Does one C=C increment fit every alkene? Two preregistered tests of Witkowski and co-workers' correlation energy per bond

Abstract

Witkowski, Śmiga, Hirata, Dral, and Grabowski, in Ultrafast Correlation Energy Estimator (J. Phys. Chem. A 129 (2025) 8877–8890), estimate a molecule’s correlation energy as a sum of fitted bond-type increments — “correlation energy per bond” (CEPB) — and state that the assignment holds regardless of bond length, bond angle, hybridization, or π-electron conjugation.¹ This note is an independent reanalysis of their published data plus an independent extension, under a protocol frozen before the first calculation. Arm A reanalyzes the source’s own published per-molecule correlation energies: three pairs in its tables differ by exactly the same bond-count swap, C=C → C–C + 2 C–H, so CEPB assigns all three the same correlation change, and the spread across the three measured changes tests the transferability claim on the source’s own numbers. Arm B is new quantum chemistry: frozen-core DF-CCSD(T) correlation energies for the four C₄H₈ positional isomers, which carry identical CEPB bond counts and therefore identical predicted correlation energies, at two basis sets. Under the registered decision rules both arms returned inconclusive. The Arm A spread is 0.51 kcal/mol at the source’s CBS reference level but 1.78 and 2.45 kcal/mol at the two finite bases, so the basis levels disagree about the registered 1.0 kcal/mol threshold; in Arm B, one pairwise difference changed sign between the two bases, which the frozen rule treats as disqualifying. Two findings are nonetheless stable across everything run here. All three Arm A contrasts lie on the same side of the CEPB prediction, by -3.66 to -4.17 kcal/mol at CBS — a systematic offset in the swap’s price rather than an environment dependence. And isobutene is the most strongly correlated C₄H₈ isomer at both bases, separated from trans-2-butene by -1.357 and -1.329 kcal/mol against a predicted difference of exactly zero.

Introduction

The idea that electron correlation is approximately local, and therefore approximately additive over chemical substructures, is old and productive: correlation energies of homologous series grow nearly linearly, and pair theories make the locality precise.² The open question is never whether such additivity roughly holds but how far it can be pushed before chemistry pushes back. Two recent papers push it far. Witkowski, Śmiga, Hirata, Dral, and Grabowski fit 33 bond-type increments to CCSD(T)-level correlation energies of 84 molecules and estimate any molecule’s correlation energy as the sum over its dominant Lewis structure’s bonds — a scheme they name correlation energy per bond (CEPB) — with the abstract-level claim that each increment applies “regardless of the bond length, bond angle, sp-hybridization, π-electron conjugation, ionicity, noncovalent interactions, etc.”¹ Independently, Vincent and Popelier report that fragment-level correlation energies at CCSD(T) are transferable within a few percent and recommend them as machine-learning targets.³ Transferability of correlation increments is, in other words, a live working assumption in the current literature, not a settled fact.

A one-number-per-bond-type model makes two sharp, parameter-free predictions. First, any two molecules that differ by the same bond-count change must differ by the same correlation energy: the swap C=C → C–C + 2 C–H costs \(e_{\mathrm{CC}} + 2e_{\mathrm{CH}} - e_{\mathrm{C=C}}\) wherever it occurs, whether the C=C sits in ethene, in a conjugated ring, or next to a saturated ring. Second, molecules with identical bond counts must have identical correlation energies: the four C₄H₈ positional isomers — 1-butene, cis-2-butene, trans-2-butene, and isobutene — each carry one C=C, two C–C, and eight C–H bonds, so CEPB assigns them one correlation energy and predicts a correlation contribution of exactly zero to every isomerization among them. The source states this second consequence plainly, noting that CEPB “fails to differentiate positional isomers” and that such isomerization energies collapse to the Hartree–Fock values; what it does not do is measure the size of the resulting error, and its published accuracy statistics are whole-molecule percentages, which do not answer either question. A previous note in this series found that reaction-level correlation contributions to hydrogenation change substantially between reactions whose starting bond orders differ, which left the sharper within-one-bond-order-class question open — and the source’s own tables contain the data to ask it.

