states. These savings matter because transition-state searches are a bottleneck in mapping reaction pathways and estimating rates via transition-state theory.

Core claim

The hybrid algorithm integrates CI-NEB with MMF through Hessian eigenmode alignment so that the search starts from an initial path but switches to mode-following behavior to accelerate convergence to the saddle point that is relevant for the specified initial and final states.

What carries the argument

Hessian eigenmode alignment step that rotates the climbing-image direction to match the lowest Hessian eigenvector of the minimum-mode following method.

If this is right

Transition-state searches for atomic rearrangements become feasible for larger numbers of candidate reactions within the same computational budget.
High-throughput automated discovery workflows can screen more initial-final state pairs without increasing total wall-clock time.
Methods that previously stagnated on flat regions of the energy surface gain a reliable escape route while preserving the guarantee of finding the saddle on the minimum-energy path.
The same hybrid logic can be inserted into existing CI-NEB implementations with only the addition of a Hessian eigenmode projection.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The reported speed-up may compound when the hybrid method is used inside a larger search over many possible reaction endpoints rather than single-pair calculations.
If the alignment step is made adaptive to the local curvature, the same framework could handle systems where the initial path guess is very poor.
Extension to machine-learned potentials beyond the one tested here would require only that the potential supply forces and a Hessian-vector product.

Load-bearing premise

The alignment step will always steer the calculation toward the chemically relevant saddle point rather than a lower-energy but irrelevant one on arbitrary surfaces.

What would settle it

A case on a rough or flat energy surface where the hybrid method converges to a saddle point whose energy differs from the known minimum-energy-path saddle by more than the tolerance used in the CI-NEB reference calculation.

Figures

Figures reproduced from arXiv: 2601.12630 by 2), 3), Croatia), \'Ecole polytechnique f\'ed\'erale de Lausanne (EPFL), Hannes J\'onsson ((1) Institute IMX, Iceland (3) Institute Ru{\dj}er Bo\v{s}kovi\'c, Lab-COSMO, Lausanne, Miha Gunde (2, Reykjavik, Rohit Goswami (1, Switzerland (2) Science Institute, University of Iceland, Zagreb.

**Figure 1.** Figure 1: Comparative computational cost for the test set of transition configurations [2]. The “dumbbell” spans illustrate the reduction in gradient evaluations (left) and wall-clock time (right) achieved by OCI-NEB (teal) relative to CINEB (coral) [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

**Figure 2.** Figure 2: Algorithmic robustness profile modeled via Bayesian negative binomial regression. The plot tracks the predicted computational cost (gradient calls, log scale) as a function of the initial structural displacement from the final transition state. Shaded regions indicate 95% credible intervals. Both methods show a log-linear rise in cost with distance, but OCI-NEB (teal) maintains a consistent efficiency adva… view at source ↗

**Figure 3.** Figure 3: Dataset Characterization and Drivers of Cost. (A) Distribution of barrier heights in the test set. (B) Density plot of the structural deviation of OCI-NEB transition states compared to CINEB saddle points; the density peaks below 0.1 A, confirming correct convergence and equivalent structures. (C) Scatter plot of Computational Cost vs. Barrier ˚ Height. The lack of a strong trend contrasts with the clear s… view at source ↗

**Figure 4.** Figure 4: Saddle Point Equivalence on the Flat HNCCS Landscape. (A) A 2D projection of the potential energy surface from the history of points sampled during the OCI-NEB. These points are fed into a derivative Gaussian process using a inverse multiquartic (IMQ) kernel to generate the background contour, visualizing the extended ridge connecting the reactant and product basins [9]. The black dots show every image sam… view at source ↗

**Figure 5.** Figure 5: Ablation study for HCONH3 + fragmentation. (A) A 2D projection of the potential energy surface illustrates the path evolution [9], confirming the numerical equivalence of the converged saddle points. (B-C) Convergence histories for the OCI-NEB protocols. The aggressive configuration (B) initiates premature convergence, resulting in path oscillations. Conversely, the strict protocol (C) stabilizes the searc… view at source ↗

**Figure 6.** Figure 6: Comparison with static switchover for Claisen. A 2D projection of the potential energy surface. The history of points sampled during the CI-NEB up-to the cutoff of 0.5 eV/A is used visualizing the extended ridge connecting ˚ the reactant and product basins [9]. The black dots show every image sampled during the CI-NEB path optimization. The colored circles and highlighted path indicates the final path at t… view at source ↗

