Transition States and the Dimer Method

A transition state is a stationary point of the potential-energy surface with one, and only one, unstable internal direction, not merely the highest-energy geometry in a collection of structures. That distinction determines how a transition-state search must be designed, how its convergence must be interpreted, and which evidence is required before a candidate can support a mechanistic or kinetic claim.

The dimer method is a local first-order saddle-point search. It uses forces, rather than a complete Hessian, to identify the direction of lowest curvature and then translates a structure uphill along that direction while relaxing it in all orthogonal directions. This makes the method attractive when analytic second derivatives are unavailable or expensive. It does not, however, remove the need for a chemically meaningful starting structure or for an independent vibrational verification.

Tako’s combined transition-state calculation uses a double-ended path to generate that starting structure: it constructs a climbing-image nudged elastic band (CI-NEB), selects the highest-energy image, optionally refines that image with a dimer, and optionally performs a finite-difference vibration check. The exact runtime contract lives in Transition-State Search with Tako Script and the generated tako.transitionState settings, and the current implementation defects are registered in Known Limitations.

Stationary points, curvature, and Hessian index

Let the Born–Oppenheimer potential energy be $E(\mathbf R)$, where $\mathbf R$ is the vector of nuclear Cartesian coordinates. A stationary point satisfies

Vanishing forces identify a stationary point but do not classify it. Local classification is provided by the Hessian,

or, more physically, by the eigenvalues of its mass-weighted form. If

then the sign of $\lambda_i$ gives the local curvature along normal coordinate $\mathbf q_i$. Positive curvature is locally restoring, negative curvature is locally unstable, and an exactly zero eigenvalue describes a flat direction. The index of a stationary point is the number of negative Hessian eigenvalues after translations, rotations, constraints, and numerical zero modes have been treated appropriately.

A minimum has index zero. A conventional transition state for an elementary reaction is an index-one saddle:

within the unconstrained internal-coordinate space. A point with two negative eigenvalues is an index-two saddle, even if its geometry looks chemically plausible and its energy lies between reactant and product. A constrained optimization can produce a stationary point only in the allowed subspace; it is not necessarily stationary in the full coordinate space.

For an isolated nonlinear molecule, three translations and three rotations ideally have zero frequency. A linear molecule has only two rotational zero modes. Periodic systems have a different acoustic-mode structure. Finite-difference error, imperfect force convergence, finite calculator precision, and incomplete projection shift these zero modes away from zero and can make them weakly imaginary. Consequently, the sign of a tiny frequency is not, by itself, decisive evidence of a reactive instability.

From negative curvature to imaginary frequency

Vibrational analysis diagonalizes a mass-weighted Hessian. For a stable mode, $\omega_i^2$ is positive and the harmonic frequency is real. For a negative Hessian eigenvalue, $\omega_i^2<0$ and $\omega_i$ is imaginary. Programs usually encode an imaginary frequency as a negative real number or as a positive magnitude followed by $i$. The reporting convention does not change the physical statement: the harmonic potential falls, rather than rises, for a small displacement along that eigenvector.

The index-one criterion is necessary but not sufficient for a chemical assignment. The unstable eigenvector must describe the intended event. A torsional saddle, surface diffusion saddle, and bond-breaking saddle can each have exactly one imaginary mode. The normal-mode displacement should be inspected, and small forward and backward displacements should lead, after relaxation or an intrinsic reaction coordinate calculation, toward the expected neighboring minima.

Local and global statements

Every saddle classification is local to a potential-energy model and a set of coordinates. A semiempirical calculator and a machine-learned potential can place the same nominal saddle at different geometries, predict different indices, or disagree about whether a barrier exists. Constraints change the space in which curvature is measured. A supercell, charge state, spin state, and boundary condition are therefore part of the scientific definition of the result, not incidental software settings.

A transition state also does not prove that a mechanism dominates. Rate constants depend on free-energy differences, partition functions, tunneling, solvent or environment, and competing paths. The electronic saddle is one component of that argument. It should be reported with the calculator, convergence thresholds, endpoint preparation, mode verification, and an estimate of uncertainty appropriate to the intended conclusion.

Why ordinary minimization cannot find a saddle

A geometry optimizer follows a force or an approximate Newton step toward lower energy in every direction. Near a stationary point, a Newton step is

At a minimum this is locally well behaved when the Hessian is positive definite. At an index-one saddle, an optimizer that minimizes in every direction is repelled along the negative-curvature mode and generally falls into one of the neighboring basins. Saddle optimizers must reverse that behavior in one selected direction.

