Density Functional Theory and the Basis Set Truncation Problem with Correlation Consistent Basis Sets: Elephant in the Room or Mouse in the Closet?

Two recent papers in this journal called into question the suitability of the correlation consistent basis sets for density functional theory (DFT) calculations, because the sets were designed for correlated methods such as configuration interaction, perturbation theory, and coupled cluster theory. These papers focused on the ability of the correlation consistent and other basis sets to reproduce total energies, atomization energies, and dipole moments obtained from “quasi-exact” multiwavelet results. Undesirably large errors were observed for the correlation consistent basis sets. One of the papers argued that basis sets specifically optimized for DFT methods were “essential” for obtaining high accuracy. In this work we re-examined the performance of the correlation consistent basis sets by resolving problems with the previous calculations and by making more appropriate basis set choices for the alkali and alkaline-earth metals and second-row elements. When this is done, the statistical errors with respect to the benchmark values and with respect to DFT optimized basis sets are greatly reduced, especially in light of the relatively large intrinsic error of the underlying DFT method. When judged with respect to high-quality Feller-Peterson-Dixon coupled cluster theory atomization energies, the PBE0 DFT method used in the previous studies exhibits a mean absolute deviation more than a factor of 50 larger than the quintuple zeta basis set truncation error

Low-Lying Electronic States of Ir_<i>n</i> Clusters with <i>n</i> = 2–8 Predicted at the DFT, CASSCF, and CCSD(T) Levels

Low-lying structures of the small iridum clusters Ir_<i>n</i> (<i>n</i> = 2–8) were optimized using DFT methods. Ir₂ and Ir₃ were also optimized using the CASSCF method. MRCI-SD (for Ir₂) energies and CCSD­(T) (for Ir₂ and Ir₃) energies of the leading configurations from the CASSCF calculations were done to predict the low-lying states. The normalized atomization energies (⟨AE⟩) for Ir_<i>n</i> (<i>n</i> = 2–8) were calculated at the CCSD­(T) level up to the complete basis set (CBS) limit in some cases using the B3LYP optimized geometries. The ground state for Ir₂ is predicted to be ⁵Δ_g, and the ground state of Ir₃ is linear ²Δ_g, with the <i>D</i>_3<i>h</i> ⁴A″₁ state ∼10 kcal/mol higher in energy at the CASSCF level without core–valence corrections and ∼15 kcal/mol higher at the CCSD­(T)/CBS level with spin–orbit and core–valence corrections. Inclusion of the spin orbit corrections in the normalized bond dissociation energies ⟨AE⟩ for Ir_<i>n</i> is critical and will decrease the ⟨AE⟩ by ∼15 kcal/mol for <i>n</i> ≥ 4. The ⟨AE⟩ for Ir_<i>n</i> increases as <i>n</i> increases in general, and the ⟨AE⟩ is far from convergence to the bulk value at <i>n</i> = 8. The average coordination number (CN) and average bond length for the low energy Ir_<i>n</i> clusters are far from being converged to the bulk values by <i>n</i> = 8

Machine-Learning Approach for the Development of Structure–Energy Relationships of ZnO Nanoparticles

The structure–energy relationships for the zinc oxide morphologies were investigated using a newly developed fragment-based energy decomposition approach. In this approach, the local chemical compositions of a material are abstracted as fragment types that serve as the material’s genes with respect to its thermodynamic properties. A machine learning-based fragment recognition scheme was developed to learn about the fragment-related knowledge from a relatively small training set consisting of computationally viable ultrasmall nanoparticles. The knowledge gained including the fragment geometries and fragment energy parameters can be used for the classification and energy expression of the test sets consisting of different polymorphs and morphologies of that material at various scales. The stabilities of ZnO nanoparticles with different morphologies were expressed explicitly as functions of the particle size. The size-related phase transitions among various morphologies including wurtzite prisms, wurtzite octahedrons, body-centered tetragonal particles, sodalite-like particles, single-layered cages, multilayered cages, and nonpolar hexagonal prisms were predicted. The multilayered cages with nonpolar surfaces exhibit superior stability among the low-energy morphologies, but wurtzite nanoparticles are more favorable under practical synthesis and growth conditions under the control of the kinetics. The growth mechanism for ZnO clusters, ultrasmall nanoparticles, nanocrystals, and bulk-sized particles is proposed based on the synergy between the size-related phase transitions and external factors that affect the surface energies of the particles. Our interpretation of the Wulff theorem at a fragment-sized resolution provides new chemical insight for understanding the structural phase transition and particle growth for ZnO at various scales

Computational Study of H₂ and O₂ Production from Water Splitting by Small (MO₂)_<i>n</i> Clusters (M = Ti, Zr, Hf)