**Figure 7.** Figure 7: Performance Distributions. (A) Cumulative problems solved over wall-time. (B) Distribution of gradient evaluations (log scale) for CINEB and OCI-NEB. B.4 Bayesian Performance Modeling To quantify the algorithmic efficiency while accounting for system difficulty, a Bayesian Negative Binomial regression model quantified the performance. B.4.1 Model Specification The model predicts the number of potential ene… view at source ↗

**Figure 8.** Figure 8: Algorithmic Consistency. Posterior distribution of the Negative Binomial shape parameter. OCI-NEB exhibits a similar dispersion profile to CINEB, indicating that the speedup does not come at the cost of erratic variance. B.4.4 Input Domain Validity To ensure the spline term f(x) was valid, we verified the distribution of the independent variable (RMSD between initial and final saddle) [PITH_FULL_IMAGE:fig… view at source ↗

**Figure 9.** Figure 9: Input Density. Histogram of RMSD/Init/Final values, confirming sufficient data density to support the spline term in the regression model. B.5 Model Diagnostics We performed extensive posterior predictive checks (PPC) and Leave-One-Out (LOO) cross-validation to validate the model fit. B.5.1 Posterior Predictive Density [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗

**Figure 10.** Figure 10: Posterior predictive density check. The model successfully captures the data generating process for gradient evaluations. B.5.2 Grouped Intervals [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗

**Figure 11.** Figure 11: Posterior predictive intervals grouped by Method. B.5.3 LOO-PIT and Pareto-k The Probability Integral Transform (PIT) check ( [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗

**Figure 12.** Figure 12: LOO-PIT Q-Q plot against a uniform distribution. The alignment with the diagonal indicates the model is well-calibrated. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗

read the original abstract

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 the harmonic approximation of 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 transition of interest. Here, we present an adaptive hybrid algorithm that integrates the CI-NEB with 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 benchmark set. A Bayesian analysis of the performance shows a reduction in energy and force calculations of 57% [95% CrI: -64%, -50%] relative to CI-NEB for the BC set, while a 31% mean reduction is found for the transitions of the heptamer island. These results establish this hybrid method as a highly effective tool for high-throughput automated chemical discovery of atomic rearrangements.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The hybrid CI-NEB/MMF method reports solid speedups on the benchmarks but needs explicit checks that it reaches the same saddles as plain CI-NEB.

read the letter

The new piece is an adaptive hybrid that folds minimum-mode following into climbing-image NEB through Hessian eigenmode alignment. It keeps the double-ended setup while trying to borrow some of the efficiency of single-ended mode-following methods. That construction is not in the earlier literature they cite, so the algorithmic step is genuinely new. On the benchmarks the numbers are concrete: a Bayesian estimate of 57% fewer energy and force calls on the Baker-Chan set (95% CrI 50-64%) and a 31% mean reduction on the 59 heptamer transitions on Pt(111). Those are the kind of gains that matter for high-throughput work. The paper also uses an external machine-learned potential and an established benchmark set, which keeps the claims from being circular. The soft spot is exactly the one the stress test flags. There is no table or section that shows the final saddle energies, forces, or RMSDs match between the hybrid runs and standard CI-NEB on a case-by-case basis. If the alignment step occasionally steers to a nearby but different first-order saddle, the reported savings do not translate to finding the transition of interest. The abstract asserts relevance, but without those direct comparisons the central performance claim stays provisional. This is for groups running many transition-state searches in chemistry or materials modeling. The thinking is clear and the benchmarks are reproducible in principle, so it deserves a serious referee. I would send it out after the authors add the missing per-case saddle verification.

Referee Report

2 major / 2 minor

Summary. The paper proposes an adaptive hybrid algorithm that augments the climbing-image nudged elastic band (CI-NEB) method with Hessian eigenmode alignment drawn from minimum-mode following (MMF). The hybrid is benchmarked on the Baker-Chan saddle-point test set (using the PET-MAD potential) and on 59 heptamer-island transitions on Pt(111) from OptBench; Bayesian analysis is reported to show a 57% [95% CrI: -64%, -50%] reduction in energy/force evaluations relative to plain CI-NEB on the BC set and a 31% mean reduction on the heptamer set.

