Detailed Pair Natural Orbital-Based Coupled Cluster Studies of Spin Crossover Energetics

Discovery of Spin-Crossover Materials with Equivariant Graph Neural Networks and Relevance-Based Classification

Swift discovery of spin-crossover materials for their potential application in electronic and quantum devices requires techniques that enable efficient identification of suitable candidates. To this end, we screened the Cambridge Structural Database to develop a specialized database of 1439 materials and computed spin-switching energies from density functional theory for each material. The database was used to train an equivariant graph convolution neural network to predict the magnitude of the spin-conversion energy. A test mean absolute error was 360 meV. For candidate identification, we equipped the system with a relevance-based classifier. This approach leads to a nearly 4-fold improvement in identifying potential spin-crossover systems of interest as compared to conventional high-throughput screening.

Beyond CCSD(T) Accuracy at Lower Scaling with Auxiliary Field Quantum Monte Carlo

We introduce a black-box auxiliary field quantum Monte Carlo (AFQMC) approach to perform highly accurate electronic structure calculations using configuration interaction singles and doubles (CISD) trial states. This method consistently provides more accurate energy estimates than coupled cluster singles and doubles with perturbative triples (CCSD(T)), often regarded as the gold standard in quantum chemistry. This level of precision is achieved at a lower asymptotic computational cost, scaling as O(N6) compared to the O(N7) scaling of CCSD(T). We provide numerical evidence supporting these findings through results for challenging main group and transition metal-containing molecules.

Projector-Based Quantum Embedding Study of Iron Complexes

Projection-based embedding theory (PBET) is used to calculate and assess the challenging spin-crossover energies for a selection of small Fe-containing systems by embedding the metal center into the frozen potential of the ligands. MP2, CCSD, and CCSD(T) are embedded in potentials from the SCAN and r2SCAN functionals and compared with the canonical values for the constituent methods and previously reported reference values. Considering the PBET calculations as a correction for the underlying DFT, the embedding calculations are able to provided improvement for most cases. In some cases, the PBET methods are able to compensate for limitations in the wave function methods and produce results similar to more rigorous calculations from the literature. For the systems with spin-crossover energies near zero, the current methodology fails to provide consistent improvement. The isolated recalculation of the electronic structure around the metal center when embedded into a DFT treatment of the ligand field shows promise as a pragmatic and lower cost treatment compared to the canonical treatment of the whole system of the difficult class of spin-crossover complexes.

Performance of quantum chemistry methods for a benchmark set of spin-state energetics derived from experimental data of 17 transition metal complexes (SSE17)

Accurate prediction of spin-state energetics for transition metal (TM) complexes is a compelling problem in applied quantum chemistry, with enormous implications for modeling catalytic reaction mechanisms and computational discovery of materials. Computed spin-state energetics are strongly method-dependent and credible reference data are scarce, making it difficult to conduct conclusive computational studies of open-shell TM systems. Here, we present a novel benchmark set of first-row TM spin-state energetics, which is derived from experimental data of 17 complexes containing FeII, FeIII, CoII, CoIII, MnII, and NiII with chemically diverse ligands. The estimates of adiabatic or vertical spin-state splittings, which are obtained from spin crossover enthalpies or energies of spin-forbidden absorption bands, suitably back-corrected for the vibrational and environmental effects, are employed as reference values for benchmarking density functional theory (DFT) and wave function methods. The results demonstrate a high accuracy of the coupled-cluster CCSD(T) method, which features the mean absolute error (MAE) of 1.5 kcal mol-1 and maximum error of -3.5 kcal mol-1, and outperforms all the tested multireference methods: CASPT2, MRCI+Q, CASPT2/CC and CASPT2+δMRCI. Switching from Hartree-Fock to Kohn-Sham orbitals is not found to consistently improve the CCSD(T) accuracy. The best performing DFT methods are double-hybrids (PWPB95-D3(BJ), B2PLYP-D3(BJ)) with the MAEs below 3 kcal mol-1 and maximum errors within 6 kcal mol-1, whereas the DFT methods so far recommended for spin states (e.g., B3LYP*-D3(BJ) and TPSSh-D3(BJ)) are found to perform much worse with the MAEs of 5-7 kcal mol-1 and maximum errors beyond 10 kcal mol-1. This work is the first such extensive benchmark study of quantum chemistry methods for TM spin-state energetics making use of experimental reference data. The results are relevant for the proper choice of methods to characterize TM systems in computational catalysis and (bio)inorganic chemistry, and may also stimulate new developments in quantum-chemical or machine learning approaches.

Mechanistic Principles of Hydrogen Evolution in the Membrane-Bound Hydrogenase

The membrane-bound hydrogenase (Mbh) from Pyrococcus furiosus is an archaeal member of the Complex I superfamily. It catalyzes the reduction of protons to H2 gas powered by a [NiFe] active site and transduces the free energy into proton pumping and Na+/H+ exchange across the membrane. Despite recent structural advances, the mechanistic principles of H2 catalysis and ion transport in Mbh remain elusive. Here, we probe how the redox chemistry drives the reduction of the proton to H2 and how the catalysis couples to conformational dynamics in the membrane domain of Mbh. By combining large-scale quantum chemical density functional theory (DFT) and correlated ab initio wave function methods with atomistic molecular dynamics simulations, we show that the proton transfer reactions required for the catalysis are gated by electric field effects that direct the protons by water-mediated reactions from Glu21L toward the [NiFe] site, or alternatively along the nearby His75L pathway that also becomes energetically feasible in certain reaction steps. These local proton-coupled electron transfer (PCET) reactions induce conformational changes around the active site that provide a key coupling element via conserved loop structures to the ion transport activity. We find that H2 forms in a heterolytic proton reduction step, with spin crossovers tuning the energetics along key reaction steps. On a general level, our work showcases the role of electric fields in enzyme catalysis and how these effects are employed by the [NiFe] active site of Mbh to drive PCET reactions and ion transport.

