Ammonia, high-temperature shift and methanol equilibria tolerate only 0.30-0.65 kJ/mol of reaction-enthalpy error per point of conversion

Thinh D. Ho | CheAI Research Inc. | Research article | November 2025

Abstract

1. Introduction

An equilibrium reactor model computes K(T) from a reaction enthalpy, an entropy and heat capacities. For the large syngas and ammonia reactions, the Active Thermochemical Tables (ATcT) give the 298 K reaction enthalpy to within 0.033-0.144 kJ/mol [1, 2]. Where evaluated data do not exist, quantum chemistry can estimate the enthalpy. W1 theory reports a mean absolute error of 0.30 kcal/mol on its training set [3], and W4 and HEAT approach 1 kJ/mol [4, 5]. Such benchmarks usually target 1 kcal/mol (4.184 kJ/mol).

A process engineer needs a different measure, the conversion error at the operating point. We looked for the cheapest level of quantum chemistry that keeps the conversion of four reactions within 1 percentage point (pp). None of the 16 tested levels met that target. The sensitive cases tolerate an enthalpy error of only 0.30-0.65 kJ/mol.

2. Methods

Reference enthalpies of formation came from ATcT version 1.220 [2], with reaction uncertainties (about 95 % confidence) from the full ATcT covariance. All quantum chemistry ran in PySCF 2.14.0 [6]. Geometries, frequencies and thermal terms came from B3LYP/cc-pVTZ at every level, so the levels differ only in the electronic energy. The levels ran from density functionals through Hartree-Fock, MP2 and CCSD to eight CCSD(T) levels, with single basis sets or complete-basis-set (CBS) extrapolations. The top rung added core-valence and scalar-relativistic corrections to CCSD(T)/CBS. We call it W1-type because it departs from W1 [3] in five choices, all fixed before any comparison with ATcT. The error δ of a level is its 298 K reaction enthalpy minus the ATcT value, per mole of reaction as written (ammonia synthesis per 2 mol NH₃).

The reference equilibrium constant used NASA 7-coefficient polynomials [7] re-anchored to ATcT. Each error was applied as a constant enthalpy offset,

ln K_level(T) = ln K_ref(T) - δ/(RT),

and the ideal-gas equilibrium was re-solved without linearization. The eight cases are prescribed-temperature equilibria from three earlier process studies (see Related studies). They cover ammonia synthesis (755 K, 300 bar) and steam methane reforming (SMR, 1273 K, 20 bar) alone and with the shift. The others are high-temperature shift (HTS, 673 K), low-temperature shift (LTS, 483 K) and methanol synthesis (483 K, 76 bar) alone and with the reverse water-gas shift (RWGS). In the two coupled cases, both reaction constants were shifted.

A level was adequate if |δ| ≤ 4.184 kJ/mol for all four reactions and the conversion error |ΔX| ≤ 1 pp at every point. The tolerance of a point is the smaller-magnitude enthalpy error that moves conversion by exactly 1 pp, found by bisection. A second solver and Cantera 3.2 [8] reproduced the equilibria, and an independent re-derivation matched all 128 level-point equilibria within 1.25 × 10^-9 pp.

3. Results

3.1 Only two levels met 1 kcal/mol, by error cancellation

The density functionals missed by 16 to 33 kJ/mol on their worst reaction. Only CCSD(T)/haVTZ (largest error 3.91 kJ/mol) and CCSD(T)/haVQZ (3.65 kJ/mol) kept all four errors within 1 kcal/mol. Extrapolating the same energies to the basis-set limit made ammonia (-1.90 to -5.12 kJ/mol) and methanol synthesis (-3.65 to -5.59 kJ/mol) worse. The two passes therefore rest on basis-set error that offsets the missing terms.

The W1-type composite had errors of +3.40 (reforming), +1.14 (shift), -4.88 (ammonia) and -4.99 kJ/mol (methanol). It missed 1 kcal/mol for ammonia and methanol synthesis by 0.69 and 0.81 kJ/mol, and checks found no implementation error behind these misses. Across the four best levels, the reaction-enthalpy errors were 8 to 117 times the corresponding ATcT uncertainty.

