Estimating Systematic Error and Uncertainty in Ab Initio Thermochemistry: II. ATOMIC(hc) Enthalpies of Formation for a Large Set of Hydrocarbons

High-Throughput Predictions of Accurate Enthalpies of Formation for Larger Molecules Utilizing the Bond Difference Correction Method

The prediction of standard enthalpies of formation (EOFs) for larger molecules involves a trade-off between accuracy and cost, often resulting in non-negligible errors. The connectivity-based hierarchy (CBH) and simple bond additivity correction (BAC) are two promising means for evaluating EOFs, although they cannot achieve strict chemical accuracy. Calculated errors in the CBH are confirmed from accumulated systematic errors associated with bond differences in chemical environments. On the basis of a new set of bond descriptors, our developed bond difference correction (BDC) method effectively solves incremental errors with molecular size and inability applications for aromatic molecules. To balance the accuracy between non-aromatic and aromatic molecules, a more accurate BAC-based method with unpaired electrons and p hybrid orbitals (BAC-EP) is developed. With the incorporation of the two methods above, strict chemical accuracy by the largest deviation is achieved at low costs. These universal, ultrafast, and high-throughput methods greatly contribute to self-consistent thermodynamic parameters in combustion mechanisms.

Potential Energy Surfaces Sampled in Cremer-Pople Coordinates and Represented by Common Force Field Functionals for Small Cyclic Molecules

The complex conformations of the cyclic moieties impact the physical and chemical properties of molecules. In this work, we chose 22 molecules of four-, five-, and six-membered rings and performed a thorough conformational sampling using Cremer-Pople coordinates. With consideration of symmetries, we obtained a total of 1504 conformational structures for four-membered, 5576 for five-membered, and 13509 for six-membered rings. All well-known and many less well-known conformers for each molecule were identified. We represented the potential energy surfaces (PESs) by fitting the data to common analytical force field (FF) functional forms. We found that the general features of PESs can be described by the essential FF functional forms; however, the accuracy of representation can be improved remarkably by including the torsion-bond and torsion-angle coupling terms. The best fit yields R-squared (R²) values close to 1.0 and mean absolute errors in energy less than 0.3 kcal/mol.

ATOMIC-2 Protocol for Thermochemistry

ATOMIC is a midlevel thermochemistry protocol that uses Pople's concept of bond separation reactions (BSRs) as a theoretical framework to reduce computational demands in the evaluation of atomization energies and enthalpies of formation. Various composite models are available that approximate bond separation energies at the complete-basis-set limit of all-electron CCSD(T), each balancing computational cost with achievable accuracy. Evaluated energies are then combined with very high-level, precomputed atomization energies of all auxiliary molecules appearing in the BSR to obtain the atomization energy of the molecule under study. ATOMIC-2 is a new version of the protocol that retains the overall concept and all previously defined composite models but improves on ATOMIC-1 in various other ways: Geometry optimization and zero-point-energy evaluation are performed at the density functional level (PBE0-D3/6-311G(d)), which shows significant computational savings and better accuracy than the previously employed RI-MP2/cc-pVTZ. The BSR framework is improved, using more accurate complete-basis-set (CBS) extrapolations toward the Full CI limit for the atomization energies of all auxiliary molecules. Finally, and most importantly, an error and uncertainty model termed ATOMIC-2_um is added that estimates average bias and uncertainty for each of the atomization energy contributions that arise from the simplified treatment of some contributions to bond separation energies (CCSD(T)) and the neglect of others (such as higher order, scalar relativistic, or diagonal Born-Oppenheimer corrections) or from residual error in the energies of auxiliary molecules. Large and diverse benchmarks including up to 1179 molecules are used to evaluate necessary reference data and to correlate the observed error for each of the contributions with appropriate proxies that are available without additional quantum-chemical calculations for a particular molecule and represent its size and type. The implementation of ATOMIC-2 considers neutral, closed-shell molecules containing H, C, N, O, and F atoms; compared to ATOMIC-1, the framework has been extended to cover a few challenging but rare bond topologies. In comparison to highly accurate reference data for 184 molecules taken from the ATcT database (V. 1.122r), regular ATOMIC-2 shows noticeable underbinding, but the bias-corrected protocol ATOMIC-2_um is found to be more accurate than either ATOMIC-1 or standard Gaussian-4 theory, and the uncertainty model is consistent with statistics of actually observed errors. Problems arising from ambiguous or challenging Lewis-valence structures defining BSRs are discussed, and computational efficiency is demonstrated. Computer code is made available to perform ATOMIC-2_um analyses.

