Do Practical Standard Coupled Cluster Calculations Agree Better than Kohn–Sham Calculations with Currently Available Functionals When Compared to the Best Available Experimental Data for Dissociation Energies of Bonds to 3d Transition Metals?

Chelate Complexes of 3d Transition Metal Ions─A Challenge for Electronic-Structure Methods?

Different electronic-structure methods were assessed for their ability to predict two important properties of the industrially relevant chelating agent nitrilotriacetic acid (NTA): its selectivity with respect to six different first-row transition metal ions and the spin-state energetics of its complex with Fe(III). The investigated methods encompassed density functional theory (DFT), the random phase approximation (RPA), coupled cluster (CC) theory, and the auxiliary-field quantum Monte Carlo (AFQMC) method, as well as the complete active space self-consistent field (CASSCF) method and the respective on-top methods: second-order N-electron valence state perturbation theory (NEVPT2) and multiconfiguration pair-density functional theory (MC-PDFT). Different strategies for selecting active spaces were explored, and the density matrix renormalization group (DMRG) approach was used to solve the largest active spaces. Despite somewhat ambiguous multi-reference diagnostics, most methods gave relatively good agreement with experimental data for the chemical reactions connected to the selectivity, which only involved transition-metal complexes in their high-spin state. CC methods yielded the highest accuracy followed by range-separated DFT and AFQMC. We discussed in detail that even higher accuracies can be obtained with NEVPT2, under the prerequisite that consistent active spaces along the entire chemical reaction can be selected, which was not the case for reactions involving Fe(III). A bigger challenge for electronic-structure methods was the prediction of the spin-state energetics, which additionally involved lower spin states that exhibited larger multi-reference diagnostics. Conceptually different, typically accurate methods ranging from CC theory via DMRG-NEVPT2 in combination with large active spaces to AFQMC agreed well that the high-spin state is energetically significantly favored over the other spin states. This was in contrast to most DFT functionals and RPA which yielded a smaller stabilization and some common DFT functionals and MC-PDFT even predicting the low-spin state to be energetically most favorable.

Improving gas adsorption modeling for MOFs by local calibration of Hubbard U parameters

While computational screening with density functional theory (DFT) is frequently employed for the screening of metal-organic frameworks (MOFs) for gas separation and storage, commonly applied generalized gradient approximations (GGAs) exhibit self-interaction errors, which hinder the predictions of adsorption energies. We investigate the Hubbard U parameter to augment DFT calculations for full periodic MOFs, targeting a more precise modeling of gas molecule-MOF interactions, specifically for N2, CO2, and O2. We introduce a calibration scheme for the U parameter, which is tailored for each MOF, by leveraging higher-level calculations on the secondary building unit (SBU) of the MOF. When applied to the full periodic MOF, the U parameter calibrated against hybrid HSE06 calculations of SBUs successfully reproduces hybrid-quality calculations of the adsorption energy of the periodic MOF. The mean absolute deviation of adsorption energies reduces from 0.13 eV for a standard GGA treatment to 0.06 eV with the calibrated U, demonstrating the utility of the calibration procedure when applied to the full MOF structure. Furthermore, attempting to use coupled cluster singles and doubles with perturbative triples calculations of isolated SBUs for this calibration procedure shows varying degrees of success in predicting the experimental heat of adsorption. It improves accuracy for N2 adsorption for cases of overbinding, whereas its impact on CO2 is minimal, and ambiguities in spin state assignment hinder consistent improvements of O2 adsorption. Our findings emphasize the limitations of cluster models and advocate the use of full periodic MOF systems with a calibrated U parameter, providing a more comprehensive understanding of gas adsorption in MOFs.

Scalar Relativistic All-Electron and Pseudopotential Ab Initio Study of a Minimal Nitrogenase [Fe(SH)4H]- Model Employing Coupled-Cluster and Auxiliary-Field Quantum Monte Carlo Many-Body Methods

Nitrogenase is the only enzyme that can cleave the triple bond in N2, making nitrogen available to organisms. The detailed mechanism of this enzyme is currently not known, and computational studies are complicated by the fact that different density functional theory (DFT) methods give very different energetic results for calculations involving nitrogenase models. Recently, we designed a [Fe(SH)4H]- model with the fifth proton binding either to Fe or S to mimic different possible protonation states of the nitrogenase active site. We showed that the energy difference between these two isomers (ΔE) is hard to estimate with quantum-mechanical methods. Based on nonrelativistic single-reference coupled-cluster (CC) calculations, we estimated that the ΔE is 101 kJ/mol. In this study, we demonstrate that scalar relativistic effects play an important role and significantly affect ΔE. Our best revised single-reference CC estimates for ΔE are 85-91 kJ/mol, including energy corrections to account for contributions beyond triples, core-valence correlation, and basis-set incompleteness error. Among coupled-cluster approaches with approximate triples, the canonical CCSD(T) exhibits the largest error for this problem. Complementary to CC, we also used phaseless auxiliary-field quantum Monte Carlo calculations (ph-AFQMC). We show that with a Hartree-Fock (HF) trial wave function, ph-AFQMC reproduces the CC results within 5 ± 1 kJ/mol. With multi-Slater-determinant (MSD) trials, the results are 82-84 ± 2 kJ/mol, indicating that multireference effects may be rather modest. Among the DFT methods tested, τ-HCTH, r2SCAN with 10-13% HF exchange with and without dispersion, and O3LYP/O3LYP-D4, and B3LYP*/B3LYP*-D4 generally perform the best. The r2SCAN12 (with 12% HF exchange) functional mimics both the best reference MSD ph-AFQMC and CC ΔE results within 2 kJ/mol.