The spin-forbidden transition in iron(IV)-oxo catalysts relevant to two-state reactivity

Quintet oxoiron(IV) intermediates are often invoked in nonheme iron enzymes capable of performing selective oxidation, while most well-characterized synthetic model oxoiron(IV) complexes have a triplet ground state. These differing spin states lead to the proposal of a two-state reactivity model, where the complexes cross from the triplet to an excited quintet state. However, the energy of this quintet state has never been measured experimentally. Here, magnetic circular dichroism is used to assign the singlet and triplet excited states in a series of triplet oxoiron(IV) complexes. These transition energies are used to determine the energies of the quintet state via constrained fitting of 2p3d resonant inelastic x-ray scattering. This allowed for a direct correlation between the quintet energies and substrate C─H oxidation rates.

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.

C-H Bond Oxidation by MnIV-Oxo Complexes: Hydrogen-Atom Tunneling and Multistate Reactivity

The reactivity of six MnIV-oxo complexes in C-H bond oxidation has been examined using a combination of kinetic experiments and computational methods. Variable-temperature studies of the oxidation of 9,10-dihydroanthracene (DHA) and ethylbenzene by these MnIV-oxo complexes yielded activation parameters suitable for evaluating electronic structure computations. Complementary kinetic experiments of the oxidation of deuterated DHA provided evidence for hydrogen-atom tunneling in C-H bond oxidation for all MnIV-oxo complexes. These results are in accordance with the Bell model, where tunneling occurs near the top of the transition-state barrier. Density functional theory (DFT) and DLPNO-CCSD(T1) computations were performed for three of the six MnIV-oxo complexes to probe a previously predicted multistate reactivity model. The DFT computations predicted a thermal crossing from the 4B1 ground state to a 4E state along the C-H bond oxidation reaction coordinate. DLPNO-CCSD(T1) calculations further confirm that the 4E transition state offers a lower energy barrier, reinforcing the multistate reactivity model for these complexes. We discuss how this multistate model can be reconciled with recent computations that revealed that the kinetics of C-H bond oxidation by this set of MnIV-oxo complexes can be well-predicted on the basis of the thermodynamic driving force for these reactions.

Approaching the Complete Basis Set Limit for Spin-State Energetics of Mononuclear First-Row Transition Metal Complexes

Convergence to the complete basis set (CBS) limit is analyzed for the problem of spin-state energetics in mononuclear first-row transition metal (TM) complexes by taking under scrutiny a benchmark set of 18 energy differences between spin states for 13 chemically diverse TM complexes. The performance of conventional CCSD(T) and explicitly correlated CCSD(T)-F12a/b calculations in approaching the CCSD(T)/CBS limits is systematically studied. An economic computational protocol is developed based on the CCSD-F12a approximation and (here proposed) modified scaling of the perturbative triples term (T#). This computational protocol recovers the relative spin-state energetics of the benchmark set in excellent agreement with the reference CCSD(T)/CBS limits (mean absolute deviation of 0.4, mean signed deviation of 0.2, and maximum deviation of 0.8 kcal/mol) and enables performing canonical CCSD(T) calculations for mononuclear TM complexes sized up to ca. 50 atoms, which is illustrated by application to heme-related metalloporphyrins. Furthermore, a good transferability of the basis set incompleteness error (BSIE) is demonstrated for spin-state energetics computed using CCSD(T) and other wave function methods (MP2, CASPT2, CASPT2/CC, NEVPT2, and MRCI + Q), which justifies efficient focal-point approximations and simplifies the construction of multimethod benchmark studies.

Cluster-in-Molecule Approach with Explicitly Correlated Methods for Large Molecules

In this article, we present a series of explicitly correlated local correlation methods developed under the cluster-in-molecule (CIM) framework, including explicitly correlated second-order Møller-Plesset perturbation (MP2), coupled-cluster singles and doubles (CCSD), domain-based local pair natural orbital CCSD (DLPNO-CCSD), and DLPNO-CCSD with perturbative triples (DLPNO-CCSD(T)). In these methods, F12 correction is decomposed into contributions from each occupied local molecular orbital and then evaluated independently in a given cluster, which consists of a subset of localized orbitals. These newly developed methods allow F12 calculations of large molecules (up to 145 atoms for quasi-one-dimensional systems) on a single node. We use these methods to investigate the relative stability between extended and folded alkane C30H62, the relative stability of four secondary structures of a polyglycine Ace(Gly)10NH2, and the binding energies of two host-guest complexes. The results demonstrate that the combination of CIM with F12 methods is a promising way to investigate large molecules with small basis set errors.

Benchmarking Density Functional Theory Methods for Metalloenzyme Reactions: The Introduction of the MME55 Set

We present a new benchmark set of metalloenzyme model reaction energies and barrier heights that we call MME55. The set contains 10 different enzymes, representing eight transition metals, both open and closed shell systems, and system sizes of up to 116 atoms. We use four DLPNO-CCSD(T)-based approaches to calculate reference values against which we then benchmark the performance of a range of density functional approximations with and without dispersion corrections. Dispersion corrections improve the results across the board, and triple-ζ basis sets provide the best balance of efficiency and accuracy. Jacob's ladder is reproduced for the whole set based on averaged mean absolute (percent) deviations, with the double hybrids SOS0-PBE0-2-D3(BJ) and revDOD-PBEP86-D4 standing out as the most accurate methods for the MME55 set. The range-separated hybrids ωB97M-V and ωB97X-V also perform well here and can be recommended as a reliable compromise between accuracy and efficiency; they have already been shown to be robust across many other types of chemical problems, as well. Despite the popularity of B3LYP in computational enzymology, it is not a strong performer on our benchmark set, and we discourage its use for enzyme energetics.

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.

Chemically accurate singlet-triplet gaps of organic chromophores and linear acenes by the random phase approximation and σ-functionals

Predicting the energy differences between different spin-states is challenging for many widely used ab initio electronic structure methods. We here assess the ability of the direct random phase approximation (dRPA), dRPA plus two different screened second-order exchange (SOX) corrections, and σ-functionals to predict adiabatic singlet-triplet gaps. With mean absolute deviations of below 0.1 eV to experimental reference values, independent of the Kohn-Sham starting point, dRPA and σ-functionals accurately predict singlet-triplet gaps of 18 organic chromophores. The addition of SOX corrections to dRPA considerably worsens agreement with experiment, adding to the mounting evidence that dRPA+SOX methods are not generally applicable beyond-RPA methods. Also for a series of linear acene chains with up to ten fused rings, dRPA, and σ-functionals are in excellent agreement with coupled-cluster single double triple reference data. In agreement with advanced multi-reference methods, dRPA@PBE and σ-functional@PBE predict a singlet ground state for all chain lengths, while dRPA@PBE0 and σ-functional@PBE0 predict a triplet ground state for longer acenes. Our work shows dRPA and σ-functionals to be reliable methods for calculating singlet-triplet gaps in aromatic molecules.