The gas-phase infrared spectra of the 2-methylallyl radical and its high-temperature reaction products

The resonance-stabilized 2-methylallyl radical, 2-MA, is considered as a possible intermediate in the formation of polycyclic aromatic hydrocarbons (PAHs) in combustion processes. In this work, we report on its contribution to molecular growth in a high-temperature microreactor and provide mass-selective IR/UV ion dip spectra of the radical, as well as the various jet-cooled reaction products, employing free electron laser radiation in the mid-infrared region. Small (aromatic) hydrocarbons such as fulvene, benzene, styrene, or para-xylene, as well as polycyclic molecules, like (methylated) naphthalene, were identified with the aid of ab initio DFT computations. Several reaction products differ by one or more methyl groups, suggesting that molecular growth is dominated by (de)methylation in the reactor.

The long road to calibrated prediction uncertainty in computational chemistry

Uncertainty quantification (UQ) in computational chemistry (CC) is still in its infancy. Very few CC methods are designed to provide a confidence level on their predictions, and most users still rely improperly on the mean absolute error as an accuracy metric. The development of reliable UQ methods is essential, notably for CC to be used confidently in industrial processes. A review of the CC-UQ literature shows that there is no common standard procedure to report or validate prediction uncertainty. I consider here analysis tools using concepts (calibration and sharpness) developed in meteorology and machine learning for the validation of probabilistic forecasters. These tools are adapted to CC-UQ and applied to datasets of prediction uncertainties provided by composite methods, Bayesian ensembles methods, and machine learning and a posteriori statistical methods.

Gas-phase thermochemistry of polycyclic aromatic hydrocarbons: an approach integrating the quantum chemistry composite scheme and reaction generator

We introduce a protocol aimed at predicting the accurate gas-phase enthalpies of formation of polycyclic aromatic hydrocarbons (PAHs). Automatic generation of a dataset of equilibrated chemical reactions preserving the number of carbon atoms in each hybridization state on each side of equations is at the core of our scheme. The performed tests suggest the recommended enthalpy of formation to be derived via a two-step scheme. First, we consider the reactions with a minimal sum of the total number of particles involved, N, and the absolute difference between the total number of products and reactants, |ΔN|. Second, among these reactions, we identify the one with the smallest absolute reaction enthalpy change, . This approach has been applied to predict the gas-phase enthalpies of formation of 113 PAHs via the Feller-Peterson-Dixon approach. Our calculated values provide the mean absolute deviations of 1.7, 1.9, 4.2, 8.1, and 18.5 kJ mol^-1 with respect to the literature group-based error corrected (GBEC) G3MP2B3, ATOMIC (HC), group equivalent M06-2X, GBEC B3LYP, and G4MP2 values. Our predicted values give the mean signed and mean absolute errors of -7.5 and 12.9 kJ mol^-1 with respect to the experimental enthalpies of formation. The combination of our predicted and the experimental values provide the solid-state enthalpies of formation, , which are not available for a few species. Approaching these values as well as , producing large discrepancies from the experimental side, would be indispensable for testing and further tuning of computational chemistry approaches.

Enthalpy formation of fluorene: a challenging problem for theory or experiment?

The large discrepancy between the experimental enthalpy of formation of fluorene and theoretical value calculated by the G3(MP2) method was revealed more than ten years ago. Three years later, a new experimental study of this compound was undertaken to ascertain whether there is any significant error in the thermochemical data. However, after this research, the agreement between theory and experiment was improved only slightly. In this work we decided to calculate the enthalpy of formation of fluorene using the high-level DLPNO-CCSD(T₁)/CBS method which shows better results compared to Gn theories. To examine the accuracy of the available experimental data, the calculations were performed not only for fluorene but also for eleven fluorene derivatives. The discrepancy of about 9 kJ mol^-1 between the experimental and theoretical enthalpies of formation of fluorene was confirmed by the present calculations, whereas good agreement was observed for the fluorene derivatives. It is highly unlikely that this discrepancy may disappear when using a higher-level theory. The possible reason for such inconsistency might be the experimental difficulty associated with the glass transition discovered in the stable crystalline state of fluorene. In this case, new experiments using the latest methods, such as differential scanning calorimetry combined with X-ray powder diffraction, are needed to gain deeper insight into the solid phase transformations of fluorene.

