How electron correlation survives a hydrogenation enthalpy subtraction
The question
A previous note in this series measured electron correlation as a total energy — the gap between Hartree–Fock and CCSD(T) for one water molecule. A total correlation energy is large, but chemistry runs on energy differences, where much of it cancels. This post turns the same subtraction on a reaction and follows it in three steps: form the reaction enthalpy at each electronic-structure level, cancel the shared thermal correction, and inspect the small correlation residual that remains.
The worked example uses HF and CCSD(T)/cc-pVTZ single points (frozen core) at B3LYP/def2-TZVP geometries for 4 closed-shell gas-phase hydrogenations. For the two reactions that each remove one C–C π bond, the residual is 5.9 kJ/mol (acetylene → ethylene) and -1.7 kJ/mol (ethylene → ethane). They differ by 7.6 kJ/mol and have opposite signs. The point is not to localise correlation onto either bond—the calculation does not do that—but to see why a reaction-level residual cannot automatically be promoted into a transferable bond property.
From molecular correlation to a reaction residual
The mean field misses a lot. For a single water molecule the correlation energy — the difference between the exact non-relativistic energy and the Hartree–Fock limit — runs to hundreds of kJ/mol, and the preceding note measured it directly by marching a basis-set ladder to its limit and dropping CCSD(T) below it. That earlier post, and the one before it on the Hartree–Fock machinery itself, both stopped at a total energy of a single system. But no experiment measures the total energy of a molecule; it measures the energy released or absorbed when one molecule turns into another. The question that matters for chemistry is therefore not how large correlation is, but how much of it survives a reaction.
When a reaction conserves the number and type of chemical bonds, the large per-molecule correlation energies on the two sides can cancel substantially. This is the reasoning behind isodesmic and bond-separation reaction schemes, which are built so that method error cancels and even modest methods give useful reaction energies.1 A hydrogenation is not isodesmic—it trades a π bond and an H–H bond for two C–H bonds—so complete cancellation is not built into the reaction.
“Small,” though, is not a number, and “the correlation of one C–C π bond” is a transferability shortcut that the cancellation argument neither states nor justifies. Hydrogenating acetylene to ethylene removes one π bond from a carbon–carbon triple bond; hydrogenating ethylene to ethane removes the one π bond of a carbon–carbon double bond. Both reactions hydrogenate exactly one C–C π bond. If the correlation contribution to a hydrogenation enthalpy were a transferable per-π-bond quantity, these two reaction-level residuals would be similar.
That comparison gives the explanation a useful scale. Chemical accuracy, 4.184 kJ/mol, is the tolerance within which the two residuals would behave like the same practical increment. The ammonia and methanol reactions then show how the same subtraction behaves for N–N and C–O multiple bonds. Together these examples separate two ideas that are easy to conflate: correlation can cancel strongly in a reaction while the small remainder is still not transferable between reactions.
What cancels, and what remains in the model
Composite and why the thermal correction cancels. For each species a geometry optimisation and harmonic-frequency calculation at B3LYP/def2-TZVP supplies the geometry and the ideal-gas rigid-rotor/harmonic-oscillator thermal and zero-point enthalpy correction to 298.15 K and 1 atm.2,3 At that geometry a single CCSD(T)/cc-pVTZ calculation with frozen core supplies two electronic energies at once: its converged self-consistent-field total is the Hartree–Fock baseline, and its CCSD(T) total is the correlated value.4,5 A reaction enthalpy at each level adds the same B3LYP thermal correction to the electronic reaction energy, so in the difference
\[ \Delta_{\mathrm{corr}} \;=\; \Delta H_{\mathrm{CCSD(T)}} - \Delta H_{\mathrm{HF}} \]
the thermal correction cancels exactly, and \(\Delta_{\mathrm{corr}}\) is the correlation part of the electronic reaction energy — no more, no less. This is a composite recipe (a coupled-cluster energy on a DFT geometry with a DFT thermal correction) and it is stated plainly rather than hidden: the quantity it isolates is a correlation difference. The thermal layer drops out algebraically. The common geometry is held fixed between methods, so geometry relaxation does not enter the subtraction, but that geometry remains part of the model and can affect the residual.
