Dioxygen Binding to all 3d, 4d, and 5d Transition Metals from Coupled-Cluster Theory.

Understanding how transition metals bind and activate dioxygen (O2 ) is limited by experimental and theoretical uncertainties, making accurate quantum mechanical descriptors of interest. Here we report coupled-cluster CCSD(T) energies with large basis sets and vibrational and relativistic corrections for 160 3d, 4d, and 5d metal-O2 systems. We define four reaction energies (120 in total for the 30 metals) that quantify O-O activation and reveal linear relationships between metal-oxygen and O-O binding energies. The CCSD(T) data can be combined with thermochemical cycles to estimate chemisorption and physisorption energies for each metal from metal oxide embedding energies, in good correlation with atomization enthalpies (R2 =0.75). Spin-geometry variations can break the linearities, of interest to circumventing the Sabatier principle. Pt, Pd, Co, and Fe form a distinct group with the weakest O2 binding. R2 up to 0.84 between surface adsorption energies and our energies for MO2 systems indicate relevance also to real catalytic systems.

Using electronegativity and hardness to test density functionals.

Density functional theory (DFT) is used in thousands of papers each year, yet lack of universality reduces DFT's predictive capacity, and functionals may produce energy-density imbalances. The absolute electronegativity (χ) and hardness (η) directly reflect the energy-density relationship via the chemical potential ∂E/∂N and we thus hypothesized that they probe universality. We studied χ and η for atoms Z = 1-36 using 50 diverse functionals covering all major classes. Very few functionals describe both χ and η well. η benefits from error cancellation, whereas χ is marred by error propagation from IP and EA; thus, almost all standard GGA and hybrid functionals display a plateau in the MAE at ∼0.2 eV-0.3 eV for η. In contrast, variable performance for χ indicates problems in describing the chemical potential by DFT. The accuracy and precision of a functional is far from linearly related, yet for a universal functional, we expect linearity. Popular functionals such as B3LYP, PBE, and revPBE perform poorly for both properties. Density sensitivity calculations indicate large density-derived errors as occupation of degenerate p- and d-orbitals causes "non-universality" and large dependency on exact exchange. Thus, we argue that performance for χ for the same systems is a hallmark of an important aspect of universality by probing ∂E/∂N. With this metric, B98, B97-1, PW6B95D3, MN-15, rev-TPSS, HSE06, and APFD are the most "universal" among the tested functionals. B98 and B97-1 are accurate for very diverse metal-ligand bonds, supporting that a balanced description of ∂E/∂N and ∂E2/∂N2, via χ and η, is probably a first simple probe of universality.

Performance of Density Functional Theory for Transition Metal Oxygen Bonds.

The accuracy of density functional theory (DFT) limits predictions in theoretical catalysis, and strong chemical bonds between transition metals and oxygen pose a particular challenge. We benchmarked 30 diverse density functionals against the bond dissociation enthalpies (BDE) of the 30 MO and 30 MO+ diatomic systems of all the 3d, 4d, and 5d metals, to test universality across the d-block as required in comparative studies. Seven functionals, B98, B97-1, B3P86, B2PLYP, TPSSh, B3LYP, and B97-2, display mean absolute errors (MAE) <30 kJ/mol. In contrast, many commonly used functionals such as PBE and RPBE overestimate M-O bonding by +30 kJ/mol and display MAEs from 48-76 kJ/mol. RPBE and OPBE reduce the over-binding of PBE but remain very inaccurate. We identify a linear relationship (p-value 7.6 ⋅ 10-5 ) between the precision and accuracy of DFT, i. e. inaccurate functionals tend to produce larger, unpredictable random errors. Some functionals commonly deviate from this relationship: Thus, M06-2X is very precise but not very accurate, whereas B3LYP* and MN15-L are more accurate but less precise than M06-2X. The best-performing hybrids have 10-30 % HF exchange, but this can be relieved by double hybrids (B2PLYP). Most functionals describe trends well, but errors comparing 5d to 4d/3d are ∼10 kJ/mol larger than group-wise errors, due to uncertainties in the spin-orbit coupling corrections for effective core potentials, affecting e. g. Pt/Pd or Au/Ag comparisons.

The Metal Hydride Problem of Computational Chemistry: Origins and Consequences.