Benchmarks for transition metal spin-state energetics: why and how to employ experimental reference data?

Accurate prediction of energy differences between alternative spin states of transition metal complexes is essential in computational (bio)inorganic chemistry-for example, in characterization of spin crossover materials and in the theoretical modeling of open-shell reaction mechanisms-but it remains one of the most compelling problems for quantum chemistry methods. A part of this challenge is to obtain reliable reference data for benchmark studies, as even the highest-level applicable methods are known to give divergent results. This Perspective discusses two possible approaches to method benchmarking for spin-state energetics: using either theoretically computed or experiment-derived reference data. With the focus on the latter approach, an extensive general review is provided for the available experimental data of spin-state energetics and their interpretations in the context of benchmark studies, targeting the possibility of back-correcting the vibrational effects and the influence of solvents or crystalline environments. With a growing amount of experience, these effects can be now not only qualitatively understood, but also quantitatively modeled, providing the way to derive nearly chemically accurate estimates of the electronic spin-state gaps to be used as benchmarks and advancing our understanding of the phenomena related to spin states in condensed phases.

How important is the amount of exact exchange for spin-state energy ordering in DFT? Case study of molybdenum carbide cluster, Mo4C2

Since the form of the exact functional in density functional theory is unknown, we must rely on density functional approximations (DFAs). In the past, very promising results have been reported by combining semi-local DFAs with exact, i.e. Hartree-Fock, exchange. However, the spin-state energy ordering and the predictions of global minima structures are particularly sensitive to the choice of the hybrid functional and to the amount of exact exchange. This has been already qualitatively described for single conformations, reactions, and a limited number of conformations. Here, we have analyzed the mixing of exact exchange in exchange functionals for a set of several hundred isomers of the transition metal carbide, Mo4C2. The analysis of the calculated energies and charges using PBE0-type functional with varying amounts of exact exchange yields the following insights: (1) The sensitivity of spin-energy splitting is strongly correlated with the amount of exact exchange mixing. (2) Spin contamination is exacerbated when correlation is omitted from the exchange-correlation functional. (3) There is not one ideal value for the exact exchange mixing which can be used to parametrize or choose among the functionals. Calculated energies and electronic structures are influenced by exact exchange at a different magnitude within a given distribution; therefore, to extend the application range of hybrid functionals to the full periodic table the spin-energy splitting energies should be investigated.

Ionization energies of metallocenes: a coupled cluster study of cobaltocene

Open-shell transition metal chemistry presents challenges to contemporary electronic structure methods, based on either density functional or wavefunction theory. While CCSD(T) is the well-trusted gold standard for maingroup thermochemistry, the accuracy and robustness of the method is less clear for open-shell transition metal chemistry, requiring benchmarking of CCSD(T)-based protocols against either higher-level theory or experiment. Ionization energies (IEs) of metallocenes provide an interesting test case with metallocenes being common redox reagents as well as playing roles as redox mediators and cocatalysts in redox catalysis. Using highly accurate ZEKE-MATI experimental measurements of gas phase adiabatic (5.3275 ± 0.0006 eV) and vertical (5.4424 ± 0.0006 eV) ionization energies of cobaltocene, we systematically assessed the accuracy of the local coupled-cluster method DLPNO-CCSD(T) with respect to geometry, reference determinant, basis set size and extrapolation schemes, PNO cut-off and extrapolation, local triples approximation, relativistic effects and core-valence correlation. We show that PNO errors are controllable via the recently introduced PNO extrapolation schemes and that the expensive iterative triples (T₁) contribution can be made more manageable by calculating it as a smaller-basis/smaller PNO-cutoff correction. The reference determinant turns out to be a critical aspect in these calculations with the HF determinant resulting in large DLPNO-CCSD(T) errors, likely due to the qualitatively flawed molecular orbital spectrum. The BP86 functional on the other hand was found to provide reference orbitals giving small DLPNO-CCSD(T) errors, likely due to more realistic orbitals as suggested by the more consistent MO spectrum compared to HF. A protocol including complete basis set extrapolations with correlation-consistent basis sets, complete PNO space extrapolations, iterative triples- and core-valence correlation corrections was found to give errors of -0.07 eV and -0.03 eV for adiabatic- and vertical-IE of cobaltocene, respectively, giving close to chemical accuracy for both properties. A computationally efficient DLPNO-CCSD(T) protocol was devised and tested against adiabatic ionization energies of 6 different metallocenes (V, Cr, Mn, Fe, Co, Ni). For the other metallocenes, the iterative triples (T₁) and PNO extrapolation contributions turn out to be even more important. The results give errors close to the experimental uncertainty, similar to recent auxiliary-field quantum Monte Carlo results. The quality of the reference determinant orbitals is identified as the main source of uncertainty in CCSD(T) calculations of metallocenes.

The performance of CCSD(T) for the calculation of dipole moments in diatomics

This work analyzes the accuracy of the coupled cluster with single, double, and perturbative triple excitation [CCSD(T)] method for predicting dipole moments. In particular, we benchmark CCSD(T) predictions for the equilibrium bond length, vibrational frequency, and dipole moment versus accurate experimental data. As a result, we find that CCSD(T) leads to accurate dipole moments. However, in some cases, it disagrees with the experimental values, and the disagreement can not be satisfactorily explained via relativistic or multi-reference effects. Therefore, our results indicate that benchmark studies for energy and geometry properties do not accurately describe other electron density magnitudes.