Understanding Antiferromagnetic and Ligand Field Effects on Spin Crossover in a Triple-Decker Dimeric Cr(II) Complex

Two possible explanations for the temperature dependence of spin-crossover (SCO) behavior in the dimeric triple-decker Cr(II) complex ([(η5-C5Me5)Cr(μ2:η5-P5)Cr(η5-C5Me5)]+) have been offered. One invokes variations in antiferromagnetic interactions between the two Cr(II) ions, whereas the other posits the development of a strong ligand-field effect favoring the low-spin ground state. We perform multireference electronic structure calculations based on the multiconfiguration pair-density functional theory to resolve these effects. We find quintet, triplet, and singlet electronic ground states, respectively, for the experimental geometries at high, intermediate, and low temperatures. The ground-state transition from quintet to triplet at an intermediate temperature derives from increased antiferromagnetic interactions between the two Cr(II) ions. By contrast, the ground-state transition from triplet to singlet at low temperature can be attributed to increased ligand-field effects, which dominate with continued variations in antiferromagnetic coupling. This study provides quantitative detail for the degree to which these two effects can act in concert for the observed SCO behavior in this complex and others subject to temperature-dependent variations in geometry.

High-throughput screening of spin states for transition metal complexes with spin-polarized extended tight-binding methods

The semiempirical GFNn-xTB ( n = 1 , 2 ) tight-binding methods are extended with a spin-dependent energy term (spin-polarization), enabling the fast and efficient screening of different spin states for transition metal complexes. While GFNn-xTB methods inherently can not differentiate properly between high-spin (HS) and low-spin (LS) states, this shortcoming is corrected with the presented methods termed spGFNn-xTB. The performance of spGFNn-xTB methods for spin state energy splittings is evaluated on a newly compiled benchmark set of 90 complexes (27 HS and 63 LS complexes) containing 3d, 4d, and 5d transition metals (termed TM90S) employing DFT references at the TPSSh-D4/def2-QZVPP level of theory. The challenging TM90S set contains complexes with charges between - 4 and +3, spin multiplicities between 1 and 6, and spin-splitting energies that range from - 47.8 to 146.6 kcal/mol with a mean average of 32.2 kcal/mol. On this set the (sp)GFNn-xTB methods, the PM6-D3H4 method, and the PM7 method are evaluated with spGFN1-xTB yielding the lowest MAD of 19.6 kcal/mol followed by spGFN2-xTB with 24.8 kcal/mol. While for the 4d and 5d subsets small or no improvements are observed with spin-polarization, large improvements are obtained for the 3d subset with spGFN1-xTB yielding the smallest MAD of 14.2 kcal/mol followed by spGFN2-xTB with 17.9 kcal/mol and PM6-D3H4 with 28.4 kcal/mol. The correct sign of the spin state splittings is obtained with spGFN2-xTB in 89% of all cases closely followed by spGFN1-xTB with 88%. On the full set, a pure semiempirical vertical spGFN2-xTB//GFN2-xTB-based workflow for screening purposes yields a slightly better MAD of 22.2 kcal/mol due to error compensation, while being qualitative correct for one additional case. In combination with their low computational cost (scanning spin states in seconds), the spGFNn-xTB methods represent robust tools for pre-screening steps of spin state calculations and high-throughput workflows.

High-Level Quantum Chemical Prediction of C-F Bond Dissociation Energies of Perfluoroalkyl Substances

In this study, 550 C-F bond dissociation energies (BDEs) of a variety of per- and polyfluoroalkyl substances (PFASs) obtained from high-level DLPNO-CCSD(T)/CBS calculations were used to assess the accuracy of contemporary density functional theory (DFT) and semiempirical methods. DLPNO-CCSD(T)/CBS gas phase C-F BDEs fall between 404.9-550.7 kJ mol-1 and M06-2X and ωB97M-V in conjunction with the aug-cc-pVTZ basis set predicted BDEs closest to the benchmark level with a mean absolute deviation (MAD) of 7.3 and 8.3 kJ mol-1, respectively. It was observed that DFT prediction errors increase with the degree of fluorination and system size. As such, previous model chemistry recommendations based on smaller nonfluorinated systems may not be carried over to modeling the energetics of PFASs and related systems.

MP2-Based Correction Scheme to Approach the Limit of a Complete Pair Natural Orbitals Space in DLPNO-CCSD(T) Calculations

The domain-based local pair natural orbital (PNO) coupled-cluster DLPNO-CCSD(T) method has been proven to provide accurate single-point energies at a fraction of the cost of canonical CCSD(T) calculations. However, the desired "chemical accuracy" can only be obtained with a large PNO space and extended basis set. We present a simple yet accurate and efficient correction scheme based on a perturbative approach. Here, in addition to DLPNO-CCSD(T) energy, one calculates DLPNO-MP2 correlation energy with the same settings as in the preceding coupled-cluster calculation. In the next step, the canonical MP2 correlation energy is obtained in the same orbital basis. This can be efficiently performed for essentially all molecule sizes accessible with the DLPNO-CCSD(T) method. By taking the difference between the canonical MP2 and DLPNO-MP2 energies, we obtain a correction term that can be added to the DLPNO-CCSD(T) correlation energy. This way, one can obtain the total correlation energy close to the limit of the complete PNO space (cPNO). The presented approach allows us to significantly increase the accuracy of the DLPNO-CCSD(T) method for both closed- and open-shell systems. The latter are known to be especially challenging for locally correlated methods. Unlike the previously developed PNO extrapolation procedure by Altun, Neese, and Bistoni ( J. Chem. Theory Comput. 2020, 16, 6142-6149), this strategy allows us to get the DLPNO-CCSD(T) correlation energy at the cPNO limit in a cost-efficient way, resulting in a minimal overall increase in calculation time as compared to the uncorrected method.