Eigenvector-following methods obtain one or more Hessian eigenvectors and maximize along a selected mode while minimizing along the remainder. They can be efficient when a trustworthy Hessian or Hessian update is available. Their cost and robustness depend on obtaining the correct mode. A full finite-difference Hessian requires a number of force evaluations that grows with the number of active Cartesian degrees of freedom, and repeating that construction during a search can be expensive.

Min-mode following asks for less information. It seeks only the lowest eigenpair $(\lambda_1,\mathbf q_1)$ and approximates it using directional force differences. The dimer method is the best-known realization of this idea. It replaces a complete second-derivative calculation with repeated rotations of a short displacement vector.

Dimer construction and directional curvature

At a center geometry $\mathbf R_0$, define a normalized orientation $\hat{\mathbf n}$ and two nearby images,

The two images constitute the dimer. They are not reaction-path images and are not physical replicas. Their small separation is a numerical probe of local curvature. From the endpoint forces,

This is a Rayleigh quotient. Minimizing it over all normalized orientations aligns $\hat{\mathbf n}$ with the lowest-curvature Hessian eigenvector. Near an index-one saddle, the converged curvature is negative. Near a minimum, all curvatures are positive, so the dimer must climb far enough to enter a region where the lowest curvature becomes negative.

The separation $\Delta R$ balances truncation error against force noise. A large displacement samples a nonlocal, anharmonic surface; a very small displacement subtracts nearly equal forces and amplifies numerical noise. The useful scale depends on calculator precision. It is not a physical bond displacement.

Rotational force

The force difference contains a component that attempts to rotate the dimer. Removing the component parallel to its current orientation gives a rotational quantity proportional to

Rotation proceeds in the plane defined by the current orientation and the rotational force. Improved dimer algorithms estimate the curvature as a trigonometric function of the trial rotation angle and choose an angle near its minimum. The orientation is normalized after rotation. Repeated rotations track the lowest mode as the center geometry moves.

Near-degenerate low modes are difficult. When $\lambda_1$ and $\lambda_2$ are similar, small force errors can rotate the estimated mode within their subspace. The energy may change smoothly while the orientation changes abruptly. Such behavior is not necessarily a bug; it can reflect weakly separated physical modes. It does mean that convergence should be assessed through the final Hessian and connectivity, not through a visually smooth orientation history alone.

Central-force variants

Some dimer implementations evaluate the midpoint and one displaced image, then estimate the opposite image force by symmetry. This reduces calculator calls but introduces an approximation whose quality depends on the local smoothness of the surface and the dimer separation. Tako’s current dimer control uses central forces. That implementation choice should be distinguished from the general two-ended derivation above.

Translating toward an index-one saddle

Once the orientation approximates the minimum mode, decompose the physical midpoint force into parallel and perpendicular parts,

In a negative-curvature region, the effective translation force reverses the parallel component,

An optimizer following $\mathbf F_{\mathrm{eff}}$ moves downhill in every direction orthogonal to the lowest mode and uphill along the lowest mode. At an index-one saddle, both the true force and the effective force vanish.

When the lowest curvature is still positive, implementations may suppress perpendicular relaxation and climb primarily along the current softest direction. This helps escape a basin. It also exposes the local character of the method: from a relaxed minimum, the chosen initial mode can determine which basin boundary and which saddle are encountered.

Translation optimizer and step control

The effective force can be passed to BFGS, FIRE, MDMin, or another optimizer. The optimizer is not the dimer method itself; it determines how successive effective-force observations are converted into displacements. A quasi-Newton optimizer may converge rapidly on a smooth surface but can inherit poor curvature information after mode switching. Inertial methods may be more forgiving but require more steps.

A maximum displacement prevents one force estimate from generating an unphysically large change. It is a stability control, not a convergence threshold. Reducing it can rescue oscillatory searches but increases the number of steps. Increasing it does not repair a wrong mode and can move the structure into an unrelated basin.

What convergence should mean

For a general dimer search, a defensible convergence assessment includes all of the following:

1. The effective translational force is below the selected threshold.
2. The dimer orientation is sufficiently converged that rotational uncertainty does not dominate the translation.
3. The lowest local curvature is negative.
4. An independent vibrational analysis finds exactly one significant imaginary internal mode.
5. The unstable mode connects the intended neighboring minima.