Mechanistic views and computational studies on transition-metal-catalyzed reductive coupling reactions

Transition-metal-catalyzed reductive coupling reactions have been considered as a powerful tool to convert two electrophiles into value-added products. Numerous related reports have shown the fascinating potential. Mechanistic studies, especially theoretical studies, can provide important implications for the design of novel reductive coupling reactions. In this review, we summarize the representative advancements in theoretical studies on transition-metal-catalyzed reductive coupling reactions and systematically elaborate the mechanisms for the key steps of reductive coupling reactions. The activation modes of electrophiles and the deep insights of selectivity generation are mechanistically discussed. In addition, the mechanism of the reduction of high-oxidation-state catalysts and further construction of new chemical bonds are also described in detail.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Route to Chemical Accuracy for Computational Uranium Thermochemistry

Benchmark spinor-based relativistic coupled-cluster calculations for the ionization energies of the uranium atom, the uranium monoxide molecule (UO), and the uranium dioxide molecule (UO₂) and for the bond dissociation energies of UO and UO₂ are reported. The accuracy of these calculations in the treatments of relativistic, electron-correlation, and basis-set effects is analyzed. The intrinsic convergence of the computed results and the favorable comparison with the experimental values demonstrate the unique applicability of the spinor representation of quantum-chemical methods to open-shell uranium-containing atomic and molecular species with uranium oxidation states ranging from U(0) to U(V).

Calculation of Metallocene Ionization Potentials via Auxiliary Field Quantum Monte Carlo: Toward Benchmark Quantum Chemistry for Transition Metals

The accurate ab initio prediction of ionization energies is essential to understanding the electrochemistry of transition metal complexes in both materials science and biological applications. However, such predictions have been complicated by the scarcity of gas phase experimental data, the relatively large size of the relevant molecules, and the presence of strong electron correlation effects. In this work, we apply all-electron phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) utilizing multideterminant trial wave functions to six metallocene complexes to compare the computed adiabatic and vertical ionization energies with experimental results. We find that ph-AFQMC yields mean absolute errors (MAEs) of 1.69 ± 1.02 kcal/mol for the adiabatic energies and 2.85 ± 1.13 kcal/mol for the vertical energies. We also carry out density functional theory (DFT) calculations using a variety of functionals, which yields MAEs of 3.62-6.98 kcal/mol and 3.31-9.88 kcal/mol, as well as one variant of localized coupled cluster calculations (DLPNO-CCSD(T₀) with moderate PNO cutoffs), which has MAEs of 4.96 and 6.08 kcal/mol, respectively. We also test the reliability of DLPNO-CCSD(T₀) and DFT on acetylacetonate (acac) complexes for adiabatic energies measured in the same manner experimentally, and we find higher MAEs, ranging from 4.56 to 10.99 kcal/mol (with a different ordering) for DFT and 6.97 kcal/mol for DLPNO-CCSD(T₀). Finally, by utilizing experimental solvation energies, we show that accurate reduction potentials in solution for the metallocene series can be obtained from the AFQMC gas phase results.

The MOBH35 Metal–Organic Barrier Heights Reconsidered: Performance of Local-Orbital Coupled Cluster Approaches in Different Static Correlation Regimes

We have revisited the MOBH35 (Metal–Organic Barrier Heights, 35 reactions) benchmark [Iron, Janes, J. Phys. Chem. A, 2019, 123 ( (17), ), 3761−378130973722; ibid. 2019, 123, 6379–6380] for realistic organometallic catalytic reactions, using both canonical CCSD(T) and localized orbital approximations to it. For low levels of static correlation, all of DLPNO-CCSD(T), PNO-LCCSD(T), and LNO-CCSD(T) perform well; for moderately strong levels of static correlation, DLPNO-CCSD(T) and (T₁) may break down catastrophically, and PNO-LCCSD(T) is vulnerable as well. In contrast, LNO-CCSD(T) converges smoothly to the canonical CCSD(T) answer with increasingly tight convergence settings. The only two reactions for which our revised MOBH35 reference values differ substantially from the original ones are reaction 9 and to a lesser extent 8, both involving iron. For the purpose of evaluating density functional theory (DFT) methods for MOBH35, it would be best to remove reaction 9 entirely as its severe level of static correlation makes it just too demanding for a test. The magnitude of the difference between DLPNO-CCSD(T) and DLPNO-CCSD(T₁) is a reasonably good predictor for errors in DLPNO-CCSD(T₁) compared to canonical CCSD(T); otherwise, monitoring all of T₁, D₁, max|t_i^A|, and 1/(ε_LUMO – ε_HOMO) should provide adequate warning for potential problems. Our conclusions are not specific to the def2-SVP basis set but are largely conserved for the larger def2-TZVPP, as they are for the smaller def2-SV(P): the latter may be an economical choice for calibrating against canonical CCSD(T). Finally, diagnostics for static correlation are statistically clustered into groups corresponding to (1) importance of single excitations in the wavefunction; (2a) the small band gap, weakly separated from (2b) correlation entropy; and (3) thermochemical importance of correlation energy, as well as the slope of the DFT reaction energy with respect to the percentage of HF exchange. Finally, a variable reduction analysis reveals that much information on the multireference character is provided by T₁, I_ND/I_tot, and the exchange-based diagnostic A₁₀₀[TPSS].