The worked reaction set. Four closed-shell singlet gas-phase hydrogenations make the cancellation concrete. Two form the C–C comparison, and two provide multiple-bond context:
| slug | reaction | C–C π bonds | role |
|---|---|---|---|
| acetylene_hydrogenation | C₂H₂ + H₂ → C₂H₄ | 1 | transferability rung A |
| ethylene_hydrogenation | C₂H₄ + H₂ → C₂H₆ | 1 | transferability rung B |
| ammonia_synthesis | N₂ + 3 H₂ → 2 NH₃ | — (N–N) | triple-bond context |
| methanol_synthesis | CO + 2 H₂ → CH₃OH | — (C–O) | triple-bond context |
The two C–C rungs each hydrogenate one π bond. Their per-π-bond value is therefore the reaction contribution itself. The ammonia and methanol reactions reduce N–N and C–O triple bonds, respectively, and provide context rather than additional C–C comparisons.
Four hydrogenations as a worked example
Table 1 lists, for each reaction, the CCSD(T) enthalpy at 298.15 K, the correlation contribution ΔH(CCSD(T)) − ΔH(HF), and that contribution as a fraction of the CCSD(T) enthalpy.
| reaction | ΔH(CCSD(T)) (kJ/mol) | Δcorr (kJ/mol) | Δcorr / ΔH(CCSD(T)) |
|---|---|---|---|
| C₂H₂ + H₂ → C₂H₄ | -178.6 | 5.9 | -3.3% |
| C₂H₄ + H₂ → C₂H₆ | -139.0 | -1.7 | 1.2% |
| N₂ + 3 H₂ → 2 NH₃ | -75.1 | -9.3 | 12.3% |
| CO + 2 H₂ → CH₃OH | -80.1 | -21.6 | 27.0% |
Table 1. CCSD(T)/cc-pVTZ hydrogenation enthalpies at 298.15 K, the correlation contribution ΔH(CCSD(T)) − ΔH(HF) to each, and that contribution as a fraction of the CCSD(T) enthalpy. Δcorr carries the sign of ΔH(CCSD(T)) − ΔH(HF). Across the four reactions the mean absolute correlation contribution is 9.6 kJ/mol and the largest absolute value is 21.6 kJ/mol.
For the two hydrocarbon rungs the correlation contribution is 5.9 kJ/mol (acetylene → ethylene, against a CCSD(T) enthalpy of -178.6 kJ/mol) and -1.7 kJ/mol (ethylene → ethane, against -139.0 kJ/mol). Each reaction hydrogenates one C–C π bond, and these values are the per-π-bond contributions. Their difference is 7.6 kJ/mol, against a chemical-accuracy comparison scale of 4.184 kJ/mol, and the two values carry opposite signs. The corresponding within-threshold check is false.
For the two context reactions the correlation contribution is -9.3 kJ/mol (N₂ + 3 H₂ → 2 NH₃) and -21.6 kJ/mol (CO + 2 H₂ → CH₃OH), the latter 27.0% of its CCSD(T) enthalpy. Every optimised species had no imaginary vibrational mode (true).
Why the residual is reaction-specific
The narrow comparison is straightforward. A reusable per-π-bond increment would have to describe both C–C rungs within 4.184 kJ/mol. Instead, the residuals sit 7.6 kJ/mol apart and on opposite sides of zero. On the acetylene rung correlation makes hydrogenation less exothermic (Δcorr positive); on the ethylene rung it makes hydrogenation slightly more exothermic (Δcorr negative). These are reaction-level balances. The calculation does not partition correlation energy among bonds, so their opposite signs do not establish that the triple bond itself is the cause. What they establish is narrower: a single per-π-bond increment cannot reproduce both rungs; calibrating on one would miss the other by 7.6 kJ/mol.