The hypothesis, frozen with its thresholds before any number was computed: the transferability fails at chemical accuracy. Concretely, Arm A predicts that the three published instances of the C=C → C–C + 2 C–H swap do not share one value — their spread exceeds 1.0 kcal/mol; Arm B predicts that at least one pairwise correlation-energy difference among the four C₄H₈ isomers exceeds 1.0 kcal/mol in magnitude. The falsifiers were fixed in advance: the hypothesis dies in Arm A if the contrast spread is at or below 1.0 kcal/mol at every basis level the source publishes, and in Arm B if all six pairwise differences stay below 1.0 kcal/mol at the registered level and under its basis-sensitivity check. The 1.0 kcal/mol threshold is this experiment’s operational bar for a chemically material difference, chosen before any calculation; the source does not claim chemical accuracy and states no accuracy budget of its own. Either outcome was worth publishing — a spread within chemical accuracy would strengthen the source’s assignment on molecules it never fitted, and a measured spread would bound how far the assumption can be pushed. If the registered basis levels disagreed about a threshold, or a registered method-fidelity gate failed, the result was to be reported as inconclusive rather than repaired, and both of those rules ended up binding.

Computational Methods

Source inputs. All Arm A quantities are arithmetic on values transcribed from the source: the fitted bond-type increments (main-text Table 2) and the per-molecule correlation energies of the training and test sets at three levels — CBS, aug-cc-pVQZ, and aug-cc-pVTZ (main-text Tables 3 and 4; Supporting Information Tables S1, S2, S4, S5). Every transcribed value is all-electron, as those tables declare; the source’s frozen-core table covers only its training set and is not used. The transcription was frozen in the experiment’s inputs.json before analysis, with each value carrying its declared basis level, and increments are only ever combined with molecular energies of the same declared level. The SHA-256 fingerprint of the frozen transcription is recorded in the committed metrics provenance. Nothing in this note reruns, refits, or re-extrapolates any calculation of the source’s; the source authors’ code and data were used only as published numbers.

Arm A. The registered molecule set is every unsaturated hydrocarbon in the source’s tables whose Lewis structure contains at least one C=C and only C–C, C=C, and C–H bond types: ethene, cyclopropene, allene, 1,3-butadiene, cyclobutene, cyclohexene, and 1,4-cyclohexadiene, with methane, ethane, and cyclohexane as saturated references. Benzene is analysed separately and never pooled, because dominant-Lewis-structure bond counts are convention-dependent for an aromatic ring. The originally registered Arm A statistic was the raw effective increment

\[ e^{\mathrm{eff}}_{\mathrm{C=C}}(X) \;=\; \frac{E_{\mathrm{corr}}(X) - n_{\mathrm{CH}}\,e_{\mathrm{CH}} - n_{\mathrm{CC}}\,e_{\mathrm{CC}}}{n_{\mathrm{C=C}}}, \]

evaluated with the source’s own fitted increments. During design review — with the CBS value of that statistic already hand-evaluated, a fact disclosed in the research journal and in the preregistration’s amendment before any contrast was computed — an objection was raised independently by an automated reviewer and by the review itself: this quantity attributes a molecule’s entire CEPB residual to its C=C bonds, so its spread grows with molecule size whether or not the C=C assignment itself varies. The preregistration was therefore amended before any contrast value existed: the raw statistic was demoted to a secondary, descriptive result, and the primary Arm A metric became a contrast in which every other bond class cancels exactly. Three pairs in the source’s tables differ by precisely the swap C=C → C–C + 2 C–H — ethene → ethane, 1,4-cyclohexadiene → cyclohexene, and cyclohexene → cyclohexane — and CEPB assigns all three the identical correlation change \(e_{\mathrm{CC}} + 2e_{\mathrm{CH}} - e_{\mathrm{C=C}}\). The primary statistic is the spread of the three measured changes, required to clear the threshold at every published basis level. Both statistics are reported below in the order they were derived.

Arm B. New calculations, run with Psi4 1.9.1:⁴ each of the four C₄H₈ isomers (with ethene and propene as secondary members that do not enter the verdict) was optimized with frozen-core density-fitted MP2/cc-pVTZ^5,6 from a committed starting structure, followed by frozen-core density-fitted CCSD(T)⁷ at that geometry, with the correlation energy taken from Psi4’s CCSD(T) CORRELATION ENERGY variable and all six pairwise differences formed. The registered basis-sensitivity check repeats the whole procedure at cc-pVDZ. The Arm B verdict stands only if every pairwise difference keeps its sign between the two bases and no pair crosses the 1.0 kcal/mol threshold in opposite directions. Registered method-fidelity gates: every optimization and CCSD(T) run converges, every optimized structure retains the intended isomer’s heavy-atom connectivity, and the largest T1 diagnostic stays at or below 0.02.⁸

Boundary. The source’s reference level — all-electron CCSD(T) at aug-cc-pVTZ and aug-cc-pVQZ with a two-point CBS extrapolation — was not reproduced: aug-cc-pVTZ on a C₄H₈ isomer is 368 basis functions, out of budget on the hardware below. Arm B therefore measures whether the exactly-zero prediction survives at a consistent correlated level, not what the source’s own protocol would return for these molecules. No conformer search and no frequency confirmation were part of the frozen protocol; each isomer is represented by the single stationary point reached from its committed starting structure. One post-hoc diagnostic outside the frozen protocol is reported as such in the Discussion.