Assessing Density Functional Theory for Chemically Relevant Open-Shell Transition Metal Reactions

Due to the principle lack of systematic improvement possibilities of density functional theory, careful assessment of the performance of density functional approximations (DFAs) on well-designed benchmark sets, for example, for reaction energies and barrier heights, is crucial. While main-group chemistry is well covered by several available sets, benchmark data for transition metal chemistry is sparse. This is especially the case for larger, chemically relevant molecules. Addressing this issue, we recently introduced the MOR41 benchmark which covers chemically relevant reactions of closed-shell complexes. In this work, we extend these efforts to single-reference open-shell systems and introduce the "reactions of open-shell single-reference transition metal complexes" (ROST61) benchmark set. ROST61 includes accurate coupled-cluster reference values for 61 reaction energies with a mean reaction energy of -42.8 kcal mol^-1. Complexes with 13-93 atoms covering 20 d-block elements are included, but due to the restriction to single-reference open-shell systems, important elements such as iron or platinum could not be taken into account, or only to a small extent. We assess the performance of 31 DFAs in combination with three London dispersion (LD) correction schemes. Further, DFT-based composite methods, MP2, and a few semiempirical quantum chemical methods are evaluated. Consistent with the results for the MOR41 closed-shell benchmark, we find that the ordering of DFAs according to Jacob's ladder is preserved and that adding an LD correction is crucial, clearly improving almost all tested methods. The recently introduced r²SCAN-3c composite method stands out with a remarkable mean absolute deviation (MAD) of only 2.9 kcal mol^-1, which is surpassed only by hybrid DFAs with low amounts of Fock exchange (e.g., 2.3 kcal mol^-1 for TPSS0-D4/def2-QZVPP) and double-hybrid (DH) DFAs but at a significantly higher computational cost. The lowest MAD of only 1.6 kcal mol^-1 is obtained with the DH DFA PWPB95-D4 in the def2-QZVPP basis set approaching the estimated accuracy of the reference method. Overall, the ROST61 set adds important reference data to a sparsely sampled but practically relevant area of chemistry. At this point, it provides valuable orientation for the application and development of new DFAs and electronic structure methods in general.

C-H Bond Activation by the Excited Zinc Atom: Gas-Phase Formation of Methylzinc Hydride (HZnCH3) Based on Multireference Second-Order Perturbation Theory and Coupled Cluster Calculations

The pioneering spectroscopic observations of the methylzinc hydride [HZnCH₃(X¹A₁)] molecule were reported previously by the Ziurys group [J. Am. Chem. Soc. 2010, 132, 17186-17192], and the possible formation mechanisms were suggested therein, including those with the participation of excited zinc atoms in reaction with methane. Herein, the ground singlet state and the lowest excited triplet state potential energy surfaces of the Zn + CH₄ reaction have been explored using high-level electronic structure calculations with multireference second-order perturbation theory and coupled cluster singles and doubles with perturbative triples (CCSD(T)) methods in conjunction with all-electron basis sets (up to aug-cc-pV5Z) and scalar relativistic effects incorporated via the second-order Douglas-Kroll-Hess (DK) method. Based on the ab initio results, a plausible scenario for the formation of HZnCH₃(X¹A₁) is proposed involving the activation of the C-H bond of methane by the lowest excited ³P state atomic zinc. Calculations also highlight the importance of an agostic-like Zn···H-C interactions in the pre-activation complex and good agreement between the structure of the HZnCH₃(X¹A₁) molecule predicted at the DK-CCSD(T)/aug-cc-pVQZ-DK level of theory and that derived from rotational spectroscopy, as well as the discrepancies between the ab initio and density functional theory predictions.

Redox Isomerism in the S3 State of the Oxygen-Evolving Complex Resolved by Coupled Cluster Theory

The electronic and geometric structures of the water-oxidizing complex of photosystem II in the steps of the catalytic cycle that precede dioxygen evolution remain hotly debated. Recent structural and spectroscopic investigations support contradictory redox formulations for the active-site Mn4 CaOx cofactor in the final metastable S3 state. These range from the widely accepted MnIV 4 oxo-hydroxo model, which presumes that O-O bond formation occurs in the ultimate transient intermediate (S4 ) of the catalytic cycle, to a MnIII 2 MnIV 2 peroxo model representative of the contrasting "early-onset" O-O bond formation hypothesis. Density functional theory energetics of suggested S3 redox isomers are inconclusive because of extreme functional dependence. Here, we use the power of the domain-based local pair natural orbital approach to coupled cluster theory, DLPNO-CCSD(T), to present the first correlated wave function theory calculations of relative stabilities for distinct redox-isomeric forms of the S3 state. Our results enabled us to evaluate conflicting models for the S3 state of the oxygen-evolving complex (OEC) and to quantify the accuracy of lower-level theoretical approaches. Our assessment of the relevance of distinct redox-isomeric forms for the mechanism of biological water oxidation strongly disfavors the scenario of early-onset O-O formation advanced by literal interpretations of certain crystallographic models. This work serves as a case study in the application of modern coupled cluster implementations to redox isomerism problems in oligonuclear transition metal systems.

Computational Discovery of Transition-metal Complexes: From High-throughput Screening to Machine Learning

Transition-metal complexes are attractive targets for the design of catalysts and functional materials. The behavior of the metal-organic bond, while very tunable for achieving target properties, is challenging to predict and necessitates searching a wide and complex space to identify needles in haystacks for target applications. This review will focus on the techniques that make high-throughput search of transition-metal chemical space feasible for the discovery of complexes with desirable properties. The review will cover the development, promise, and limitations of "traditional" computational chemistry (i.e., force field, semiempirical, and density functional theory methods) as it pertains to data generation for inorganic molecular discovery. The review will also discuss the opportunities and limitations in leveraging experimental data sources. We will focus on how advances in statistical modeling, artificial intelligence, multiobjective optimization, and automation accelerate discovery of lead compounds and design rules. The overall objective of this review is to showcase how bringing together advances from diverse areas of computational chemistry and computer science have enabled the rapid uncovering of structure-property relationships in transition-metal chemistry. We aim to highlight how unique considerations in motifs of metal-organic bonding (e.g., variable spin and oxidation state, and bonding strength/nature) set them and their discovery apart from more commonly considered organic molecules. We will also highlight how uncertainty and relative data scarcity in transition-metal chemistry motivate specific developments in machine learning representations, model training, and in computational chemistry. Finally, we will conclude with an outlook of areas of opportunity for the accelerated discovery of transition-metal complexes.