Density Functional Geometries and Zero-Point Energies in Ab Initio Thermochemical Treatments of Compounds with First-Row Atoms (H, C, N, O, F)

Density functionals are often used in ab initio thermochemistry to provide optimized geometries for single-point evaluations at a high level and to supply estimates of anharmonic zero-point energies (ZPEs). Their use is motivated by relatively high accuracy at a modest computational expense, but a thorough assessment of geometry-related error seems to be lacking. We have benchmarked 53 density functionals, focusing on approximations of the first four rungs and on relatively small basis sets for computational efficiency. Optimized geometries of 279 neutral first-row molecules (H, C, N, O, F) are judged by energy penalties relative to the best available geometries, using the composite model ATOMIC/B5 as energy probe. Only hybrid functionals provide good accuracy with root-mean-square errors around 0.1 kcal/mol and maximum errors below 1.0 kcal/mol, but not all of them do. Conspicuously, first-generation hybrids with few or no empirical parameters tend to perform better than highly parameterized ones. A number of them show good accuracy already with small basis sets (6-31G(d), 6-311G(d)). As is standard practice, anharmonic ZPEs are estimated from scaled harmonic values. Statistics of the latter show less performance variation among functionals than observed for geometry-related error, but they also indicate that ZPE error will generally dominate. We have selected PBE0-D3/6-311G(d) for the next version of the ATOMIC protocol (ATOMIC-2) and studied it in more detail. Empirical expressions have been calibrated to estimate bias corrections and 95% uncertainty intervals for both geometry-related error and scaled ZPEs.

Detailed kinetics of hydrogen abstraction from trans-decalin by OH radicals: the role of hindered internal rotation treatment

In this study, the detailed kinetic mechanism of the trans-decalin + OH reaction is firstly investigated for a wide range of conditions (i.e., T = 200-2000 K & P = 0.76-76 000 Torr) using the M06-2X/aug-cc-pVTZ level and stochastic Rice-Ramsperger-Kassel-Marcus based master equation (RRKM-ME) rate model, which includes corrections of the hindered internal rotor (HIR) and tunneling effects. Our predicted global rate constant excellently matches with the scarce experimental measurement (R. Atkinson, et al. Int. J. Chem. Kinet., 1983, 15, 37-50). The H-abstraction channel from Cα of trans-decalin is found to be dominant at low temperatures. A U-shaped temperature-dependent behavior and slightly positive pressure-dependence at low temperatures (e.g., T ≤ 400 K & P = 760 Torr) of the total rate constants are also observed. Detailed analysis reveals that the HIR treatment is essential to capture the kinetic behavior while the tunneling correction only plays a minor role.

Fullerene Thermochemical Stability: Accurate Heats of Formation for Small Fullerenes, the Importance of Structural Deformation on Reactivity, and the Special Stability of C60

We have used quantum chemistry computations, in conjunction with isodesmic-type reactions, to obtain accurate heats of formation (HoFs) for the small fullerenes C₂₀ (2358.2 ± 8.0 kJ mol^-1), C₂₄ (2566.2 ± 7.6), and the lowest-energy isomers of C₃₂ (2461.1 ± 15.4), C₄₂ (2629.0 ± 20.5), and C₅₄ (2686.2 ± 25.3). As part of this endeavor, we have also obtained accurate HoFs for several medium-sized molecules, namely 216.6 ± 1.4 for fulvene, 375.5 ± 1.5 for pentalene, 670.8 ± 2.9 for acepentalene, and 262.7 ± 2.5 for acenaphthylene. We combine the energies of the small fullerenes and previously obtained energies for larger fullerenes (from C₆₀ to C₆₀₀₀) into a full picture of fullerene thermochemical stability. In general, the per-carbon energies can be reasonably approximated by the "R+D" model that we have previously developed [Chan et al. J. Chem. Theory Comput. 2019, 15, 1255-1264], which takes into account Resonance and structural Deformation factors. In a case study on C₅₄, we find that most of the high-deformation-energy atoms correspond to the sites of the C-Cl bond in the experimentally captured C₅₄Cl₈. In another case study, we find that C₆₀ has the lowest value for the maximum local-deformation energy when compared with similar-sized fullerenes, which is consistent with its "special stability". These results are indicative of structural deformation playing an important role in the reactivity of fullerenes.

Probabilistic performance estimators for computational chemistry methods: Systematic improvement probability and ranking probability matrix. I. Theory

The comparison of benchmark error sets is an essential tool for the evaluation of theories in computational chemistry. The standard ranking of methods by their mean unsigned error is unsatisfactory for several reasons linked to the non-normality of the error distributions and the presence of underlying trends. Complementary statistics have recently been proposed to palliate such deficiencies, such as quantiles of the absolute error distribution or the mean prediction uncertainty. We introduce here a new score, the systematic improvement probability, based on the direct system-wise comparison of absolute errors. Independent of the chosen scoring rule, the uncertainty of the statistics due to the incompleteness of the benchmark datasets is also generally overlooked. However, this uncertainty is essential to appreciate the robustness of rankings. In the present article, we develop two indicators based on robust statistics to address this problem: P_inv, the inversion probability between two values of a statistic, and P_r, the ranking probability matrix. We demonstrate also the essential contribution of the correlations between error sets in these scores comparisons.