On the potentially transformative role of auxiliary-field quantum Monte Carlo in quantum chemistry: A highly accurate method for transition metals and beyond

Approximate solutions to the ab initio electronic structure problem have been a focus of theoretical and computational chemistry research for much of the past century, with the goal of predicting relevant energy differences to within "chemical accuracy" (1 kcal/mol). For small organic molecules, or in general, for weakly correlated main group chemistry, a hierarchy of single-reference wave function methods has been rigorously established, spanning perturbation theory and the coupled cluster (CC) formalism. For these systems, CC with singles, doubles, and perturbative triples is known to achieve chemical accuracy, albeit at O(N7) computational cost. In addition, a hierarchy of density functional approximations of increasing formal sophistication, known as Jacob's ladder, has been shown to systematically reduce average errors over large datasets representing weakly correlated chemistry. However, the accuracy of such computational models is less clear in the increasingly important frontiers of chemical space including transition metals and f-block compounds, in which strong correlation can play an important role in reactivity. A stochastic method, phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC), has been shown to be capable of producing chemically accurate predictions even for challenging molecular systems beyond the main group, with relatively low O(N3 - N4) cost and near-perfect parallel efficiency. Herein, we present our perspectives on the past, present, and future of the ph-AFQMC method. We focus on its potential in transition metal quantum chemistry to be a highly accurate, systematically improvable method that can reliably probe strongly correlated systems in biology and chemical catalysis and provide reference thermochemical values (for future development of density functionals or interatomic potentials) when experiments are either noisy or absent. Finally, we discuss the present limitations of the method and where we expect near-term development to be most fruitful.

Spin-crossover complexes: Self-interaction correction vs density correction

Complexes containing a transition metal atom with a 3d⁴-3d⁷ electron configuration typically have two low-lying, high-spin (HS) and low-spin (LS) states. The adiabatic energy difference between these states, known as the spin-crossover energy, is small enough to pose a challenge even for electronic structure methods that are well known for their accuracy and reliability. In this work, we analyze the quality of electronic structure approximations for spin-crossover energies of iron complexes with four different ligands by comparing energies from self-consistent and post-self-consistent calculations for methods based on the random phase approximation and the Fermi-Löwdin self-interaction correction. Considering that Hartree-Fock densities were found by Song et al., J. Chem. Theory Comput. 14, 2304 (2018), to eliminate the density error to a large extent, and that the Hartree-Fock method and the Perdew-Zunger-type self-interaction correction share some physics, we compare the densities obtained with these methods to learn their resemblance. We find that evaluating non-empirical exchange-correlation energy functionals on the corresponding self-interaction-corrected densities can mitigate the strong density errors and improves the accuracy of the adiabatic energy differences between HS and LS states.

Spin-state gaps and self-interaction-corrected density functional approximations: Octahedral Fe(II) complexes as case study

Accurate prediction of a spin-state energy difference is crucial for understanding the spin crossover phenomena and is very challenging for density functional approximations, especially for local and semi-local approximations due to delocalization errors. Here, we investigate the effect of the self-interaction error removal from the local spin density approximation (LSDA) and Perdew-Burke-Ernzerhof generalized gradient approximation on the spin-state gaps of Fe(II) complexes with various ligands using recently developed locally scaled self-interaction correction (LSIC) by Zope et al. [J. Chem. Phys. 151, 214108 (2019)]. The LSIC method is exact for one-electron density, recovers the uniform electron gas limit of the underlying functional, and approaches the well-known Perdew-Zunger self-interaction correction (PZSIC) as a particular case when the scaling factor is set to unity. Our results, when compared with reference diffusion Monte Carlo results, show that the PZSIC method significantly overestimates spin-state gaps favoring low spin states for all ligands and does not improve upon density functional approximations. The perturbative LSIC-LSDA using PZSIC densities significantly improves the gaps with a mean absolute error of 0.51 eV but slightly overcorrects for the stronger CO ligands. The quasi-self-consistent LSIC-LSDA, such as coupled-cluster single double and perturbative triple [CCSD(T)], gives a correct sign of spin-state gaps for all ligands with a mean absolute error of 0.56 eV, comparable to that of CCSD(T) (0.49 eV).

Computational NMR of the iron pyrazolylborate complexes [Tp2Fe]+ and Tp2Fe including solvation and spin-crossover effects

Transition metal complexes have important roles in many biological processes as well as applications in fields such as pharmacy, chemistry and materials science. Paramagnetic nuclear magnetic resonance (pNMR) is a valuable tool in understanding such molecules, and theoretical computations are often advantageous or even necessary in the assignment of experimental pNMR signals. We have employed density functional theory (DFT) and the domain-based local pair natural orbital coupled-cluster method with single and double excitations (DLPNO-CCSD), as well as a number of model improvements, to determine the critical hyperfine part of the chemical shifts of the iron pyrazolylborate complexes [Tp₂Fe]⁺ and Tp₂Fe using a modern version of the Kurland-McGarvey theory, which is based on parameterising the hyperfine, electronic Zeeman and zero-field splitting interactions via the parameters of the electron paramagnetic resonance Hamiltonian. In the doublet [Tp₂Fe]⁺ system, the calculations suggest a re-assignment of the ¹³C signal shifts. Consideration of solvent via the conductor-like polarisable continuum model (C-PCM) versus explicit solvent molecules reveals C-PCM alone to be insufficient in capturing the most important solvation effects. Tp₂Fe exhibits a spin-crossover effect between a high-spin quintet (S = 2) and a low-spin singlet (S = 0) state, and its recorded temperature dependence can only be reproduced theoretically by accounting for the thermal Boltzmann distribution of the open-shell excited state and the closed-shell ground-state occupations. In these two cases, DLPNO-CCSD is found, in calculating the hyperfine couplings, to be a viable alternative to DFT, the demonstrated shortcomings of which have been a significant issue in the development of computational pNMR.