High-Level ab Initio Predictions for the Ionization Energies, Bond Dissociation Energies, and Heats of Formation of Vanadium Methylidene, Vanadium Methyl Species, and Their Cations (VCH2/VCH2+, VCH3/VCH3+)

The ionization energies of VCH₂ and VCH₃, the various 0 K bond dissociation energies (D₀s) in their neutrals and cations, and their respective heats of formation at 0 and 298 K are computed by the single-reference, wave function-based CCSDTQ/CBS procedure. The core of the composite method is the approximation to the complete basis set (CBS) limit at the coupled cluster (CC) level which includes up to full quadruple excitations. The zero-point vibrational energy, core-valence correlation, spin-orbit coupling, and scalar relativistic effects have their contributions incorporated in an additive manner. For the species in the current study, this protocol requires geometry optimizations and harmonic frequency calculations practically no higher than the CCSD(T)/aug-cc-pwCVTZ and CCSD(T)/aug-cc-pVTZ levels, respectively. The present calculations successfully predict D₀(V⁺-CH₃) = 2.126 eV and D₀(V⁺-CH₂) = 3.298 eV in remarkable agreement with the data recently measured by a spin-orbit state selected V⁺ + CH₄ collision experiment (Phys. Chem. Chem. Phys. 2021, 23, 273-286). The good accord encourages the use of CCSDTQ/CBS protocol in thermochemical predictions of various feasible product channels identified in methane activation by transition metal species.

Revealing the nature of electron correlation in transition metal complexes with symmetry breaking and chemical intuition

In this work, we provide a nuanced view of electron correlation in the context of transition metal complexes, reconciling computational characterization via spin and spatial symmetry breaking in single-reference methods with qualitative concepts from ligand-field and molecular orbital theories. These insights provide the tools to reliably diagnose the multi-reference character, and our analysis reveals that while strong (i.e., static) correlation can be found in linear molecules (e.g., diatomics) and weakly bound and antiferromagnetically coupled (monometal-noninnocent ligand or multi-metal) complexes, it is rarely found in the ground-states of mono-transition-metal complexes. This leads to a picture of static correlation that is no more complex for transition metals than it is, e.g., for organic biradicaloids. In contrast, the ability of organometallic species to form more complex interactions, involving both ligand-to-metal σ-donation and metal-to-ligand π-backdonation, places a larger burden on a theory's treatment of dynamic correlation. We hypothesize that chemical bonds in which inter-electron pair correlation is non-negligible cannot be adequately described by theories using MP2 correlation energies and indeed find large errors vs experiment for carbonyl-dissociation energies from double-hybrid density functionals. A theory's description of dynamic correlation (and to a less important extent, delocalization error), which affects relative spin-state energetics and thus spin symmetry breaking, is found to govern the efficacy of its use to diagnose static correlation.

Spin-state energetics of metallocenes: How do best wave function and density functional theory results compare with the experimental data?

We benchmark the accuracy of quantum-chemical methods, including wave function theory methods [coupled cluster theory at the CCSD(T) level, multiconfigurational perturbation-theory (CASPT2, NEVPT2) and internally contracted multireference configuration interaction (MRCI)] and 30 density functional theory (DFT) approximations, in reproducing the spin-state splittings of metallocenes. The reference values of the electronic energy differences are derived from the experimental spin-crossover enthalpy for manganocene and the spectral data of singlet-triplet transitions for ruthenocene, ferrocene, and cobaltocenium. For ferrocene and cobaltocenium we revise the previous experimental interpretations regarding the lowest triplet energy; our argument is based on the comparison with the lowest singlet excitation energy and herein reported, carefully determined absorption spectrum of ferrocene. When deriving vertical energies from the experimental band maxima, we go beyond the routine vertical energy approximation by introducing vibronic corrections based on simulated vibrational envelopes. The benchmarking result confirms the high accuracy of the CCSD(T) method (in particular, for UCCSD(T) based on Hartree-Fock orbitals we find for our dataset: maximum error 0.12 eV, weighted mean absolute error 0.07 eV, weighted mean signed error 0.01 eV). The high accuracy of the single-reference method is corroborated by the analysis of a multiconfigurational character of the complete active space wave function for the triplet state of ferrocene. On the DFT side, our results confirm the non-universality problem with approximate functionals. The present study is an important step toward establishing an extensive and representative benchmark set of experiment-derived spin-state energetics for transition metal complexes.

Accurate ground state potential of Cu2 up to the dissociation limit by perturbation assisted double-resonant four-wave mixing

Perturbation facilitated double-resonant four-wave mixing is applied to access high-lying vibrational levels of the X ¹Σ_g ⁺ (0_g ⁺) ground state of Cu₂. Rotationally resolved transitions up to v^″ = 102 are measured. The highest observed level is at 98% of the dissociation energy. The range and accuracy of previous measurements are significantly extended. By applying the near dissociation equation developed by Le Roy [R. J. Le Roy, J. Quant. Spectrosc. Radiat. Transfer 186, 197 (2017)], a dissociation energy of D_e = 16 270(7) hc cm^-1 is determined, and an accurate potential energy function for the X ¹Σ_g ⁺ (0_g ⁺) ground state is obtained. Molecular constants are determined from the measured transitions and by solving the radial Schrödinger equation using this function and are compared with results from earlier measurements. In addition, benchmark multi-reference configuration interaction computations are performed using the Douglas-Kroll-Hess Hamiltonian and the appropriate basis of augmented valence quadruple ζ type. Coupled-cluster single, double, and perturbative triple calculations were performed for comparison.