An optimizer’s converged flag usually addresses only part of this list. A small projected force does not by itself establish index one, and negative curvature does not establish the correct reaction. Conversely, a run that reaches its step budget may still yield a useful candidate, but that candidate requires further refinement before quantitative use.

Initial modes and starting structures

The dimer method is local. Its greatest practical determinant is the quality of the initial geometry and orientation.

An approximate transition structure obtained from a path method is usually the best start. The tangent of a converged or nearly converged climbing-image band provides a chemically informed initial mode. A structure generated by constrained scans, interpolation, coordinate driving, or chemical intuition can also work if it lies near the correct ridge. Starting from a fully relaxed minimum with a random direction is an exploratory saddle search, not a targeted transition-state refinement.

The initial mode should emphasize the atoms involved in the event and avoid domination by irrelevant soft motion. In large systems, surface translations, ligand torsions, acoustic modes, and unconstrained spectators can have lower curvature than the desired bond-making or bond-breaking coordinate. Mass weighting, localized displacement, selective active regions, or a path-derived tangent can reduce that ambiguity, although each choice changes the search space and must be reported.

Random initialization is reproducible only when the seed, displacement distribution, mask, and reference geometry are recorded. Multiple seeds can reveal nearby saddles, but agreement across seeds is not proof that the lowest barrier has been found. Conversely, different saddles from different seeds may be scientifically valuable evidence of competing mechanisms.

Endpoint preparation for a double-ended handoff

When NEB supplies the dimer start, reactant and product endpoints must represent local minima at the same calculator settings, charge, spin, composition, atom identity mapping, boundary conditions, and constraints. Equivalent atoms still require a chemically consistent mapping. Validation that checks only elements by array position cannot detect the exchange of two atoms of the same element.

Endpoint relaxation should not destroy the intended states. For reactions involving weak complexes, symmetry breaking, surfaces, defects, or metastable conformers, unconstrained optimization may collapse two nominal endpoints into one basin. The investigator should compare geometry, energy, bonding, magnetization where relevant, and periodic images before constructing the band.

Linear Cartesian interpolation can produce atom overlap, concerted motion that should be sequential, or the wrong periodic displacement. IDPP and chemically designed intermediate images can improve a difficult path; the combined Tako operation currently offers only linear interpolation, so difficult paths belong in the standalone reaction-path workflow.

NEB-to-dimer refinement

NEB and dimer methods solve complementary problems. NEB is double-ended and describes a path between known basins. A dimer is single-ended and refines a local saddle without preserving a complete path. A typical hybrid workflow uses NEB to identify the relevant transition region and the dimer to reduce the force on one candidate geometry.

For a band of images $\mathbf R_0,\ldots,\mathbf R_{N-1}$, NEB projects the true force perpendicular to the path and retains spring forces primarily along the path. A climbing image reverses the parallel component of the true force at the selected highest-energy image and removes its spring force. The climbing image therefore moves toward a saddle constrained by the path tangent. A subsequent dimer releases the explicit band and follows the locally lowest mode.

The handoff contains several assumptions. The highest-energy image must lie in the attraction region of the intended saddle. The band tangent must approximate its unstable mode. The calculator must be smooth enough for both image forces and directional force differences. If the band cuts a corner, crosses a discontinuity, changes electronic state, or converges to the wrong mechanism, dimer refinement can produce a well-converged but irrelevant saddle.

The NEB and dimer convergence thresholds need not be identical in principle. A moderately converged band may be adequate to seed a stricter local refinement. Step budgets should also be interpreted independently: exhausting the NEB budget says the path search did not meet its threshold; exhausting the dimer budget says local refinement did not meet its threshold. Neither fact is erased by successful completion of the other stage.

Practical interpretation checklist

Before calling a combined result a transition state, answer these questions:

1. Are both endpoints distinct, converged minima under identical physical conditions?
2. Is the atom mapping chemically correct, including equivalent atoms?
3. Is linear fixed-cell Cartesian interpolation adequate for this event?
4. Did neb.converged return true, and is the spring-constant unit limitation (Known Limitations) acceptable or explicitly accounted for?
5. If dimer ran, did dimer.converged return true?
6. Are the reported energy and force associated with the geometry being discussed, rather than copied from another stage?
7. Was vibration checking actually performed (checked: true)?
8. Is there exactly one significant internal imaginary mode?
9. Does that mode describe the intended chemistry?
10. Do displacements or IRC trajectories connect the candidate to the intended reactant and product?
11. Is the calculator credible for the stretched-bond and high-energy configurations on the path?
12. Are numerical, path, vibrational, and model uncertainties small enough for the claim?

