A new hybrid Hartree−Fock−density functional (HF-DF) model called the modified Perdew−Wang 1-parameter model for kinetics (MPW1K) is optimized against a database of 20 forward barrier heights, 20 reverse barrier heights, and 20 energies of reaction. The results are compared to other hybrid HF-DF methods with the 6-31+G(d,p) basis. The new method reduces the mean unsigned error in reaction barrier heights by a factor of 2.4 over MPW1PW91 and by a factor of 3 over B3LYP.

Magnetic resonance imaging (MRI) has transformed our understanding of the human brain through well-replicated mapping of abilities to specific structures (for example, lesion studies) and functions1–3 (for example, task functional MRI (fMRI)). Mental health research and care have yet to realize similar advances from MRI. A primary challenge has been replicating associations between inter-individual differences in brain structure or function and complex cognitive or mental health phenotypes (brain-wide association studies (BWAS)). Such BWAS have typically relied on sample sizes appropriate for classical brain mapping4 (the median neuroimaging study sample size is about 25), but potentially too small for capturing reproducible brain–behavioural phenotype associations5,6. Here we used three of the largest neuroimaging datasets currently available—with a total sample size of around 50,000 individuals—to quantify BWAS effect sizes and reproducibility as a function of sample size. BWAS associations were smaller than previously thought, resulting in statistically underpowered studies, inflated effect sizes and replication failures at typical sample sizes. As sample sizes grew into the thousands, replication rates began to improve and effect size inflation decreased. More robust BWAS effects were detected for functional MRI (versus structural), cognitive tests (versus mental health questionnaires) and multivariate methods (versus univariate). Smaller than expected brain–phenotype associations and variability across population subsamples can explain widespread BWAS replication failures. In contrast to non-BWAS approaches with larger effects (for example, lesions, interventions and within-person), BWAS reproducibility requires samples with thousands of individuals.

A new hybrid Hartree−Fock-density functional model called the Becke88−Becke95 1-parameter model for kinetics (BB1K) was optimized against a database of three forward barrier heights, three reverse barrier heights, and three energies of reaction for the reactions in the BH6 representative barrier height database. We then assessed the newly developed BB1K method against a saddle point geometries database, a database of 42 barrier heights, the AE6 representative atomization energy database, a molecular geometries data set, and a set of 13 zero point energies. The results show that BB1K can give excellent saddle point geometries and barrier heights, and its performance for calculating atomization energies is 40% better than MPW1K. Using a mean mean unsigned error criterion that equally weights the errors in barrier heights and in bond energies, the new BB1K method outperforms all other DFT and hybrid DFT methods by a large margin, and we therefore conclude that it is the best density functional-type method for thermochemical kinetics.

The addition of diffuse functions to a double-ζ basis set is shown to be more important than increasing to a triple-ζ basis when calculating reaction energies, reaction barrier heights, and conformational energies with density functional theory, in particular with the modified Perdew−Wang density functional. It is shown that diffuse basis functions are vital to describe the relative energies between reactants, products, and transition states in isogyric reactions, and they provide enormous improvement in accuracy for conformational equilibria, using 1, 2-ethanediol and butadiene as examples. As a byproduct of the present study, we present a one-parameter hybrid density functional method optimized for sugars and sugar-like molecules; this is called MPW1S.

We compare hybrid Hartree-Fock density-functional theory to ab initio approaches for locating saddle point geometries and calculating barrier heights on a Born-Oppenhiemer potential energy surface. We located reactant, product, and saddle point stationary points for 22 reactions by the MP2 and QCISD ab initio methods and the B3LYP, BH&HLYP, mPW1PW91, and MPW1K hybrid Hartree-Fock DFT methods. We examined all of these methods with two basis sets, 6-31+G(d,p) and MG3. By comparison to calculations on five systems where the saddle point has been optimized at a high level of theory, we determined that the best saddle point geometries were obtained using the MPW1K and QCISD levels of theory. Of the methods tested, mPW1PW91 and B3LYP are the least effective for determining saddle point geometries and have mean unsigned error in barrier heights of 3.4-4.2 kcal/mol, depending on the basis set. In contrast, the MPW1K level of theory predicts the most accurate saddle point geometries and has a mean unsigned error of only 1.5 kcal/mol for either basis set. For even better accuracy, the combination of MPW1K/6-31+G(d,p) geometry calculations with QCISD(T)/MG3 or CCSD(T)/MG3 single-point energy calculations is shown to have an excellent performance-to-cost ratio. As a side product of this work, we report optimized scale factors for computing zero point energies by MPW1K.