Quantifying and Understanding Errors in Molecular Geometries

Electronic structure calculations are ubiquitous in most branches of chemistry, but all have errors in both energies and equilibrium geometries. Quantifying errors in possibly dozens of bond angles and bond lengths is a Herculean task. A single natural measure of geometric error is introduced, the geometry energy offset (GEO). GEO links many disparate aspects of geometry errors: a new ranking of different methods, quantitative insight into errors in specific geometric parameters, and insight into trends with different methods. GEO can also reduce the cost of high-level geometry optimizations and shows when geometric errors distort the overall error of a method. Results, including some surprises, are given for both covalent and weak interactions.

Predicting Bond Dissociation Energies and Bond Lengths of Coordinatively Unsaturated Vanadium-Ligand Bonds

Understanding the electronic structure of coordinatively unsaturated transition-metal compounds and predicting their physical properties are of great importance for catalyst design. Bond dissociation energy D_e and bond length r_e are two of the fundamental quantities for which good predictions are important for a successful design strategy. In the present work, recent experimentally measured bond energies and bond lengths of VX diatomic molecules (X = C, N, S) are used as a gauge to consider the utility of a number of electronic structure methods. Single-reference methods are one focus because of their efficiency and utility in practical calculations, and multireference configuration interaction (MRCISD) methods and a composite coupled cluster (CCC) method are a second focus because of their potential high accuracy. The comparison is especially challenging because of the large multireference M diagnostics of these molecules, in the range 0.15-0.19. For the single-reference methods, Kohn-Sham density functional theory (KS-DFT) has been tested with a variety of approximate exchange-correlation functionals. Of these, MOHLYP provides the bond dissociation energies in best agreement with experiments, and BLYP provides the bond lengths that are in best agreement with experiments; but by requiring good performance for both the D_e and r_e of the vanadium compounds, MOHLYP, MN12-L, MGGA_MS1, MGGA_MS0, O3LYP, and M06-L are the most highly recommended functionals. The CCC calculations include up to connected pentuple excitations for the valence electrons and up to connected quadruple excitations for the core-valence terms; this results in highly accurate dissociation energies and good bond lengths. Averaged over the three molecules, the mean unsigned deviation of CCC bond energies from experimental ones is only 0.4 kcal/mol, demonstrating excellent convergence of theory and experiments.

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.

Remarkable Accuracy of an O(N6) Perturbative Correction to Opposite-Spin CCSD: Are Triples Necessary for Chemical Accuracy in Coupled Cluster?

The focus of this work is OS-CCSD-SPT(2), which is a second-order similarity transformed perturbation theory correction to opposite spin coupled cluster singles doubles, where in the latter the same-spin amplitudes are removed and the opposite-spin ones are solved self-consistently. OS-CCSD-SPT(2) is free of empirical parameters, has an instrinsic scaling of O(N⁶), and makes no use of triples. We demonstrate that, for non-multireference molecules, OS-CCSD-SPT(2) produces relative energies whose accuracy is significantly higher than what is generally expected of a triples-free model. For example, using PBE0 orbitals in the reference, OS-CCSD-SPT(2) exhibits a mean absolute deviation (MAD) of 1.13 kcal/mol with respect to CCSD(2_F) benchmark values for the non-multireference subset of W4-08 atomization energies (cf. a MAD > 6.5 kcal/mol for CCSD) and a MAD of 0.68 kcal/mol for the energies of reactions generated from the W4-08 molecules. These MADs are reduced to 0.61 and 0.63 kcal/mol, respectively, by a simple one-parameter spin-component scaling of the OS-CCSD-SPT(2) same-spin correlation energy. OS-CCSD is also naturally amenable to higher order corrections: the associated third-order correction, OS-CCSD-SPT(3), which does involve connected triples and quadruples, exhibits a MAD of 0.44 kcal/mol for the same atomization-energy benchmark.

What Levels of Coupled Cluster Theory Are Appropriate for Transition Metal Systems? A Study Using Near-Exact Quantum Chemical Values for 3d Transition Metal Binary Compounds

Transition metal compounds are traditionally considered to be challenging for standard quantum chemistry approximations like coupled cluster (CC) theory, which are usually employed to validate lower level methods like density functional theory (DFT). To explore this issue, we present a database of bond dissociation energies (BDEs) for 74 spin states of 69 diatomic species containing a 3d transition metal atom and a main group element, in the moderately sized def2-SVP basis. The presented BDEs appear to have an (estimated) 3σ error less than 1 kJ/mol relative to the exact solutions to the nonrelativistic Born-Oppenheimer Hamiltonian. These benchmark values were used to assess the performance of a wide range of standard single reference CC models, as the results should be beneficial for understanding the limitations of these models for transition metal systems. We find that interactions between metals and monovalent ligands like hydride and fluoride are well described by CCSDT. Similarly, CCSDTQ appears to be adequate for bonds between metals and nominally divalent ligands like oxide and sulfide. However, interactions with polyvalent ligands like nitride and carbide are more challenging, with even CCSDTQ(P)_Λ yielding errors on the scale of a few kJ/mol. We also find that many perturbative and iterative approximations to higher order terms either yield disappointing results or actually worsen the performance relative to the baseline low level CC method, indicating that complexity does not always guarantee accuracy.