Environment. The registered run executed 2026-08-03 on Ubuntu 24.04.4 (x86_64) with CPython 3.10.13, Psi4 1.9.1, and NumPy 1.26.4 from the committed conda lock, using 7 threads and 9 GB of memory, with energy and density convergence at 10⁻⁹; the full registered set is under an hour of wall time. The analysis layer (analyze.py) regenerates deterministically from the committed run records without rerunning quantum chemistry, and its --check mode byte-compares the committed results.json. The experiment bundle — preregistration with its amendment, frozen inputs, runner, raw Psi4 outputs, canonical results, and fingerprinted metrics — is published under research/cepb-increment-spread/. The reproducibility level this earns is end-to-end reproducible on linux-x86_64 from the committed lock, and analysis-reproducible from the committed outputs elsewhere.

Results

Every registered method-fidelity gate passed: all twelve optimizations and CCSD(T) calculations terminated normally, every optimized structure retains its intended isomer’s heavy-atom connectivity, and the largest T1 diagnostic across the C₄H₈ set is 0.0098 at cc-pVTZ and 0.0091 at cc-pVDZ, against the registered ceiling of 0.02.

Table 1 reports the Arm A contrasts. The spread across the three measured swap values is 0.51 kcal/mol at CBS, 1.78 kcal/mol at aug-cc-pVQZ, and 2.45 kcal/mol at aug-cc-pVTZ, against the registered threshold of 1.0 kcal/mol: at or below the threshold at CBS, above it at both finite bases. At every level, all three measured values are more negative than the CEPB prediction, by -3.66 to -4.17 kcal/mol at CBS, by -3.78 to -5.56 kcal/mol at aug-cc-pVQZ, and by -3.94 to -6.39 kcal/mol at aug-cc-pVTZ.

Table 1. Correlation-energy change of the C=C → C–C + 2 C–H swap in the three published pairs that realize it, computed from the source’s per-molecule correlation energies at each of its three published levels, with the single CEPB-predicted value and the spread across the three measured changes.

The secondary, descriptive statistic — the raw effective C=C increment of the amendment, which assigns each molecule’s whole CEPB residual to its C=C bonds — spans 10.38 kcal/mol across the seven-molecule set at CBS, 10.35 kcal/mol at aug-cc-pVQZ, and 11.38 kcal/mol at aug-cc-pVTZ, with cyclohexene at one extreme and ethene at the other at every level. Benzene’s Kekulé-counted effective increment falls inside the range spanned by the nonaromatic set at each level.

Table 2 reports the Arm B pairwise differences. The largest magnitude is 1.329 kcal/mol at cc-pVTZ and 1.357 kcal/mol at cc-pVDZ. Isobutene has the most negative correlation energy of the four isomers at both bases, and trans-2-butene the least negative. Five of the six pairs keep their sign between the two bases; the cis-2-butene − 1-butene pair is 0.168 kcal/mol at cc-pVDZ and -0.329 kcal/mol at cc-pVTZ.

Table 2. Pairwise frozen-core DF-CCSD(T) correlation-energy differences among the four C₄H₈ positional isomers at the registered basis and its sensitivity check. CEPB predicts every entry to be exactly zero.

Discussion

Both registered verdicts are inconclusive, and they stay that way. Arm A is inconclusive because the registered rule required the contrast spread to sit on one side of the threshold at every published basis level, and the levels disagree: the spread is within 1.0 kcal/mol at the source’s CBS reference and above it at both finite bases. Arm B is inconclusive because the frozen rule required every pairwise difference to keep its sign between cc-pVDZ and cc-pVTZ, and one pair — cis-2-butene − 1-butene — did not. The preregistration says a failed gate is reported, not repaired, so no goalpost moves; the paragraphs below interpret the parts of the data that are stable across every level run here, and say so explicitly when they step beyond the registered statistics.