Formation and breaking of metal-hydrogen bonds are central to many important catalytic processes such as transition-metal catalyzed ammonia synthesis, hydrogenation reactions, and water splitting, and thus, they require an adequate theoretical description. We studied a data set of all 30 M-H and 30 M+-H bonds of the 3d, 4d, and 5d transition series; 50 of these systems have experimentally known bond dissociation enthalpies (BDE). To probe both the limit of low and high coordination number, we also studied a data set of 19 ML nH complexes. The BDEs were computed using Hartree-Fock (HF), MP2, CCSD, CCSD(T), and 10 diverse density functionals including local, GGA, hybrid GGA, meta hybrid, range-separated, and double hybrids. Our ten most important findings are as follows: (1) HF fails completely to describe the metal hydrogen bond due to its lack of static correlation; (2) this makes post-HF methods such as MP2 and even CCSD(T) perform worse than many density functionals; (3) DFT requires much more HF exchange (∼35% on average) to describe the pure M-H bonds than to describe other metal ligand bonds (0-20%); (4) we design a test to determine if self-interaction error (SIE) is important by correlating DFT errors against a one-electron SIE metric; (5) we show that SIE correlates directly with the DFT errors and thus causes most of the problem; (6) HF-DFT cannot handle these systems because the HF method is too pathological already at the density level; (7) instead, we define and apply a simple metric of electronic abnormality as the difference in PBE energy computed at the self-consistent PBE0 and SVWN densities, and this metric gives appropriate spread and effectively captures density-derived errors; (8) the low electronegativity of the metal enforces a diffuse hydride-like electron density, which make the metal hydrides primary examples of many-electron systems exhibiting SIE already at equilibrium geometries; (9) in the coordinatively saturated ML nH systems, much less HF exchange is required; i.e., the HF exchange requirements vary drastically with coordination number. Accordingly, DFT is unbalanced for any catalytic process involving both M-H and M-L bonds and changing coordination numbers; (10) importantly, the range-separated and double-hybrid functionals CAM-B3LYP and B2PLYP alone perform well for both M-H and M-L systems and in both limits of low and high coordination number, and at least as well as CCSD(T). This lends hope to a balanced treatment of computational chemistry for all types of M-L bonds at variable coordination number, as required for real catalytic reactions.

Electron Transfer of Hydrated Transition-Metal Ions and the Electronic State of Co3+(aq).

Electron transfer (ET) is broadly described by Marcus-type theories and plays a central role in many materials and catalytic systems and in biomolecules such as cytochromes. Classic ET processes are the self-exchange reactions between hydrated transition-metal ions such as Fe2+(aq) + Fe3+(aq) → Fe3+(aq) + Fe2+(aq). A well-known anomaly of Marcus theory is Co2+/Co3+ exchange, which proceeds ∼105 times faster than predicted. Co3+(aq) is a complex and reactive system widely thought to feature low-spin Co3+. We studied the self-exchange process systematically for Cr2+/Cr3+, V2+/V3+, Fe2+/Fe3+, and Co2+/Co3+ using six distinct density functionals. We identify directly the ∼105 anomaly of Co2+/Co3+ from the electronic reorganization energies without the use of empirical cross-relations. Furthermore, when Co3+ is modeled as high-spin, the anomaly disappears, bringing all four processes on a linear trend within the uncertainty of the experiments and theory. We studied both the acid-independent [Co(H2O)6]3+ species that dominates at low pH and the acid-dependent [Co(OH)(H2O)5]2+ species that becomes important at higher pH and used two distinct explicit second-sphere hydration models and models of perchlorate anion association. The high-spin state with weaker Co-O bonds is stabilized by vibrational energy and entropy by ∼11 and ∼12 kJ mol-1, correcting the gap estimates from absorption spectroscopy. High-spin Co3+(aq) explains the full experimental data series of the M(aq) systems. Low-spin Co3+ and high-spin Co2+ involve changes in the eg occupation upon ET with associated M-O bond changes and increased reorganization energy. In contrast, with high-spin Co3+(aq), the redox-active electrons shuffle between t2g orbitals to minimize structural changes, producing a relative rate in excellent agreement with the experiments. This eg occupation effect explains most of the experimental differences in the rate constants, with the remaining part explained by second-sphere hydration and anion effects. Our results consistently suggest that some high-spin Co3+(aq) is active during the experiments.

Chemical Bond Energies of 3d Transition Metals Studied by Density Functional Theory.

Despite their vast importance to inorganic chemistry, materials science, and catalysis, the accuracy of modeling the formation or cleavage of metal-ligand (M-L) bonds depends greatly on the chosen functional and the type of bond in a way that is not systematically understood. In order to approach a state of high-accuracy DFT for rational prediction of chemistry and catalysis, such system-dependencies need to be resolved. We studied 30 different density functionals applied to a "balanced data set" of 60 experimental diatomic M-L bond energies; this data set has no bias toward any dq configuration, metal, bond type, or ligand as all of these occur to the same extent, and we can therefore identify accuracy bottlenecks. We show that the performance of a functional is very dependent on data set choice, and we dissect these effects into system type. In addition to the use of balanced data sets, we also argue that the precision (rather than just accuracy) of a functional is of interest, measured by standard deviations of the errors. There are distinct system dependencies both in the ligand and metal series: Hydrides are best described by a very large HF exchange percentage, possibly due to self-interaction error, whereas halides are best described by very small (0-10%) HF exchange fractions, and double-bond enforcing oxides and sulfides favor 10-25% HF exchange, as is also average for the full data set. Thus, average HF requirements hide major system-dependent requirements. For late transition metals Co-Zn, HF percentage of 0-10% is favored, whereas for the early transition metals Sc-Fe hybrid functionals with 20% HF exchange or higher are commonly favored. Accordingly, B3LYP is an excellent choice for early d-block but a poor choice for late transition metals. We conclude that DFT intrinsically underestimates the bond strengths of late vs early transition metals, correlating with increased effective nuclear charge. Thus, the revised RPBE, which reduces the overbinding tendency of PBE, is mainly an advantage for the early and mid transition metals and not very much for the late transition metals, i.e. there is a metal-dependent effect of the relative performance of RPBE vs PBE, which are widely used to study adsorption energetics on metal surfaces. Overall, the best performing functionals are PW6B95, the MN15 and MN15-L functionals, and the double hybrid B2PLYP.