High-Level Ab Initio Predictions for the Ionization Energy, Bond Dissociation Energies, and Heats of Formation of Vanadium Methylidyne Radical and Its Cation (VCH/VCH+)

The ionization energy (IE) of VCH, the 0 K V-CH/VC-H bond dissociation energies (D₀s), and the heats of formation at 0 K (ΔH_f0^°) and 298 K (ΔH_f298^°) for VCH/VCH⁺ are predicted by the wave function-based CCSDTQ/CBS approach. This composite-coupled cluster method includes full quadruple excitations in conjunction with the approximation to the complete basis set (CBS) limit. The contributions of zero-point vibrational energy, core-valence (CV) correlation, spin-orbit coupling, and scalar relativistic corrections are taken into account. The present calculations show that adiabatic IE(VCH) = 6.785 eV and demonstrate excellent agreement with an IE value of 6.774 7 ± 0.000 1 eV measured with two-color laser-pulsed field ionization-photoelectron spectroscopy. The CCSDT and MRCI+Q methods which include CV correlations give the best predictions of harmonic frequencies: ω₂ (ω₂⁺) (bending) = 534 (650) and 564 (641) cm^-1 and the V-CH stretching ω₃ (ω₃⁺) = 835 (827) and 856 (857) cm^-1 compared with the experimental values. In this work, we offer a streamlined CCSDTQ/CBS approach which shows an error limit (≤20 meV) matching with previous benchmarking efforts for reliable IE and D₀ predictions for VCH/VCH⁺. The CCSDTQ/CBS D₀(V⁺-CH) - D₀(V-CH) = -0.012 eV and D₀(VC⁺-H) - D₀(VC-H) = 0.345 eV are in good accord with the experimentally derived values of -0.028 4 ± 0.000 1 and 0.355 9 ± 0.000 1 eV, respectively. The present study has demonstrated that the CCSDTQ/CBS protocol can be readily extended to investigate triatomic molecules containing 3d-metals.

Reassessment of the Mechanisms of Thermal C-H Bond Activation of Methane by Cationic Magnesium Oxides: A Critical Evaluation of the Suitability of Different Density Functionals

The mechanisms of the thermal reactions of the two iconic magnesium oxide cations MgO^.+ and Mg₂ O₂ ^.+ with methane have been re-evaluated at the CCSD(T)/CBS//CCSD/def2-TZVP level of theory. For the reaction of MgO^.+ with CH₄ , only the classical hydrogen-atom transfer (HAT) was found; in contrast, for the Mg₂ O₂ ^.+ /CH₄ couple, both HAT and proton-coupled electron-transfer (PCET) exist as mechanistic variants. In order to evaluate the suitability of density functional theory (DFT) methods, the reactions were computed by using 27 density functionals. The results obtained demonstrate that the various DFT methods often deliver rather different results for both geometric and energetic features. As to the prediction of the apparent barriers, pure functionals give the largest mean absolute errors. BMK, ωB97XD, and the double-hybrid functional mPW2PLYP were confirmed to come closest to the results provided by CCSD(T)/CBS. Thus, mechanistic conclusions based on a single DFT method should be viewed with great caution. In summary, this study may assist in the selection of a suitable quantum chemical method to unravel the mechanistic details of C-H bond activation by charged metal oxides.

Gas Phase Silver Thermochemistry from First Principles

Domain-based local pair natural orbital coupled cluster approach with single, double, and perturbative triple excitations, DLPNO-CCSD(T), has been applied within a framework of a reduced version of the reaction-based Feller-Peterson-Dixon (FPD) scheme to predict gas phase heats of formation and absolute entropies of silver inorganic and organometallic compounds. First, we evaluated all existing experimental data currently limited by thermodynamic functions of 10 silver substances (AgH, AgF, AgBr, AgI, Ag₂, Ag₂S, Ag₂Se, Ag₂Te, AgCN, AgPO₂). The mean average deviation between computed and experimental heats of formation was found to be 1.9 kcal/mol. Notably, all predicted heats of formation turned out to be within the error bounds of their experimental counterparts. Second, we predicted heats of formation and entropies for additional 90 silver species with no experimental data available, substantially enriching silver thermochemistry. Combination of gas phase heats of formation Δ H_f and entropies S° of AgNO₂, AgSCN, Ag₂SO₄, and Ag₂SeO₄ obtained in this work, with respective solid-state information, resulted in accurate sublimation thermochemistry of these compounds. Complementation of predicted Δ H_f with heats of formation of some neutrals and positive ions produced 33 silver bond strengths of high reliability. Obtained thermochemical data are promising for developing the concepts of silver chemistry. In addition, derived heats of formation and bond dissociation enthalpies, due to their high diversity, are found to be relevant for testing and training of computational chemistry methods.

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.

Benchmarking quantum chemistry methods for spin-state energetics of iron complexes against quantitative experimental data

The accuracy of relative spin-state energetics predicted by selected quantum chemistry methods: coupled cluster theory at the CCSD(T) level, multiconfigurational perturbation theory (CASPT2, NEVPT2), multireference configuration interaction at the MRCISD+Q level, and a number of DFT methods, is quantitatively evaluated by comparison with the experimental data of four octahedral iron complexes. The available experimental data, either spin-forbidden transition energies or spin crossover enthalpies, are corrected for relevant environmental effects in order to derive the quantitative benchmark set of iron spin-state energetics. Comparison of theory predictions with the resulting reference data: (1) validates the high accuracy of the CCSD(T) method, particularly when based on Kohn-Sham orbitals, giving the maximum error below 2 kcal mol-1 and the mean absolute error (MAE) below 1 kcal mol-1; (2) corroborates the tendency of CASPT2 to systematically overstabilize higher-spin states by up to 5.5 kcal mol-1; (3) confirms that the latter problem is partly remedied by the recently proposed CASPT2/CC approach [Phung et al., J. Chem. Theory Comput., 2018, 14, 2446-2455]; (4) demonstrates that NEVPT2 performs worse than CASPT2, by giving errors up to 7 kcal mol-1; (5) shows that the accuracy of MRCISD+Q spin-state energetics strongly depends on the size-consistency correction: the Davidson-Silver and Pople corrections perform best (MAE < 3 kcal mol-1), whereas the standard Davidson correction is not recommended (MAE of 7 kcal mol-1). Only a few DFT methods (including the best performing ones identified in this study: B2PLYP-D3 and OPBE) are able to provide a balanced description of the spin-state energetics for all four studied iron complexes simultaneously, corroborating the non-universality problem of approximate density functionals.