The Arm A pattern that survives every level is not the spread but the offset. The three contrasts agree with each other to 0.51 kcal/mol at CBS — comfortably inside chemical accuracy — while all three miss the CEPB prediction in the same direction, by -3.66 to -4.17 kcal/mol at that level. On the source’s own best numbers, then, the C=C → C–C + 2 C–H swap behaves as if it has one well-defined price that is systematically different from the price the fitted increments assign it. That reading supports the source’s environment-independence claim for this swap — ring strain, homoconjugation, and substitution across these three pairs move the measured change by only half a kilocalorie at CBS — while locating the difficulty somewhere the whole-molecule error statistics cannot see it: a least-squares fit to total correlation energies can place increment combinations a few kcal/mol away from the value that reaction differences demand, because whole-molecule residuals, not reaction residuals, are what the fit minimizes. This is not a claim that the source’s fit is wrong — it is exactly what fitting to totals optimizes for — but it does mean that anyone forming reaction energies from CEPB increments inherits an offset of this size wherever this swap appears. The spread’s growth from CBS to the finite bases (1.78 and 2.45 kcal/mol), in step with the contrasts’ molecular sizes, reads as basis-set incompleteness rather than chemistry, which is why the registered every-level rule — written to be conservative — returned inconclusive rather than supported.

The amendment deserves its own accounting, with our hands up. The raw effective-increment spread — 10.38 kcal/mol at CBS — was computed before the contrast metric was registered, and the amendment that demoted it was recorded with that number known — the research journal and the preregistration both disclose this. The objection that motivated the amendment (raised independently by an automated review of a related change) is confirmed by the data: the raw spread is an order of magnitude larger than the contrast spread at every level and tracks molecule size, which is the signature of accumulated C–H and C–C residuals being booked to the C=C column, not of a variable C=C assignment. Both statistics are reported above so a reader can weigh the derivation order themselves.

In Arm B, the exactly-zero prediction misses by more than the registered threshold at both bases: the isobutene − trans-2-butene gap is -1.357 kcal/mol at cc-pVDZ and -1.329 kcal/mol at cc-pVTZ — same sign, nearly the same magnitude, and isobutene is the most strongly correlated isomer at both bases. Had the registered statistic been that gap alone, the hypothesis would have been supported; it is the smallest pair in the set, cis-2-butene − 1-butene, drifting through zero between the bases that trips the frozen sign rule. Two readings of that flip are open. It may be ordinary basis sensitivity of a difference an order of magnitude below the others. It may also be a geometry artifact on our side: the committed 1-butene starting structure has a planar anti carbon skeleton, a gradient-following optimizer cannot leave a symmetry plane, and a post-hoc frozen-core DF-MP2/cc-pVDZ frequency calculation on the committed 1-butene structure — run after the registered analysis was complete, outside the frozen protocol, and committed under the experiment’s diagnostics/ directory — finds an imaginary torsional mode, so the committed 1-butene point is a saddle of the torsional profile rather than a minimum. The three other isomers’ committed structures are unaffected, and the robust isobutene − trans-2-butene finding does not involve 1-butene; but the frozen rule makes no exception for explanations arrived at afterwards, and the Arm B verdict remains inconclusive.

The limitations are the boundary conditions stated in Methods. Arm B is frozen-core DF-CCSD(T) at cc-pVTZ and cc-pVDZ on MP2 geometries; the source’s reference is all-electron at augmented bases with a CBS extrapolation, so our pairwise differences are not predictions of what the source’s protocol would return, and correlation differences of a kilocalorie at these bases can move toward CBS. Each isomer is one stationary point, not a conformer ensemble, and for 1-butene demonstrably not the equilibrium one. Arm A inherits the source’s published rounding and its geometry choices, which differ between its training and test sets. And the contrasts isolate one swap in one bond family; nothing here measures any of the other 32 increments.

If the sign flip, the offset, or the isobutene separation has an explanation we have missed — a transcription error on our side, a known basis artifact of density-fitted CCSD(T) on branched alkenes, or prior literature that has already priced this swap — we would genuinely like to hear it, and the frozen bundle is published so that checking us is cheap.

Conclusion

Within the source’s own published data, the C=C → C–C + 2 C–H swap has a consistent price across three chemical environments at the CBS reference level — and it is not the price the fitted increments charge. In new calculations on the four butenes, the correlation energies CEPB declares identical span more than a kilocalorie per mole at both bases run here, with isobutene consistently the most correlated. Both registered verdicts are nonetheless inconclusive under their own frozen rules, and the two stable findings are narrower than the question this experiment set out to ask: a systematic offset in one swap’s price, and one robust pairwise split the model sets to zero.

The next experiment is the one the offset points at directly: reprice the swap. Take the contrast-derived value of \(e_{\mathrm{CC}} + 2e_{\mathrm{CH}} - e_{\mathrm{C=C}}\) measured here from the source’s own CBS tables, and test — by arithmetic on the source’s published tables, with no new quantum chemistry — whether that single correction reduces CEPB’s errors on the reactions in its published test set that contain this swap, without degrading the total-energy accuracy the model was built for. That question is now on the research shelf.

References