Spin-State Splittings in 3d Transition-Metal Complexes Revisited: Toward a Reliable Theory Benchmark

A new composite method for the calculation of spin-crossover energies in 3d transition-metal complexes based on multireference methods is presented. The method reduces to MRCISD+Q at the complete-basis-set (CBS) level for atomic ions, for which it gives excitation energies with a mean absolute error of only ca. 0.01 eV. For molecular complexes, the CASPT2+δMRCI composite approach corresponds to a CASPT2/CBS calculation augmented by a high-level MRCISD+Q-CASPT2 correction with a smaller ligand basis set. For a set of [Fe(He)6]n+ test complexes, the approach reproduces full MRCISD+Q/CBS results to within better than 0.04 eV, without depending on any arbitrary IPEA shifts. The high-quality CASPT2+δMRCI method has then been applied to a series of 3d transition-metal hexaqua complexes in aqueous solution, augmented by an elaborate 3D-RISM-SCF solvent treatment of the underlying structures. It provides unprecedented agreement with experiment for the lowest-lying vertical spin-flip excitation energies, except for the Fe³⁺ system. Closer examination of the latter case provides strong evidence that the observed lowest-energy excitation at 1.56 eV, which has been used frequently for evaluating quantum-chemical methods, does not arise from the iron(III) hexaqua complex in solution, but from its singly deprotonated counterpart, [Fe(H2O)5OH]2+.

Comparative oxidative ability of mononuclear and dinuclear high-valent iron-oxo species towards the activation of methane: does the axial/bridge atom modulate the reactivity?

Over the years, mononuclear Fe^IVO species have been extensively studied, but the presence of dinuclear Fe^IVO species in soluble methane monooxygenase (sMMO) has inspired the development of biomimic models that could activate inert substrates such as methane. There are some successful attempts; particularly the [(Por)(m-CBA) Fe^IV(μ-N)Fe^IV(O)(Por˙⁺)]^- species has been reported to activate methane and yield decent catalytic turnover numbers and therefore regarded as the closest to the sMMO enzyme functional model, as no mononuclear Fe^IVO analogues could achieve this feat. In this work, we have studied a series of mono and dinuclear models using DFT and ab initio DLPNO-CCSD(T) calculations to probe the importance of nuclearity in enhancing the reactivity. We have probed the catalytic activities of four complexes: [(HO)Fe^IV(O)(Por)]^- (1), [(HO)Fe^IV(O)(Por˙⁺)] (2), μ-oxo dinuclear iron species [(Por)(m-CBA)Fe^IV(μ-O)Fe^IV(O) (Por˙⁺)]^- (3) and N-bridged dinuclear iron species [(Por)(m-CBA)Fe^IV(μ-N)Fe^IV(O)(Por˙⁺)]^- (4) towards the activation of methane. Additionally, calculations were performed on the mononuclear models [(X)Fe^IV(O)(Por˙⁺)]ⁿ {X = N 4a (n = -2), NH 4b (n = -1) and NH₂4c (n = 0)} to understand the role of nuclearity in the reactivity. DFT calculations performed on species 1-4 suggest an interesting variation among them, with species 1-3 possessing an intermediate spin (S = 1) as a ground state and species 4 possessing a high-spin (S = 2) as a ground state. Furthermore, the two Fe^IV centres in species 3 and 4 are antiferromagnetically coupled, yielding a singlet state with a distinct difference in their electronic structure. On the other hand, species 2 exhibits a ferromagnetic coupling between the Fe^IV and the Por˙⁺ moiety. Our calculations suggest that the higher barriers for the C-H bond activation of methane and the rebound step for species 1 and 3 are very high in energy, rendering them unreactive towards methane, while species 2 and 4 have lower barriers, suggesting their reactivity towards methane. Studies on the system reveal that model 4a has multiple FeN bonds facilitating greater reactivity, whereas the other two models have longer Fe-N bonds and less radical character with steeper barriers. Strong electronic cooperativity is found to be facilitated by the bridging nitride atom, and this cooperativity is suppressed by substituents such as oxygen, rendering them inactive. Thus, our study unravels that apart from enhancing the nuclearity, bridging atoms that facilitate strong cooperation between the metals are required to activate very inert substrates such as methane, and our results are broadly in agreement with earlier experimental findings.

A transferable recommender approach for selecting the best density functional approximations in chemical discovery

Approximate density functional theory has become indispensable owing to its balanced cost-accuracy trade-off, including in large-scale screening. To date, however, no density functional approximation (DFA) with universal accuracy has been identified, leading to uncertainty in the quality of data generated from density functional theory. With electron density fitting and Δ-learning, we build a DFA recommender that selects the DFA with the lowest expected error with respect to the gold standard (but cost-prohibitive) coupled cluster theory in a system-specific manner. We demonstrate this recommender approach on the evaluation of vertical spin splitting energies of transition metal complexes. Our recommender predicts top-performing DFAs and yields excellent accuracy (about 2 kcal mol-1) for chemical discovery, outperforming both individual Δ-learning models and the best conventional single-functional approach from a set of 48 DFAs. By demonstrating transferability to diverse synthesized compounds, our recommender potentially addresses the accuracy versus scope dilemma broadly encountered in computational chemistry.

Influence of Number of Ligands and Point Group on the Electronic Structure of Co2+ Aqua-Complexes

The nucleation process of zeolitic imidazolate frameworks (ZIFs) is, to date, not yet completely understood, making the search for tailored materials very difficult. Recently, it has been shown that, during the formation process, the symmetry of the precursors is reduced by ligand elimination and substitution reactions. The octahedral precursors with simple ligands, such as water, methanol, and/or NO₃^-, are transformed to five- and finally four-coordinated complexes with imidazole ligands. This reduction of symmetry, caused both by the changing coordination environment and distortions from the perfect symmetry leading to another point group, will have a large influence on the electronic structure and more specifically on the d-orbital splitting. This, in turn, will affect the d-d electronic excitations, which can be followed using UV-vis spectroscopy and which can help to unravel the formation process. In this work, we systematically investigate how the lowering of the number of ligands affects the symmetry and thus the geometry and electronic structure of Co²⁺ complexes with six, five, and four aqua ligands. Therefore, we first resort to qualitative techniques, such as crystal field theory (CFT) and ligand field theory (LFT), which reveal that the orbital splitting is characteristic for the number of ligands. However, as these techniques are not capable of providing quantitative results without the use of experimental data as input, we perform various computational calculations. Both average of configuration (AOC) and unrestricted density functional theory (UDFT) are thoroughly investigated, and we will determine which technique is the best suited to properly describe the ground state of these systems. To investigate the dependency on the d-orbital occupation, we also investigated V²⁺, Mn²⁺, and Ni²⁺ hexa-aqua-complexes and compared them to the Co systems.