Predissociation Measurements of Bond Dissociation Energies

A fundamental need in chemistry is understanding the chemical bond, for which the most quantitative measure is the bond dissociation energy (BDE). While BDEs of chemical bonds formed from the lighter main group elements are generally well-known and readily calculated by modern computational chemistry, chemical bonds involving the transition metals, lanthanides, and actinides remain computationally extremely challenging. This is due to the simultaneous importance of electron correlation, spin-orbit interaction, and other relativistic effects, coupled with the large numbers of low-lying states that are accessible in systems with open d or f subshells. The development of efficient and accurate computational methods for these species is currently a major focus of the field. An obstacle to this effort has been the scarcity of highly precise benchmarks for the BDEs of M-X bonds. For most of the transition metal, lanthanide, or actinide systems, tabulated BDEs of M-X bonds have been determined by Knudsen effusion mass spectrometric measurements of high-temperature equilibria. The measured ion signals are converted to pressures and activities of the species involved in the equilibrium, and the equilibrium constants are then analyzed using a van't Hoff plot or the third-law method to extract the reaction enthalpy, which is extrapolated to 0 K to obtain the BDE. This procedure introduces errors at every step and ultimately leads to BDEs that are typically uncertain by 2-20 kcal mol^-1 (0.1-1 eV). A second method in common use employs a thermochemical cycle in which the ionization energies of the MX molecule and M atom are combined with the BDE of the M⁺-X bond, obtained via guided ion beam mass spectrometry, to yield the BDE of the neutral, M-X. When accurate values of all three components of the cycle are available, this method yields good results-but only rarely are all three values available. We have recently implemented a new method for the precise measurement of BDEs in molecules with large densities of electronic states that is based on the rapid predissociation of these species as soon as the ground separated atom limit is exceeded. When a sharp predissociation threshold is observed, its value directly provides the BDE of the system. With this method, we are able in favorable cases to determine M-X BDEs to an accuracy of ∼0.1 kcal mol^-1 (0.004 eV). The method is generally applicable to species that have a high density of states at the ground separated atom limit and has been used to measure the BDEs of more than 50 transition metal-main group MX molecules thus far. In addition, a number of metal-metal BDEs have also been measured with this method. There are good prospects for the extension of the method to polyatomic systems and to lanthanide and actinide-containing molecules. These precise BDE measurements provide chemical trends for the BDEs across the transition metal series, as well as crucial benchmarks for the development of efficient and accurate computational methods for the d- and f-block elements.

Spin-Unrestricted Self-Energy Embedding Theory

We present a new theoretical approach, unrestricted self-energy embedding theory (USEET), that is a Green's function embedding theory used to study problems in which an open, embedded system exchanges electrons with the environment. USEET has a high potential to be used in studies of strongly correlated systems with an odd number of electrons and open shell systems such as transition metal complexes important in inorganic chemistry. In this paper, we show that USEET results agree very well with common quantum chemistry methods while avoiding typical bottlenecks present in these methods.

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 d^q 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.

Bond dissociation energies of TiSi, ZrSi, HfSi, VSi, NbSi, and TaSi

Predissociation thresholds have been observed in the resonant two-photon ionization spectra of TiSi, ZrSi, HfSi, VSi, NbSi, and TaSi. It is argued that because of the high density of electronic states at the ground separated atom limit in these molecules, the predissociation threshold in each case corresponds to the thermochemical bond dissociation energy. The resulting bond dissociation energies are D₀(TiSi) = 2.201(3) eV, D₀(ZrSi) = 2.950(3) eV, D₀(HfSi) = 2.871(3) eV, D₀(VSi) = 2.234(3) eV, D₀(NbSi) = 3.080(3) eV, and D₀(TaSi) = 2.999(3) eV. The enthalpies of formation were also calculated as Δ_f,0KH°(TiSi(g)) = 705(19) kJ mol^-1, Δ_f,0KH°(ZrSi(g)) = 770(12) kJ mol^-1, Δ_f,0KH°(HfSi(g)) = 787(10) kJ mol^-1, Δ_f,0KH°(VSi(g)) = 743(11) kJ mol^-1, Δ_f,0KH°(NbSi(g)) = 879(11) kJ mol^-1, and Δ_f,0KH°(TaSi(g)) = 938(8) kJ mol^-1. Using thermochemical cycles, ionization energies of IE(TiSi) = 6.49(17) eV and IE(VSi) = 6.61(15) eV and bond dissociation energies of the ZrSi^- and NbSi^- anions, D₀(Zr-Si^-) ≤ 3.149(15) eV, D₀(Zr^--Si) ≤ 4.108(20) eV, D₀(Nb-Si^-) ≤ 3.525(31) eV, and D₀(Nb^--Si) ≤ 4.017(39) eV, have also been obtained. Calculations on the possible low-lying electronic states of each species are also reported.