Coupled cluster [CCSD­(T)] theory and density functional theory (DFT) have been used to study the production of H₂ and O₂ from hydrolysis products generated from H₂O addition to (MO₂)_<i>n</i> (M = Ti, Zr, Hf, <i>n</i> = 1–3) clusters on both the lowest singlet and triplet potential energy surfaces (PESs). H₂ production occurs via the formation of an M–H containing intermediate followed by H–H recombination and H₂ desorption from M_<i>n</i>O_2<i>n</i>(OH)₂ and M_<i>n</i>O_2<i>n</i>+2. The hydrogen transfer reactions to form the M–H bond are the rate determining steps and can be considered to be proton coupled, electron transfer (PCET) reactions with one or two electrons being transferred. Oxygen is produced by breaking two weak M–O bonds in an atomic oxygen saturated metal oxide from an M_<i>n</i>O_2<i>n</i>•O₂ intermediate. On the triplet PES, the activation energies for the first and second H transfer to the metal are calculated to be ∼10 to 50 kcal/mol and ∼75 to 90 kcal/mol depending on the size of the clusters and the metal. The barriers on the singlet surface for the first and the second H transfer are predicted to be 110 to 140 kcal/mol, in general larger than the H–O bond dissociation energy. The activation barriers for the step of H–H recombination are 15 to 50 kcal/mol, and the H₂ desorption energies are less than 10 kcal/mol on the singlet and triplet PESs. The oxygen desorption energies follow the order Ti < Zr < Hf for the triplets and Ti < Zr ≈ Hf for the singlets. The oxygen desorption energy is approximately independent of the size of the cluster for the same metal. The water splitting reactions prefer to take place on the triplet surface. A low excess potential energy is needed to generate 2H₂ and O₂ from 2H₂O after the endothermicity of the reaction is overcome on the triplet PES

Computational Study of Ethanol Conversion on Al₈O₁₂ as a Model for γ‑Al₂O₃

Correlated molecular orbital theory at the coupled cluster CCSD­(T) level with density functional theory geometries is used to study ethanol dehydration, dehydrogenation, and condensation reactions on an the Al₈O₁₂ cluster which is a model for γ<b>-</b>Al₂O₃. The Al in the active site on the cluster is a strong Lewis acid. The reactions begin with formation of a very stable Lewis acid–base ethanol–cluster adduct. Dehydration proceeds by β-H transfer to a bicoordinate oxygen leading to the direct formation of ethylene and two OH groups following an E2 mechanism. Dehydrogenation proceeds directly by α-H transfer to the active metal center and a proton transfer to a bicoordinate bridge O to form acetaldehyde plus a metal hydride and a hydroxyl, again an E2 mechanism. After addition of a second ethanol, diethyl ether is generated by an α-C transfer from the first to the second ethanol, an acid-driven S_N2 mechanism. Condensation and dehydration with two alcohols have comparable energy barriers. The addition of a second ethanol or a water molecule raises the energy barriers. Condensation and dehydration are predicted to be more likely than dehydrogenation. The computational results for the mechanism and the energetics agree well with the available experimental data

Absolute Hydration Free Energy of Small Anions and the Aqueous p<i>K</i>_a of Simple Acids

Heats of formation and gas phase acidities for the simple acids and their deprotonated anions (A– = F–, Cl–, Br–, I–, OH–, SH–, SeH–, TeH–, OCl–, OBr–, and OI–) were calculated using the Feller–Peterson–Dixon (FPD) method with large basis sets including Douglass–Kroll scalar relativistic corrections. Hydration of the neutral and anionic species was predicted using the supermolecule-continuum approach, resulting in absolute hydration free energies that, when combined with calculated gas phase acidities, produce aqueous acidities and pKa values for these simple acids that are, in general, in excellent agreement with experimental literature values. Absolute hydration free energy values converged quickly in terms of the experimental values for neutral species, requiring only four explicit H2O molecules. HI is anomalous in that it fully dissociates ionically in a water tetramer and was treated without explicit water molecules. The hydration energies of anionic species converged more slowly and were modeled with up to 16 explicit H2O molecules. Calculated values for ΔHf and ΔGgas agree with experimental values within ca. 1.2 kcal/mol, and ΔGaq and ΔΔGhyd agree with experimental values within ca. 2 kcal/mol in most cases

Electronic Structures of Small (RuO₂)_<i>n</i> (<i>n</i> = 1–4) Nanoclusters and Their Anions and the Hydrolysis Reactions with Water