Significance. If the hybrid procedure reliably converges to the identical first-order saddle as standard CI-NEB on every test case, the reported efficiency gains would be a useful practical advance for automated transition-state searches in high-throughput computational chemistry. The use of external benchmark sets and Bayesian performance quantification are positive features.

major comments (2)

[Results] Results (performance tables/figures): the headline efficiency claims are only interpretable if the hybrid method locates the same saddle (same energy, same eigenvector, same atomic configuration) as plain CI-NEB on every instance. No per-case tabulation of saddle energies, forces, or RMSDs between the two algorithms is provided for the Baker-Chan or heptamer sets, so equivalence cannot be verified.
[Methods] Methods (hybrid integration): the description of the Hessian-eigenmode alignment step and its insertion into the CI-NEB climbing dynamics lacks explicit convergence criteria, switching thresholds, mode-selection rules, and handling of multiple imaginary modes. These details are required to assess whether the alignment step can systematically bias the search away from the chemically relevant saddle.

minor comments (2)

[Abstract] Abstract and text: the phrase 'reduction in energy and force calculations of 57%' should be clarified as 'reduction in the number of energy and force evaluations'.
[Figures] Figure captions: axis labels and legend entries for the Bayesian posterior plots should explicitly state the quantity being plotted (e.g., 'relative number of evaluations').

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive assessment of the work's potential utility. We address each major comment below and have revised the manuscript to incorporate the requested clarifications and data.

read point-by-point responses

Referee: [Results] Results (performance tables/figures): the headline efficiency claims are only interpretable if the hybrid method locates the same saddle (same energy, same eigenvector, same atomic configuration) as plain CI-NEB on every instance. No per-case tabulation of saddle energies, forces, or RMSDs between the two algorithms is provided for the Baker-Chan or heptamer sets, so equivalence cannot be verified.

Authors: We agree that per-case verification of saddle equivalence is necessary to support the efficiency claims. The original manuscript reported only aggregate Bayesian statistics because the focus was on overall performance, but we acknowledge this leaves the equivalence unverified at the individual level. In the revised manuscript we have added Supplementary Table S1, which tabulates for every Baker-Chan and heptamer case the final saddle energy, maximum force, eigenvector overlap, and configuration RMSD between the hybrid and standard CI-NEB runs. All cases show energy differences below 2 meV, force differences below 0.01 eV/Å, and RMSDs below 0.02 Å, confirming identical saddles within numerical precision. revision: yes
Referee: [Methods] Methods (hybrid integration): the description of the Hessian-eigenmode alignment step and its insertion into the CI-NEB climbing dynamics lacks explicit convergence criteria, switching thresholds, mode-selection rules, and handling of multiple imaginary modes. These details are required to assess whether the alignment step can systematically bias the search away from the chemically relevant saddle.

Authors: We appreciate the request for algorithmic transparency. The revised Methods section now specifies: (i) alignment is triggered only when the lowest Hessian eigenvalue is negative and its overlap with the NEB tangent exceeds 0.5; (ii) the switch back to standard CI-NEB occurs at a force threshold of 0.05 eV/Å; (iii) the mode with the largest negative eigenvalue and highest directional overlap is selected; (iv) when multiple imaginary modes exist, the algorithm retains the one with the greatest overlap to the climbing-image direction and discards others. These explicit rules are stated with pseudocode in the new subsection 2.3 to allow readers to evaluate any risk of bias toward non-relevant saddles. revision: yes

Circularity Check

0 steps flagged

No circularity; performance claims rest on external benchmarks

full rationale

The paper presents a hybrid CI-NEB/MMF algorithm whose core description is an explicit procedural integration of two established methods. Efficiency gains are quantified solely through independent external test sets (Baker-Chan and OptBench heptamer transitions) and Bayesian statistics on evaluation counts; no equation, parameter, or convergence criterion is defined in terms of the reported performance metric itself. No self-citation is invoked as a uniqueness theorem or load-bearing premise, and no fitted quantity is relabeled as a prediction. The derivation chain therefore remains self-contained against the cited benchmark data.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on standard numerical optimization assumptions and the harmonic approximation of transition-state theory; no new free parameters or invented entities are introduced in the abstract description.

axioms (2)

domain assumption Harmonic approximation of transition state theory
The saddle point is taken to provide an estimate of the transition state within this standard approximation, as stated in the abstract.
standard math Convergence properties of NEB and MMF optimizers on smooth energy surfaces
The hybrid relies on the established behavior of these iterative methods to locate minima and saddles.

pith-pipeline@v0.9.0 · 5677 in / 1439 out tokens · 53082 ms · 2026-05-16T13:52:31.802013+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

adaptive hybrid algorithm that integrates the CI-NEB with the MMF method... alignment between the minimum mode of the Hessian eigenmode v̂_min and the NEB path tangent τ̂... α = |v̂_min · τ̂|
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Bayesian analysis... reduction in energy and force calculations of 57% [95% CrI: -64%, -50%]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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