Spin-State Splittings in 3d Transition-Metal Complexes Revisited: Benchmarking Approximate Methods for Adiabatic Spin-State Energy Differences in Fe(II) Complexes

The CASPT2+δMRCI composite approach reported in a companion paper has been extended and used to provide high-quality reference data for a series of adiabatic spin gaps (defined as ΔE = E_quintet - E_singlet) of [Fe^IIL₆]²⁺ complexes (L = CNH, CO, NCH, NH₃, H₂O), either at nonrelativistic level or including scalar relativistic effects. These highly accurate data have been used to evaluate the performance of various more approximate methods. Coupled-cluster theory with singles, doubles, and perturbative triples, CCSD(T), is found to agree well with the new reference data for Werner-type complexes but exhibits larger underestimates by up to 70 kJ/mol for the π-acceptor ligands, due to appreciable static correlation in the low-spin states of these systems. Widely used domain-based local CCSD(T) calculations, DLPNO-CCSD(T), are shown to depend very sensitively on the cutoff values used to construct the localized domains, and standard values are not sufficient. A large number of density functional approximations have been evaluated against the new reference data. The B2PLYP double hybrid gives the smallest deviations, but several functionals from different rungs of the usual ladder hierarchy give mean absolute deviations below 20 kJ/mol. This includes the B97-D semilocal functional, the PBE0* global hybrid with 15% exact-exchange admixture, as well as the local hybrids LH07s-SVWN and LH07t-SVWN. Several further functionals achieve mean absolute errors below 30 kJ/mol (M06L-D4, SSB-D, B97-1-D4, LC-ωPBE-D4, LH12ct-SsirPW92-D4, LH12ct-SsifPW92-D4, LH14t-calPBE-D4, LHJ-HFcal-D4, and several further double hybrids) and thereby also still overall outperform CCSD(T) or uncorrected CASPT2. While exact-exchange admixture is a crucial factor in favoring high-spin states, the present evaluations confirm that other aspects can be important as well. A number of the better-performing functionals underestimate the spin gaps for the π-acceptor ligands but overestimate them for L = NH₃, H₂O. In contrast to a previous suggestion, non-self-consistent density functional theory (DFT) computations on top of Hartree-Fock orbitals are not a promising path to produce accurate spin gaps in such complexes.

Cluster-in-Molecule Method Combined with the Domain-Based Local Pair Natural Orbital Approach for Electron Correlation Calculations of Periodic Systems

The cluster-in-molecule (CIM) method was extended to systems with periodic boundary conditions (PBCs) in a previous work (PBC-CIM) [J. Chem. Theory Comput.2019, 15, 2933], which is able to compute the electronic structures of periodic systems at second-order Møller-Plesset perturbation theory (MP2) and coupled cluster singles and doubles (CCSD) levels. However, the high computational costs of CCSD with respect to the size of clusters limit the usage of PBC-CIM to crystals with small or medium unit cells. In this work, we further develop the PBC-CIM method by employing the domain-based local pair natural orbital (DLPNO) methods for the electron correlation calculations of clusters to reduce the computational costs. The combined approach allows CCSD with perturbative triples, denoted as CCSD(T), to be computationally available for accurate descriptions of periodic systems. The distant-pair correction is also implemented to improve the accuracy of PBC-CIM. As in the molecular cases, the distant pair correction significantly improves the accuracy of various PBC-CIM methods with few additional costs. The PBC-CIM-DLPNO-CCSD(T) approach has been applied to investigate the optimized lattice parameter of the cubic LiCl crystal and two adsorption problems (CO on the NaCl(100) surface and H₂O on the h-BN surface). The results show that the CIM-DLPNO-CCSD(T) method offers accurate and efficient descriptions for the studied systems. Another application to the cohesive energy of the acetic acid crystal reveals that large basis sets are necessary for reliable calculations on the cohesive energies of molecular crystals.

Exploiting Ligand Additivity for Transferable Machine Learning of Multireference Character across Known Transition Metal Complex Ligands

Accurate virtual high-throughput screening (VHTS) of transition metal complexes (TMCs) remains challenging due to the possibility of high multireference (MR) character that complicates property evaluation. We compute MR diagnostics for over 5,000 ligands present in previously synthesized octahedral mononuclear transition metal complexes in the Cambridge Structural Database (CSD). To accomplish this task, we introduce an iterative approach for consistent ligand charge assignment for ligands in the CSD. Across this set, we observe that the MR character correlates linearly with the inverse value of the averaged bond order over all bonds in the molecule. We then demonstrate that ligand additivity of the MR character holds in TMCs, which suggests that the TMC MR character can be inferred from the sum of the MR character of the ligands. Encouraged by this observation, we leverage ligand additivity and develop a ligand-derived machine learning representation to train neural networks to predict the MR character of TMCs from properties of the constituent ligands. This approach yields models with excellent performance and superior transferability to unseen ligand chemistry and compositions.

Reconciling Local Coupled Cluster with Multireference Approaches for Transition Metal Spin-State Energetics

Spin-state energetics of transition metal complexes remain one of the most challenging targets for electronic structure methods. Among single-reference wave function approaches, local correlation approximations to coupled cluster theory, most notably the domain-based local pair natural orbital (DLPNO) approach, hold the promise of bringing the accuracy of coupled cluster theory with single, double, and perturbative triple excitations, CCSD(T), to molecular systems of realistic size with acceptable computational cost. However, recent studies on spin-state energetics of iron-containing systems raised doubts about the ability of the DLPNO approach to adequately and systematically approximate energetics obtained by the reference-quality complete active space second-order perturbation theory with coupled-cluster semicore correlation, CASPT2/CC. Here, we revisit this problem using a diverse set of iron complexes and examine several aspects of the application of the DLPNO approach. We show that DLPNO-CCSD(T) can accurately reproduce both CASPT2/CC and canonical CCSD(T) results if two basic principles are followed. These include the consistent use of the improved iterative (T₁) versus the semicanonical perturbative triple corrections and, most importantly, a simple two-point extrapolation to the PNO space limit. The latter practically eliminates errors arising from the default truncation of electron-pair correlation spaces and should be viewed as standard practice in applications of the method to transition metal spin-state energetics. Our results show that reference-quality results can be readily achieved with DLPNO-CCSD(T) if these principles are followed. This is important also in view of the applicability of the method to larger single-reference systems and multinuclear clusters, whose treatment of dynamic correlation would be challenging for multireference-based approaches.