Group 8 (RuO₂)_<i>n</i> (<i>n</i> = 1–4) nanoclusters, their anions, and the hydrolysis reactions of the neutral clusters have been studied with the density functional theory (DFT) as well as coupled cluster CCSD­(T) theory. The ground state is predicted to be a singlet and a doublet for the neutral RuO₂ clusters and anionic clusters, respectively. The CCSD­(T) method is required to predict the correct ground state. The calculated singlet–triplet gaps (<15 kcal/mol) and fluoride affinities (<95 kcal/mol) are smaller than those of the group 4 (MO₂)_<i>n</i> and group 6 (MO₃)_<i>n</i> metal oxide clusters. The electron affinities range from 2.2 to 3.4 eV, showing that the RuO₂ clusters are quite reducible. Clustering energies and heats of formation are calculated. The water physisorption energies are predicted to be −10 to −20 kcal/mol with the adsorption energy for the singlet being generally more exothermic than that for the triplet. The hydrolysis reactions are exothermic for the monomer and dimer clusters and are slightly endothermic or neutral for the trimer and tetramer. H₂O is readily dissociated on the monomer and dimer but not on the trimer and tetramer. The physisorption and chemisorption energies are less exothermic, and the barriers for the hydrolysis reactions are larger for RuO₂ nanoclusters than for the corresponding group 4 ZrO₂ nanoclusters

Benchmark-Quality Atomization Energies for BeH and BeH₂

The total atomization energies for BeH and BeH₂ have been calculated using the Feller–Peterson–Dixon approach to better than ±1 kcal/mol. The calculations are based on CCSD­(T) all-electron calculations extrapolated to the complete limit, and CCSDT and CCSDTQ corrections are included. A scalar relativistic correction and a diagonal Born–Oppenheimer correction are included. Accurate zero-point energies are used. The total atomization energies at 0 K are 47.7 kcal/mol for BeH and 140.0 kcal/mol for BeH₂ with error of at most ±0.3 kcal/mol

1,2-Ethanediol and 1,3-Propanediol Conversions over (MO₃)₃ (M = Mo, W) Nanoclusters: A Computational Study

The dehydration and dehydrogenation reactions of one and two 1,2-ethanediol and 1,3-propanediol molecules on (MO₃)₃ (M = Mo, W) nanoclusters have been studied computationally using density functional and coupled cluster (CCSD­(T)) theory. The reactions are initiated by the formation of a Lewis acid–base complex with an additional hydrogen bond. Dehydration is the dominant reaction proceeding via a metal bisdiolate. Acetaldehyde, the major product for 1,2-ethanediol, is produced by α-hydrogen transfer from one CH₂ group to the other. For 1,3-propanediol, the C–C bond breaking pathways to produce C₂H₄ and HCHO simultaneously and proton transfer to generate propylene oxide have comparable barrier energies. The barrier to produce propanal from the propylene oxide complex is less than that for epoxide release from the cluster. On the Mo₃O₉ cluster, a redox reaction channel for 1,2-ethanediol to break the C–C bond to form two formaldehyde molecules and then to produce C₂H₄ is slightly more favorable than the formation of acetaldehyde. For W^VI, the energy barrier for the reduction pathway is larger due to the lower reducibility of W₃O₉. Similar reduction on Mo^VI for 1,3-propanediol to form propene is not a favorable pathway compared with the other pathways as additional C–H bond breaking is required in addition to breaking a C–C bond. The dehydrogenation and dehydration activation energies for the selected glycols are larger than the reactions of ethanol and 1-propanol on the same clusters. The CCSD­(T) method is required because density functional theory with the M06 and B3LYP functionals does not predict quantitative energies on the potential energy surface. The M06 functional performs better than does the B3LYP functional

Role of Electronegative Substituents on the Bond Energies in the Grubbs Metathesis Catalysts for M = Fe, Ru, Os

Coupled cluster theory [CCSD­(T)] with the aug-cc-pVDZ/aug-cc-pVDZ-PP basis sets is used to predict the thermodynamic properties of models of the Grubbs catalyst, H₂ImM­(PH₃)­(CRR′) and H₂ImM­(C₂H₄)­(CRR′) for M = Fe, Ru, and Os and CRR′ = CH₂, CHF, and CF₂. The PH₃ and C₂H₄, imidazolinium carbene (H₂Im), and CRR′ bond dissociation energies (BDEs) are reported. Because of low metal-carbene BDEs, the M = Fe complexes are unlikely to form, so they will not be good catalysts for olefin metathesis. The metal–carbene BDE is an important component in metathesis catalyst design and correlates with the singlet–triplet splitting in the carbene. The two metallacyclobutane intermediates (cis and trans to the imidazolinium carbene) formed by reaction of the CRR′CRR′ with the 14-electron active species (H₂ImM­(CRR′)) (R and R′ either H or F) are investigated at the same level of theory. The metallacycles cis to the nitrogen heterocyclic carbene are lower in energy than the trans conformer with the exception of four M = Fe metallacycles in the gas phase and in CH₂Cl₂ solution at 298 K. The olefin π complex for the simplest CH₂ ligand plus C₂H₄ reactant combination is more stable in the gas phase, but in CH₂Cl₂ solution at 298 K, the cis metallacycles are more stable than the π complexes or the trans metallacycles on the free energy scale as a result of the large dipole moments in the cis metallacycles. The results show that the best energy balance is achieved with M = Ru, a CH₂ carbene substituent, and a C₂H₄ reactant. The energetics for the Grubbs catalysts are shown to differ from those of the Schrock catalysts