If any answer is unknown, the structure is a saddle candidate, not a verified transition state. That terminology is scientifically useful: it states exactly what the calculation has established and what remains to be tested.

The combined calculation in Tako

The combined transitionState operation implements one fixed pipeline: linear Cartesian interpolation between two atom-matched endpoints, a hardcoded improved-tangent climbing-image NEB (climbing from the first step, static-step optimizer), selection of the highest-energy interior image whose band tangent seeds an optional BFGS dimer refinement, and an optional fixed finite-difference vibration check (0.01 bohr displacement, two-point central differences) that counts internal imaginary modes.

Three properties of this pipeline drive the validation protocol below. First, the stages have independent budgets and the run completes normally even when NEB, the dimer, or both report converged: false — gate explicitly on neb.converged, dimer.converged, and vibrations.imaginaryModes === 1. Second, the result contains stage summaries and a single saddle frame but no band, image energies, barrier, mode vectors, or restart state, so mechanistic evidence requires separately archived path and vibration calculations. Third, the current runtime has several registered defects — the spring constant is passed in raw atomic units, residual fields mix units, most dimer setup controls are ignored, and the output frame mixes stage scalars with a different stage’s geometry — all catalogued in Known Limitations.

The exact call signature, endpoint validation semantics, stage budgets and clamps, result schema, and gating recipes are documented in Transition-State Search with Tako Script and the generated tako.transitionState settings; the setup dialog is documented in Refine and Verify a Transition State.

Interpreting the imaginary-mode count

Zero imaginary internal modes usually means that the candidate is a minimum or that refinement crossed into a basin. It can also reflect a calculator surface too noisy to preserve a shallow negative curvature or an inappropriate constrained subspace.

Exactly one significant imaginary internal mode is consistent with a first-order saddle. The word “consistent” is deliberate: connectivity still has to be demonstrated. The unstable displacement should produce the expected bond changes in opposite directions, and relaxation or IRC propagation from the two sides should reach the intended basins.

Two or more imaginary internal modes identify a higher-index saddle within the analyzed space. A common recovery is to displace away from the candidate along one of the unwanted modes and repeat local refinement, or to rebuild the path with a more appropriate intermediate geometry. Merely tightening the force threshold does not guarantee removal of an extra negative mode if the search has converged to the wrong stationary point.

Very small imaginary values require a numerical sensitivity test: repeat the Hessian with another displacement, tighten force convergence, increase calculator precision, enlarge the periodic cell where appropriate, and inspect whether the eigenvector resembles translation, rotation, or a delocalized acoustic motion. The combined calculation cannot vary its displacement or expose the mode, so ambiguous cases require a separate, more complete vibration workflow or an external validation calculation.

Scientific validation protocol

A robust transition-state study treats the combined result as a candidate-generation step within a larger validation chain.

Validate the endpoints

Confirm that both endpoints are genuine minima at the same calculator settings and have the intended charge, spin, composition, atom mapping, cell, periodicity, and constraints. Check that their energies and structures are stable to tighter optimization. For periodic work, inspect interaction with periodic images and ensure the same fixed cell is scientifically justified. Because the runtime endpoint validator is narrow and the band inherits initial-endpoint metadata, perform these checks outside the runtime.

Validate the path hypothesis

Inspect chemically reasonable intermediates before trusting a linear Cartesian band. Look for atom overlap, swapped equivalent atoms, discontinuous periodic motion, and concerted displacements that imply an implausible mechanism. If the reaction requires cell evolution, solvent rearrangement, spin crossing, proton shuttling, or multiple sequential events, decompose the mechanism or use a path implementation that represents those degrees of freedom correctly.

The absence of a stored combined band prevents retrospective visual inspection. For important work, run a path calculation that archives every image and its energy. Confirm that the maximum is interior, the band is continuous, and the tangent near the candidate represents the intended event.

Require stage convergence explicitly

Read neb.converged and, when dimer ran, dimer.converged. Normal completion is not enough. If NEB is unconverged, the dimer seed may move as the band continues to relax. If dimer is unconverged, its final structure may still carry substantial effective force. An unconverged stage can supply a diagnostic guess but should not supply a reported barrier or stationary-point geometry without further refinement.

Force thresholds should be chosen relative to the target accuracy. A threshold of 0.05 eV Å$^{-1}$ may locate a qualitative saddle but can be too loose for small barrier differences or reliable low-frequency modes. Tightening the threshold is useful only if calculator noise is lower than the requested force scale.