The magnitudes put that difference in perspective. Both hydrocarbon contributions are small in absolute terms: 5.9 and -1.7 kJ/mol against CCSD(T) enthalpies of -178.6 and -139.0 kJ/mol, respectively. Their signed ratios to the CCSD(T) enthalpies are -3.3% and 1.2%. That is the electron-count cancellation the isodesmic argument predicts; the transferability mismatch lives inside that small surviving residue, not in the bulk of the enthalpy. A gap of 7.6 kJ/mol is minor next to the measured reaction enthalpies and decisive next to the 4.184 kJ/mol accuracy target — which is why the per-bond question needs a number rather than an adjective.
The two non-C–C reactions show that the C–C rungs are, if anything, the benign end of the range. Ammonia synthesis carries -9.3 kJ/mol and methanol synthesis -21.6 kJ/mol — the latter 27.0% of its own CCSD(T) enthalpy. The methanol-synthesis entry has the largest absolute correlation contribution in the set, 21.6 kJ/mol. The spread across the four reactions, from 5.9 to -21.6 kJ/mol, is the measured picture: the correlation contribution is a reaction-specific residual, not a transferable per-bond increment.
This pattern is compatible with the error-cancellation logic of isodesmic schemes: those schemes conserve bond types, whereas these hydrogenations do not. The worked values make the distinction concrete: substantial cancellation does not imply that the remainder belongs to one bond. Holding the geometry fixed and cancelling the thermal layer isolates an electronic reaction difference, not a bond-energy partition.
Where the model stops
The boundaries are worth stating plainly, because they limit what the number means. The basis is finite: cc-pVTZ is not the complete-basis limit, and the correlation energy of triple bonds such as N₂ converges notoriously slowly with basis size, so the absolute contributions — especially for ammonia and methanol — would shift under extrapolation.1 The reference is CCSD(T) itself, with no higher-order or multireference check, so this measures the correlation that CCSD(T) recovers, not necessarily the exact correlation. The RRHO thermal correction cancels in the method difference, but the B3LYP geometry remains part of the model and could shift the correlation contribution. The worked comparison contains one pair of C–C rungs, not a statistical sample of C–C π bonds, and it does not partition correlation energy among atoms or bonds. A basis- and geometry-sensitivity study could test how stable the 7.6 kJ/mol gap is.
Reproducibility
Every energy was produced locally by psi4 1.9.1 with tight SCF and geometry
thresholds.6 The run was single-threaded
(OMP_NUM_THREADS=1) to avoid changes in threaded floating-point reduction
order, and it used no random seeds. Every optimised species was a true minimum
with no imaginary vibrational mode
(true across the set).
The pinned conda environment and calculate.py regenerate the species and
reaction results. A psi4-free calculate.py --check reconstructs every reaction
quantity from the committed per-species energies, and the metrics generator
fingerprints those inputs before projecting the values used here. The
calculation uses no external data, network service, paid service, or living
subjects.
What to calculate next
Turning the water note’s CCSD(T)−HF subtraction from a molecule onto a reaction gives small correlation residuals for the hydrocarbon hydrogenations. The residual is 5.9 kJ/mol for the acetylene rung and -1.7 kJ/mol for the ethylene rung—a gap of 7.6 kJ/mol. The calculation therefore supplies a useful worked warning: a small reaction-level correlation term is not automatically a transferable bond property.
The comparison also changes the starting C–C bond order along with the rest of the reaction-level correlation balance. Whether a stable increment emerges among reactions that all reduce the same bond order — a family of isolated C–C double bonds hydrogenated to single bonds, say — is a separate, and answerable, measurement. That would distinguish a genuine per-bond correlation increment from the change in starting bond order that this comparison leaves entangled.