3.2 No level kept conversion within 1 pp at every point

The smallest worst-point error was 10.00 pp, for CCSD(T)/haVQZ at HTS (Figure 1). CCSD(T)/haVTZ reached 14.65 pp, CCSD(T)/CBS(TQ) 13.23 pp and the W1-type composite 11.85 pp.

The reforming cases were the exception, because the reference CH₄ conversion is 98.43 %. No level moved it by more than 4.11 pp. Nine levels stayed within 1 pp there. Four of these passes arose from saturation and did not count toward adequacy. At the LTS fed with the CO-lean HTS outlet, three levels gave errors beyond ±100 pp, because their shift errors made the reactor produce more CO than entered.

3.3 Sensitive cases tolerate less than 1 kJ/mol

HTS tolerated 0.30 kJ/mol of enthalpy error per 1 pp of conversion, the LTS fed with the HTS outlet 0.42 kJ/mol and the two methanol cases 0.42 and 0.43 kJ/mol (Figure 2a). Ammonia synthesis tolerated 0.65 kJ/mol. A 1 kcal/mol error moved conversion by 6.3 to 15.8 pp at the four sensitive points. Only the reforming points, at 5.3-5.6 kJ/mol, are less demanding than 1 kcal/mol.

At the sensitive points, almost every error of the four best levels exceeded the tolerance, by factors of 2.7 to 13. No level was adequate at half, once or twice the thresholds. More computer time lowered the worst enthalpy error from about 58 to 3.7 kJ/mol, but the worst conversion error stayed at 10 pp or more (Figure 2b, c).

3.4 Quantum-chemistry error exceeds data and database effects

At the two high-pressure points, the ATcT uncertainty moved conversion by 0.088 pp (ammonia) and 0.34 pp (methanol). Burcat's enthalpies of formation in place of ATcT's moved it by +1.18 and +0.32 pp (Figure 3). Peng-Robinson fugacities [9] raised it by 3.18 pp at 300 bar and 5.98 pp at 76 bar. CCSD(T)/haVQZ gave errors of 2.91 and 8.72 pp, and the W1-type composite 7.45 and 11.85 pp.

4. Discussion

The tolerance is small because an enthalpy error enters ln K divided by RT. At 483 K, 1 kcal/mol changes ln K by 1.04, a factor of 2.8 in K. For the 1 pp criterion adopted, the 1 kcal/mol target of benchmark thermochemistry is an order of magnitude looser than the tolerance of the ammonia, shift and methanol cases.

The required accuracy is below what standard composite methods claim [3-5]. Error cancellation helped the haVTZ and haVQZ levels but disappeared at the basis-set limit. The terms beyond CCSD(T)/CBS that the recipe leaves out are one possible cause of the W1-type errors [10], and the zero-point-energy convention is another. Whether a W4- or HEAT-class recipe reaches 0.3 kJ/mol for these reactions remains open.

For reactions in ATcT, ATcT enthalpies are the better choice for an equilibrium model. Re-anchoring other databases to ATcT removes a data-source effect of up to 1.18 pp, and non-ideality, at 3 to 6 pp in the high-pressure cases, deserves attention next. Where an enthalpy must be computed, two one-dimensional searches give the tolerance before any quantum chemistry is run. At a sensitive point, the tested enthalpies fixed the sign and rough size of an effect, but not the conversion to within a percentage point.

5. Limitations

The conclusion holds for the 16 levels tested, for an enthalpy-only perturbation of the stated model and for the 1 pp criterion. Canonical W1, W4 and HEAT were not tested. The cases are prescribed-temperature equilibria, not whole-reactor models, and downstream feeds stay at their reference compositions. Conversions at 300 and 76 bar are ideal-gas values, and Peng-Robinson fugacities move them by 3.2 and 6.0 pp.