Verify index and connectivity

Require exactly one significant imaginary internal mode from a sufficiently converged, well-conditioned Hessian. Inspect the eigenvector. Displace the geometry by small positive and negative amplitudes along the mass-weighted unstable mode, then relax or propagate an IRC in both directions. The endpoints should correspond to the intended reactant and product basins. If both directions return to the same minimum, the mode does not establish the desired connection.

For periodic or constrained systems, define the active space explicitly. Report which atoms and Cartesian components were fixed. If partial constraints were used in the combined calculation, repeat the frequency analysis with a method whose constrained Hessian construction is understood and whose modes can be inspected.

Recompute energies consistently

Because the combined result’s top-level scalars can be paired with a different stage’s geometry, recompute single-point energies on the final candidate, reactant, and product using the same calculator and settings. For quantitative work, evaluate all three with a higher-level reference method where feasible. Report electronic barriers as

Thermal or free-energy barriers require consistent zero-point, thermal, entropy, standard-state, and environmental corrections. A harmonic correction at an unverified or unconverged saddle is not meaningful.

Diagnostics and recovery strategies

NEB does not converge

First examine whether the endpoints and mapping define one elementary event. Add images or improve interpolation when the path is under-resolved. Remove severe overlaps. For periodic motion, unwrap coordinates consistently. A path that contains several sequential rearrangements should often be split into multiple elementary paths with intermediate minima.

Remember that the combined runtime ignores the visible NEB method, optimizer, and maximum-step controls and has the spring-unit defect (Known Limitations). If those algorithmic choices are the source of failure, use the standalone reaction-path capability or another implementation rather than repeatedly changing ignored controls.

Dimer converges to the wrong saddle

The local minimum mode may be a spectator torsion, surface drift, defect translation, or another soft event. Improve the NEB path, localize the initial displacement, constrain only demonstrably irrelevant motion, or start from a different candidate image. Seed controls do not affect the combined path-derived tangent, so a random-restart diagnostic requires the standalone dimer workflow.

If the dimer oscillates, reduce the shared maximum-step value (the only step control that reaches dimer translation in the combined runtime) or improve the starting geometry.

Dimer is unconverged but vibrations show one imaginary mode

This is still an unconverged candidate. Frequency classification does not replace the stationarity requirement. Residual forces mix gradient and curvature information, shift frequencies, and can make the apparent mode count sensitive to displacement. Continue refinement or transfer the geometry to a dedicated saddle optimizer.

No imaginary modes

The structure may have relaxed into a minimum, the negative curvature may be extremely shallow, or numerical projection may have removed an ambiguous mode. Recompute with tighter forces and inspect a complete spectrum. Verify that vibration analysis was actually enabled: hasImaginaryMode: false is also returned when checking is disabled, and the checked field distinguishes that case.

More than one imaginary mode

Inspect every unstable eigenvector. If one corresponds to the intended reaction and another to an avoidable distortion, displace along the unwanted mode and reoptimize toward an index-one point. If the modes describe coupled chemistry, the assumed elementary mechanism may be incomplete. Tightening convergence can remove numerical near-zero modes but will not turn a genuine higher-order saddle into a first-order saddle.

Calculator failures during vibration analysis

Finite differences evaluate geometries displaced in positive and negative directions for many coordinates. A calculator can fail on one displaced structure even after successful saddle optimization. Check element coverage, neighbor-list continuity, charge and spin support, cell validity, and model domain. The combined call loses structured partial output when this happens, so preserve a saddle candidate independently before expensive validation.

Comparison with related methods

NEB and climbing-image NEB

NEB is the appropriate tool when two endpoints are known and the path itself matters. It can reveal intermediates and multiple maxima. CI-NEB drives the selected maximum toward a saddle along the path tangent. Its cost scales with the number of images and optimizer steps, and it can fail when interpolation or mapping is poor.

A dimer is cheaper per translation step because it acts on one center geometry plus curvature probes. It does not return a global path. The hybrid approach uses NEB for global context and dimer for local refinement, but only if both stages and their handoff are preserved and inspected. Tako’s present combined artifacts omit the global band, so a separate archived NEB remains preferable when mechanistic evidence is required.

Intrinsic reaction coordinate

An IRC integrates the mass-weighted steepest-descent path away from a verified saddle in both directions. It answers a connectivity question: which minima are locally connected to this saddle on this potential surface? It is not primarily a saddle-search method. IRC propagation requires a reliable transition structure and curvature information, and it can be sensitive to step size, bifurcations, flat valleys, and coordinate definitions.

