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

Abstract

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/B 5 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.

The long road to calibrated prediction uncertainty in computational chemistry

“…I note below PTCn (Polynomial Trend Correction of order n) the correction of the errors trend vs. V by a polynomial of degree n. For this study, the PTCn model's parameters and their uncertainty are estimated by standard least-squares, which is known to be robust to non-normal error distributions 89 . More complex PU models could be used for datasets presenting complex trends, requiring weighting schemes (e.g., in the case of heterogeneous reference values uncertainties 6 ) or the consideration of heteroscedasticity 21,42 . More specific trend correction models are defined in their application case.…”

“…They did not identify any concern about the predicted prediction intervals and conclude that their method "provides a fair estimate of 95 % confidence" 21 . This is a small dataset (N = 99), and I find interesting to check how the validation tools perform in this context.…”

“…Bakowies and von Lilienfeld 21 proposed a new method to estimate zero-point energies (ZPE) and the corresponding expanded U 95 uncertainty in the framework of the composite ATOMIC method. Interestingly, they observed a quadratic dependence of the ZPE scaling factors and their dispersion with the fraction of heteroatoms in a set of 279 molecules.…”

“…• Embedded UQ methods produce PU concurrently with property predictions. This is an heterogeneous class which presently encompasses the above-mentioned BEEF or Bayesian Ensemble approach 4 and several bottom-up correction methods with detailed uncertainty budgets (the Type B methods of Ruscic 5 ), such as the Feller-Peterson-Dixon method 17,18 or the ATOMIC protocol [19][20][21] . Some machine learning methods, such as Bayesian neural networks which are designed to provide PUs along with prediction, also belong here 22,23 .…”

“…When quantitative tools have been used, notably for the validation of a-posteriori methods, a popular test statistic is the prediction interval coverage probability (PICP) 23,41 . PICP tests have been done either on the full dataset 21,29,42 , or using a splitting scheme between calibration and validation sets 6,42 . Globally, there does not seem to be (yet) a consensus in the community on the adequate validation vocabulary, concepts and tools, which would be necessary to compare the merits of different CC-UQ strategies.…”