On the method side, the 298 K error is applied at all temperatures, with no level-specific entropy or heat-capacity error. Thermal terms are rigid-rotor harmonic-oscillator values from one B3LYP geometry, with a harmonic methanol torsion and a scaled harmonic zero-point energy. They agree with ATcT within 0.076 kJ/mol per reaction. Post-(T) correlation, the diagonal Born-Oppenheimer correction and anharmonic zero-point energies were not computed, and the core-valence term has an unquantified basis-set error. CO and CO₂ have T₁ diagnostics of 0.018-0.019, close to the 0.02 flag for multireference character. The ammonia zero-point-energy error is uncertain by about 0.6 kJ/mol, depending on the benchmark, and the methanol error could not be decomposed.

All electronic energies come from one program. The ATcT covariance response has no network version field, so its association with version 1.220 rests on the common retrieval time. Wall times come from a shared container, so close cost rankings between cheap levels are not meaningful.

6. Conclusions

Under an enthalpy-only perturbation of an ATcT-anchored model, none of 16 tested levels kept the equilibrium conversion of ammonia synthesis, the water-gas shift and CO₂-to-methanol within one percentage point in every case. The two levels that met 1 kcal/mol did so by error cancellation. The sensitive cases tolerate an error several times smaller than 1 kcal/mol, and ATcT uncertainties met that tolerance in the cases examined. A computed enthalpy should first be checked against the tolerance of its operating point.

References

1. Ruscic B, Pinzon RE, Morton ML, von Laszevski G, Bittner SJ, Nijsure SG, et al. Introduction to Active Thermochemical Tables. Several "key" enthalpies of formation revisited. Journal of Physical Chemistry A 108, 9979-9997, 2004.
2. Ruscic B, Bross DH. Active Thermochemical Tables (ATcT) values based on ver. 1.220 of the Thermochemical Network. Argonne National Laboratory, 2025. atct.anl.gov, accessed 2025-10-20.
3. Martin JML, de Oliveira G. Towards standard methods for benchmark quality ab initio thermochemistry, W1 and W2 theory. Journal of Chemical Physics 111, 1843-1856, 1999.
4. Karton A, Rabinovich E, Martin JML, Ruscic B. W4 theory for computational thermochemistry. In pursuit of confident sub-kJ/mol predictions. Journal of Chemical Physics 125, 144108, 2006.
5. Tajti A, Szalay PG, Császár AG, Kállay M, Gauss J, Valeev EF, et al. HEAT. High accuracy extrapolated ab initio thermochemistry. Journal of Chemical Physics 121, 11599-11613, 2004.
6. Sun Q, Zhang X, Banerjee S, Bao P, Barbry M, Blunt NS, et al. Recent developments in the PySCF program package. Journal of Chemical Physics 153, 024109, 2020.
7. Goos E, Burcat A, Ruscic B. Extended Third Millennium Ideal Gas Thermochemical Database with updates from Active Thermochemical Tables. File BURCAT.THR, 2023, downloaded 2025-10-20.
8. Goodwin DG, Speth RL, Moffat HK, Weber BW. Cantera. An object-oriented software toolkit for chemical kinetics, thermodynamics, and transport processes, version 3.2.0. www.cantera.org, 2025.
9. Peng DY, Robinson DB. A new two-constant equation of state. Industrial and Engineering Chemistry Fundamentals 15, 59-64, 1976.
10. Feller D, Peterson KA, Dixon DA. A survey of factors contributing to accurate theoretical predictions of atomization energies and molecular structures. Journal of Chemical Physics 129, 204105, 2008.

Related studies

• Raoult's law misplaces the cost-optimal condenser temperature of a 300 bar ammonia synthesis loop, the source of the ammonia case
• Syngas CO₂ capture on a steam methane reformer matches grey hydrogen at 54 USD/t, the source of the reforming and shift cases
• Hydrogen sets 94 % of the cost of methanol made from captured CO₂, the source of the methanol case