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

Abstract

ATOMIC is a thermochemistry protocol geared toward larger molecules with first-row atoms. It implements Pople's concept of bond separation reactions in an ab initio fashion and so enhances the accuracy of midlevel composite models for atomization energies. Recently we have introduced ATOMIC(hc), a model for applications to hydrocarbons, that estimates bias and uncertainty for each of the components contributing to the ATOMIC bottom-ofthe-well atomization energy (Bakowies, D. J. Chem. Theory Comput. 2019, 15, 5230−5251). Here we scrutinize the remaining components of the ATOMIC protocol, including midlevel composite models to approximate the complete-basis set (CBS) limit of CCSD(T) as well as zero-point energies (ZPEs) and thermal enthalpy increments that are evaluated from scaled harmonic MP2 frequencies. Potential errors relating to imperfections in MP2 geometries and ZPEs are estimated using auxiliary information obtained from geometry optimizations and frequency calculations at the density functional (B3LYP) level. Overall corrections to and uncertainties of enthalpies of formation are obtained from summation and error propagation, respectively. The error and uncertainty model is validated with accurate data from the Active Thermochemical Tables (ATcT) and compared to earlier statistical assessments for the G3/99 benchmark. The proposed model is a welcome alternative to statistical assessment, first because it does not depend on comparison with experiment, second because it recognizes the expected scaling of error with system size, and third because it provides a detailed account of the importance of various contributions to overall error and uncertainty. The evaluation of ZPEs from scaled harmonic frequencies expectedly emerges as the leading source of uncertainty if highly accurate composite models are used to treat the electronic problem, but uncertainties are usually balanced with those arising from computationally more attractive B level (B 1 ...B 6 ) models to estimate the CBS limit of CCSD(T). ATOMIC(hc) enthalpies of formation, complete with uncertainty estimates, are reported for 161 hydrocarbons ranging in size from methane (CH 4 ) to [8]circulene (C 32 H 16 ) and tetra-tert-butyltetrahedrane (C 20 H 36 ). Experimental data are available for 127 molecules but cannot be reconciled with theory in 37 cases. Theory helps to identify the more accurate among conflicting experimental values in 11 cases and emerges as a valuable complement to experiment also for larger molecules, provided that fair estimates of uncertainty are available.

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

“…For an experimental reference value r i , u(r i ) would typically be a measurement uncertainty. For a computed reference value r i and for a calculated value c i , uncertainty might come from numerical uncertainty due to the use of finite precision arithmetics and discretization errors 13,14 , statistical uncertainty (e.g., for Monte Carlo methods 15,16 ), or parametric uncertainty (e.g., for calibrated methods [16][17][18][19][20] ).…”

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

“…The procedure may be regarded as an incomplete geometry optimization at the B 5 level that uses auxiliary trial structures from lower-level approaches at no additional cost instead of generating increasingly refined geometries using expensive gradient information. We have augmented the set of trial structures with highly accurate model A geometries obtained previously for 27 hydrocarbons 38 and additional CCSD(T)/cc-pVTZ geometries (obtained here with CFOUR 123 ) for 99 smaller molecules of the benchmark. The final collection of lowest B 5 energies for each of the 279 molecules forms the database that we use to assess each of the individual 219 methods.…”

“…Using nomenclature compatible with previous ATOMIC papers, 34 , 37 , 38 we may formally define geometry-related error as the energy penalty arising from using an approximate equilibrium (“e”) geometry G̃ e k that was optimized for molecule M with a method labeled k and then introduce a series of approximations to illustrate our approach; first, the replacement of the exact energy E exact by that of a suitably chosen composite model m (here: B 5 ), second, the replacement of the exact equilibrium geometry G e exact [ M ] by that optimized with composite model m , and finally, the approximation of the latter by the best available approximate equilibrium geometry G̃ e k opt [ M ], i.e., the one optimized with the particular method k = k opt that minimizes E { m } [ M ; G̃ e k [ M ]] In Section 6.2 , we shall analyze errors for the finally selected geometry optimization method (index k henceforth dropped) and calibrate a model to estimate a bias correction C A,e geo [ M ] that annihilates the error Δ E A,e geo [ M ] in atomization energy on average (note that Δ E A,e geo [ M ] = −Δ E e geo [ M ]), as well as a corresponding uncertainty u A,e geo [ M ] for molecule M such that is fulfilled with high confidence (95% or better for a balanced test set M ∈ { M 1 , M 2 ,...}).…”

“…The second external reference (model A, 27 geometries taken from earlier work 38 ) contributes 17 “best hits” to our database and shows energy penalties of less than 0.01 kcal/mol in all remaining 10 cases ( Table S2 ). This demonstrates the close agreement of model B 5 with model A, which is an alternative, more accurate, approximation to all-electron CCSD(T)/CBS.…”

Applications of isodesmic‐type reactions for computational thermochemistry

“…Here we simply note that isodesmic‐type and fragment‐based methods can be used in a complementary manner. As an example, 71 high‐level wave‐function composite schemes, in conjunction with isodesmic‐type reactions, have been applied to the calculation of heats of formation for small to medium‐sized hydrocarbons; the “isodesmic” values are then further improved using a statistically derived group‐additivity scheme to yield the eventual best estimates.…”