Summary
The ethanol–coronene candidate structure from R02's first geometry search has now had its vibrational frequencies calculated twice: once at the working settings of the pilot and once at tighter numerical settings. To decide which vibrations can be trusted, we fixed an error bound before either calculation ran: three times the largest difference between the two calculated Hessian matrices. By that rule, 119 of 129 vibrations have clearly positive curvature and none has clearly negative curvature. The ten softest, all below about 191 cm⁻¹, cannot be classified. The structure therefore remains a candidate, not a confirmed minimum.
Why the softest vibrations are the hard part
A structure is a local energy minimum only if moving its atoms in any direction raises the energy. In a calculation this is read from the eigenvalues of the mass-weighted Hessian, the matrix of second derivatives of the energy. A positive eigenvalue corresponds to a real vibrational frequency; a negative one corresponds to an imaginary frequency, a direction in which the energy falls.
For a molecule resting loosely on a carbon surface, the lowest vibrations are very soft: ethanol sliding, tilting and turning against coronene. Their eigenvalues are small, so a small numerical error in the Hessian can change their sign. A single frequency printout does not show whether such a sign is reliable.
What was calculated
Both Hessians were calculated analytically in ORCA 6.1.1 at the r2SCAN-3c level for the same 45-atom geometry, the candidate from the first geometry search. The first calculation used the working settings of the pilot. The second repeated it with tighter convergence of the electronic structure and of the response equations and with a finer integration grid. Overall translation and rotation were removed from both matrices in the same way, leaving 129 internal vibrations.
The rule, fixed in advance
The error bound is U = 3 × ‖K₁ − K₂‖₂: the largest eigenvalue magnitude of the difference between the two mass-weighted Hessians K₁ and K₂, multiplied by three. The factor gives a margin, because the difference between two calculations can understate the error of either one. The rule and the factor were signed before either frequency calculation started; they were not chosen after the numbers were known. Each eigenvalue λ of the first Hessian is then classified as follows:
| Condition | Classification |
|---|---|
| λ − U > 0 | Positive curvature confirmed |
| λ + U < 0 | Negative curvature confirmed |
| Otherwise | Undetermined |
What we found
The largest difference between the two Hessians was 4.60 × 10⁻⁴ hartree bohr⁻² u⁻¹, about 0.08 % of the largest eigenvalue of the first Hessian. The bound is therefore U = 1.38 × 10⁻³ hartree bohr⁻² u⁻¹. An eigenvalue of that size corresponds to a frequency of about 191 cm⁻¹, so no vibration below about 191 cm⁻¹ can be classified by this rule, whatever its calculated sign.
| Vibration | Working settings | Tightened settings | Classification |
|---|---|---|---|
| 1 | 21.8 | 21.0 | Undetermined |
| 2 | 31.1 | 30.0 | Undetermined |
| 3 | 38.6 | 37.7 | Undetermined |
| 4 | 61.9 | 59.7 | Undetermined |
| 5 | 83.1 | 82.1 | Undetermined |
| 6 | 88.7 | 88.6 | Undetermined |
| 7 | 94.4 | 93.9 | Undetermined |
| 8 | 122.7 | 124.6 | Undetermined |
| 9 | 134.7 | 135.8 | Undetermined |
| 10 | 162.2 | 162.1 | Undetermined |
The next vibration, at about 222 cm⁻¹, and all 118 vibrations above it are classified as positive. In total: 119 positive, 0 negative, 10 undetermined.
In both calculations all ten soft vibrations came out real, and the two calculations agree on them to within 2.2 cm⁻¹. That agreement is encouraging, but the pre-declared rule does not use it, so it does not change the classification.
Why the bound is conservative
The difference between the two Hessians is not spread evenly. Its largest component is concentrated on one hydrogen atom of ethanol's methyl group and lies in the high-frequency C–H stretching region, around 3100 cm⁻¹, far from the soft vibrations. Because the bound uses the largest difference anywhere in the matrix, it is set by this stiff region. The direct differences between the soft eigenvalues of the two calculations are at least about 75 times smaller than the bound.
What we did not do
It would be easy to redefine the bound now, for example by measuring the error only among the soft vibrations, and report a cleaner result. We have not done that. A rule changed after the numbers are known is a post hoc choice. If the rule is ever revised, the revision will be signed separately, labeled post hoc, and reported next to the original result, which remains the primary one.
What comes next
The ten softest vibrations will be tested by a separate, targeted check of the candidate, with its rules fixed before it starts. It follows the frequency calculations for the three reference structures: isolated ethanol in two conformations and isolated coronene. Until that check is completed and reviewed, the candidate is not described as a local or global minimum.
What this note does not show
This note reports a numerical diagnostic, not an accepted molecular result. It does not establish that the candidate is a local or global minimum, reports no binding energy, and provides no evidence about formaldehyde, water, a functionalized target material, or cartridge performance. The computational work is reviewed by the project lead. It has not yet undergone independent external review.
Change history
v1.0 ·
First publication.