The dimer can locate a candidate without a Hessian; the IRC validates what that candidate connects. Neither substitutes for the other. The current combined Tako operation does not run an IRC or store connectivity evidence.

Eigenvector following

Eigenvector following uses a Hessian or Hessian approximation to select an uphill mode. It is especially effective when a good initial Hessian is available and the intended mode can be identified. It may converge faster near the saddle than repeated dimer rotation, but full Hessian construction is expensive for large systems.

The dimer estimates only the minimum mode from first derivatives. This makes it attractive for force-only calculators and large systems. It also means it naturally follows the lowest curvature, which may not be the chemically desired mode. Mode-following with a selected Hessian eigenvector can target a higher mode more directly.

Coordinate scans and constrained optimization

A relaxed scan imposes one or more reaction coordinates and minimizes the remaining degrees of freedom. It is intuitive and can produce a useful saddle guess, but its maximum is constrained and need not be a stationary point in the full space. A scan is best treated as a generator for a subsequent unconstrained saddle refinement and vibrational classification.

Growing-string and related path methods

String, growing-string, and freezing-string methods construct paths with different image coupling and growth strategies. They can reduce dependence on a complete initial interpolation or target reaction paths more adaptively. Their availability and numerical behavior differ by implementation. Method names exposed for standalone path calculations should not be assumed to configure Tako’s combined operation.

Calculator domain and numerical uncertainty

Transition states often contain stretched bonds, unusual coordination, charge redistribution, radical character, or high-energy local environments. These are exactly the configurations for which approximate calculators and machine-learned potentials may be least reliable.

GFN2-xTB can be useful for rapid molecular screening and starting structures, but barrier heights and state ordering may require higher-level electronic structure. Charge and number of unpaired electrons must match the intended state. A single-surface search cannot describe a spin crossing or nonadiabatic transition without specialized treatment.

An MLIP is reliable only within the chemical and configurational domain represented by its training and validation. Good equilibrium energies do not guarantee correct saddle curvature. Inspect model coverage for elements, charge states, local coordination, and periodic environments. Where uncertainty estimates or model ensembles are available, evaluate them along the entire path, not only at endpoints.

Force smoothness is particularly important for dimer rotation and finite-difference vibrations. Neighbor-list discontinuities, low numerical precision, or nonsmooth cutoffs contaminate force differences. A model may optimize geometries acceptably while producing unstable curvature estimates. Test displacement sensitivity and compare selected forces or Hessian modes against a reference method.

Uncertainty should be separated into at least four sources: incomplete numerical convergence, path or saddle-search uncertainty, finite-difference and mode-classification uncertainty, and calculator model error. Repeating the same calculation with a tighter threshold addresses only the first.

Reproducibility record

The shared record skeleton and reporting standard are defined in Reproducibility and Reporting. A reproducible transition-state report additionally includes enough information to reconstruct both the scientific problem and the exact runtime behavior:

• Tako and calculation-engine version or commit, because the spring conversion and result semantics may change.
• Calculator family, model identifier or weights, dispersion treatment, precision, and all method parameters.
• Total charge, number of unpaired electrons or spin state, and any electronic-state assumptions.
• Initial and final coordinates with stable atom identities and an explicit mapping.
• Cell vectors, periodic axes, coordinate wrapping convention, and supercell construction.
• Every fixed atom and fixed Cartesian component.
• Endpoint optimization method, thresholds, budgets, and convergence evidence.
• Image count, interpolation procedure, and the effective NEB algorithm and spring value with its effective unit.
• Handoff image index, geometry, energy, and tangent construction.
• The effective dimer controls, budget, convergence, and final scalar result, and whether dimer was skipped.
• Vibration displacement, stencil, active indices, constraint treatment, zero-mode projection, imaginary-mode definition, and complete frequencies and eigenvectors when available.
• Forward and reverse connectivity evidence and consistently recomputed endpoint and saddle energies.
• Every warning, failed stage, cancellation, rerun, and manual intervention.

For the current combined operation, some of this information cannot be recovered from the result artifact. Record public inputs before execution, retain endpoint files, and independently archive a full reaction path and full vibrational calculation for work intended for publication or quantitative decision-making. Do not infer ignored settings from their displayed values. Do not pair the top-level final energy with a dimer-refined geometry without an independent single-point calculation.

References

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