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.

FCIQMC-Tailored Distinguishable Cluster Approach: Open-Shell Systems

A recently proposed tailored approach based on the distinguishable cluster method and the stochastic FCI solver, FCIQMC [J. Chem. Theory Comput. 2020, 16, 5621], is extended to open-shell molecular systems. The method is employed to calculate spin gaps of various Fe(II) complexes, including a Fe(II) porphyrin model system. Both distinguishable cluster and fully relaxed CASSCF natural orbitals were used in this work as reference for the subsequent tailored distinguishable cluster calculations. The distinguishable cluster natural orbitals occupation numbers were also used as an aid to the selection of the active space. The effect of the active space sizes and of the explicit correlation correction (F12) onto the predicted spin gaps is investigated. The tailored distinguishable cluster with singles and doubles yields consistently more accurate results compared to the tailored coupled cluster with singles and doubles.

Improving Results by Improving Densities: Density-Corrected Density Functional Theory

Density functional theory (DFT) calculations have become widespread in both chemistry and materials, because they usually provide useful accuracy at much lower computational cost than wavefunction-based methods. All practical DFT calculations require an approximation to the unknown exchange-correlation energy, which is then used self-consistently in the Kohn-Sham scheme to produce an approximate energy from an approximate density. Density-corrected DFT is simply the study of the relative contributions to the total energy error. In the vast majority of DFT calculations, the error due to the approximate density is negligible. But with certain classes of functionals applied to certain classes of problems, the density error is sufficiently large as to contribute to the energy noticeably, and its removal leads to much better results. These problems include reaction barriers, torsional barriers involving π-conjugation, halogen bonds, radicals and anions, most stretched bonds, etc. In all such cases, use of a more accurate density significantly improves performance, and often the simple expedient of using the Hartree-Fock density is enough. This Perspective explains what DC-DFT is, where it is likely to improve results, and how DC-DFT can produce more accurate functionals. We also outline challenges and prospects for the field.

Ionization Energies and Redox Potentials of Hydrated Transition Metal Ions: Evaluation of Domain-Based Local Pair Natural Orbital Coupled Cluster Approaches

Hydrated transition metal ions are prototypical systems that can be used to model properties of transition metals in complex chemical environments. These seemingly simple systems present challenges for computational chemistry and are thus crucial in evaluations of quantum chemical methods for spin-state and redox energetics. In this work, we explore the applicability of the domain-based pair natural orbital implementation of coupled cluster (DLPNO-CC) theory to the calculation of ionization energies and redox potentials for hydrated ions of all first transition row (3d) metals in the 2+/3+ oxidation states, in connection with various solvation approaches. In terms of model definition, we investigate the construction of a minimally explicitly hydrated quantum cluster with a first and second hydration layer. We report on the convergence with respect to the coupled cluster expansion and the PNO space, as well as on the role of perturbative triple excitations. A recent implementation of the conductor-like polarizable continuum model (CPCM) for the DLPNO-CC approach is employed to determine self-consistent redox potentials at the coupled cluster level. Our results establish conditions for the convergence of DLPNO-CCSD(T) energetics and stress the absolute necessity to explicitly consider the second solvation sphere even when CPCM is used. The achievable accuracy for redox potentials of a practical DLPNO-based approach is, on average, 0.13 V. Furthermore, multilayer approaches that combine a higher-level DLPNO-CCSD(T) description of the first solvation sphere with a lower-level description of the second solvation layer are investigated. The present work establishes optimal and transferable methodological choices for employing DLPNO-based coupled cluster theory, the associated CPCM implementation, and cost-efficient multilayer derivatives of the approach for open-shell transition metal systems in complex environments.

CASPT2 molecular geometries of Fe(II) spin-crossover complexes

Using fully internally contracted (FIC)-CASPT2 analytical gradients, geometry optimizations of spin-crossover complexes are reported. This approach is tested on a series of Fe(II) complexes with different sizes, ranging from 13 to 61 atoms. A combination of active space and basis set choices are employed to investigate their role in determining reliable molecular geometries. The reported strategy demonstrates that a wave function-based level of theory can be used to optimize the geometries of metal complexes in reasonable times and enables one to treat the molecular geometry and electronic structure of the complexes using the same level of theory. For a series of smaller Fe(II) SCO complexes, strong field ligands in the LS state result in geometries with the largest differences between DFT and CASPT2; however, good agreement overall is observed between DFT and CASPT2. For the larger complexes, moderate sized basis sets yield geometries that compare well with DFT and available experimental data. We recommend using the (10e,12o) active space since convergence to a minimum structure was more efficient than with truncated active spaces despite having similar Fe-ligand bond distances.

Experimental and Computational Insight into the Mechanism of NO Binding to Ferric Microperoxidase. The Likely Role of Tautomerization to Account for the pH Dependence

