Summary
The first frequency check of updated isolated coronene showed how soft its lowest movements were. A focused error rule was then fixed and signed before the second Hessian was calculated. Applying both rules to that same pair gives 99 positive and 3 undetermined modes under the original rule, or 102 positive and none undetermined under the focused rule. The project accepted the structure as a local minimum under the revised setup and focused analysis. The original classification remains part of the result.
One geometry, two numerical checks
The structure is isolated coronene, C₂₄H₁₂, optimized with r2SCAN/def2-TZVPP without D4 dispersion or gCP corrections. ORCA 6.1.1 calculated two analytic Hessians at this same geometry: K₁ at the working numerical settings and K₂ with tighter electronic and response convergence and a finer integration grid. The comparison checks numerical sensitivity within this calculation level; it is not an experimental measurement.
The Hessians were symmetrized and mass-weighted using the same masses. A common projection removed overall translation and rotation, leaving 102 internal modes. Tiny negative raw-frequency entries associated with overall motion are separate from these internal modes. The warning in Story 01 concerned an earlier geometry and setup; those downhill directions are not the three unresolved modes of this updated structure.
When the rule changed
On 24 September 2026, the team calculated the first Hessian, K₁, at the updated geometry and inspected its low-frequency spectrum. Later that day, before the second Hessian existed, the focused rule and its set of soft modes were fixed from K₁ alone and signed as an amendment. The refinement was therefore informed by the first check; it was not declared before both. The team then calculated K₂ and applied both rules to the same pair of Hessians. The result was accepted on 25 September, and the reference decision was recorded on 26 September, keeping both classifications and the amendment’s limited scope.
What the two rules measure
The original rule uses U = 3‖K₁ − K₂‖₂ for every internal eigenvalue λ. A mode is positive when λ − U > 0, negative when λ + U < 0, and otherwise undetermined. The norm measures the largest change anywhere in the internal space, which can be dominated by stiff vibrations.
The focused rule selects the four lowest K₁ modes below 200 cm⁻¹: approximately 87.06, 87.07, 123.75 and 163.85 cm⁻¹. This selection is fixed before K₂. In the K₁ eigenvector basis, the analysis measures changes within the soft and stiff blocks, their coupling, and the gap between them. A positive remaining gap permits a block perturbation bound bₛ. Soft modes use Uₛ = 3 min(bₛ, ‖K₁ − K₂‖₂); other modes retain the original allowance. If the gap condition fails, the original rule is used throughout.
| Rule | Positive | Negative | Undetermined | Interpretation |
|---|---|---|---|---|
| Original global rule, v1.1 | 99 | 0 | 3 | The three softest internal modes remain unresolved. |
| Focused rule, v1.2 | 102 | 0 | 0 | Local minimum accepted under the revised setup and this analysis. |
For this pair the global allowance is approximately 5.88 × 10⁻⁴ hartree bohr⁻² u⁻¹, and the soft-block allowance approximately 9.48 × 10⁻⁶ in the same units. The perturbation inequalities bound differences between the two computed matrices. Treating the allowance, including its factor of three, as an error bound on the calculation remains an empirical project rule. It is not a rigorous bound on the exact molecular Hessian.
What the reference enables
The signed amendment applies to this isolated-coronene check. It does not retrospectively replace the ethanol or ethanol–coronene classifications or automatically apply to new structures. Accepting coronene alone does not confirm the pair as a minimum.
Before the amendment was signed, the block calculation was also tested on historical Hessian pairs. Applied retrospectively to the existing ethanol–coronene candidate, it would give 125 positive, 4 undetermined and no negative modes instead of 119 positive, 10 undetermined and no negative modes. That more favorable diagnostic was not used to revise the candidate’s accepted classification: the amendment was restricted to structures whose second Hessian did not yet exist.
The pilot now has a defined free-coronene reference before molecular-interaction comparisons. This does not identify a single cause of the earlier warning, qualify the final energy method for other systems, or validate a prefilter. Scientific acceptance is an internal project decision by the project lead; independent external scientific review of the computational work has not been performed.
Change history
v1.0 ·
First publication.