According to the current paradigm, the metal-hydroxo bond in a six-coordinate porphyrin complex is believed to be significantly less reactive in ligand substitution than the analogous metal-aqua bond, due to a much higher strength of the former bond. Here, we report kinetic studies for nitric oxide (NO) binding to a heme-protein model, acetylated microperoxidase-11 (AcMP-11), that challenge this paradigm. In the studied pH range 7.4-12.6, ferric AcMP-11 exists in three acid-base forms, assigned in the literature as [(AcMP-11)FeIII(H2O)(HisH)] (1), [(AcMP-11)FeIII(OH)(HisH)] (2), and [(AcMP-11)FeIII(OH)(His-)] (3). From the pH dependence of the second-order rate constant for NO binding (kon), we determined individual rate constants characterizing forms 1-3, revealing only a ca. 10-fold decrease in the NO binding rate on going from 1 (kon(1) = 3.8 × 106 M-1 s-1) to 2 (kon(2) = 4.0 × 105 M-1 s-1) and the inertness of 3. These findings lead to the abandonment of the dissociatively activated mechanism, in which the reaction rate can be directly correlated with the Fe-OH bond energy, as the mechanistic explanation for the process with regard to 2. The reactivity of 2 is accounted for through the existence of a tautomeric equilibrium between the major [(AcMP-11)FeIII(OH)(HisH)] (2a) and minor [(AcMP-11)FeIII(H2O)(His-)] (2b) species, of which the second one is assigned as the NO binding target due to its labile Fe-OH2 bond. The proposed mechanism is further substantiated by quantum-chemical calculations, which confirmed both the significant labilization of the Fe-OH2 bond in the [(AcMP-11)FeIII(H2O)(His-)] tautomer and the feasibility of the tautomer formation, especially after introducing empirical corrections to the computed relative acidities of the H2O and HisH ligands based on the experimental pKa values. It is shown that the "effective lability" of the axial ligand (OH-/H2O) in 2 may be comparable to the lability of the H2O ligand in 1.

Implicit solvation in domain based pair natural orbital coupled cluster (DLPNO-CCSD) theory

A nearly linear scaling implementation of coupled-cluster with singles and doubles excitations (CCSD) can be achieved by means of the domain-based local pair natural orbital (DLPNO) method. The combination of DLPNO-CCSD with implicit solvation methods allows the calculation of accurate energies and chemical properties of solvated systems at an affordable computational cost. We have efficiently implemented different schemes within the conductor-like polarizable continuum model (C-PCM) for DLPNO-CCSD in the ORCA quantum chemistry suite. In our implementation, the overhead due to the additional solvent terms amounts to less than 5% of the time the equivalent gas phase job takes. Our results for organic neutrals and open-shell ions in water show that for most systems, adding solvation terms to the coupled-cluster amplitudes equations and to the energy leads to small changes in the total energy compared to only considering solvated orbitals and corrections to the reference energy. However, when the solute contains certain functional groups, such as carbonyl or nitrile groups, the changes in the energy are larger and estimated to be around 0.04 and 0.02 kcal/mol for each carbonyl and nitrile group in the solute, respectively. For solutes containing metals, the use of accurate CC/C-PCM schemes is crucial to account for correlation solvation effects. Simultaneously, we have calculated the electrostatic component of the solvation energy for neutrals and ions in water for the different DLPNO-CCSD/C-PCM schemes. We observe negligible changes in the deviation between DLPNO-CCSD and canonical-CCSD data. Here, DLPNO-CCSD results outperform those for Hartree-Fock and density functional theory calculations.

Coupled Cluster Benchmark of New DFT and Local Correlation Methods: Mechanisms of Hydroarylation and Oxidative Coupling Catalyzed by Ru(II, III) Chloride Carbonyls

We have evaluated a set of accurate canonical CCSD(T) energies for stationary points on the potential energy surface for Ru(II, III) chloride carbonyl catalysis of two competing reactions between benzene and methyl acrylate (MA), namely, hydroarylation and oxidative coupling. We have then applied this set to evaluate the performance of localized orbital coupled-cluster methods and several new and common density functionals. We find that (a) DLPNO-CCSD(T) with TightPNO cutoffs is an acceptable substitute for full canonical CCSD(T) calculations on this system; (b) for the closed-shell systems where it could be applied, LNO-CCSD(T) with tight convergence criteria is very close to the canonical results; (c) the recent ωB97X-V and ωB97M-V functionals exhibit superior performance to commonly used DFT functionals in both closed- and open-shell calculations; (d) the revDSD-PBEP86 revision of the DSD-PBEP86 double hybrid represents an improvement over the original, even though transition metals were not involved in its parametrization; and (e) DSD-SCAN and DOD-SCAN show comparable efficiency. Most tested (meta)-GGA and hybrid density functionals perform better for open-shell than for closed-shell complexes; this is not the case for the double hybrids considered.

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.

Spin-free formulation of the multireference driven similarity renormalization group: A benchmark study of first-row diatomic molecules and spin-crossover energetics

We report a spin-free formulation of the multireference (MR) driven similarity renormalization group (DSRG) based on the ensemble normal ordering of Mukherjee and Kutzelnigg [J. Chem. Phys. 107, 432 (1997)]. This ensemble averages over all microstates of a given total spin quantum number, and therefore, it is invariant with respect to SU(2) transformations. As such, all equations may be reformulated in terms of spin-free quantities and they closely resemble those of spin-adapted closed-shell coupled cluster (CC) theory. The current implementation is used to assess the accuracy of various truncated MR-DSRG methods (perturbation theory up to third order and iterative methods with single and double excitations) in computing the constants of 33 first-row diatomic molecules. The accuracy trends for these first-row diatomics are consistent with our previous benchmark on a small subset of closed-shell diatomic molecules. We then present the first MR-DSRG application on transition-metal complexes by computing the spin splittings of the [Fe(H₂O)₆]²⁺ and [Fe(NH₃)₆]²⁺ molecules. A focal point analysis (FPA) shows that third-order perturbative corrections are essential to achieve reasonably converged energetics. The FPA based on the linearized MR-DSRG theory with one- and two-body operators and up to a quintuple-ζ basis set predicts the spin splittings of [Fe(H₂O)₆]²⁺ and [Fe(NH₃)₆]²⁺ to be -35.7 and -17.1 kcal mol^-1, respectively, showing good agreement with the results of local CC theory with singles, doubles, and perturbative triples.

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.

H2 activation by hydrogenase-inspired NiFe catalyst using frustrated Lewis pair: effect of buffer and halide ion in the heterolytic H-H bond cleavage

Hydrogen is a clean fuel alternative to fossil fuels, and it is vital to develop catalysts for its efficient activation and production. We investigate the reaction mechanism of H₂ activation in an aqueous solution by the recently developed NiFe complex (Ogo et al. Sci. Adv. 2020, 6, eaaz8181) using density functional theory (DFT) calculation. Our computational results showed that H₂ is activated using frustrated Lewis pair. That is, H₂ binds to the Fe site of the NiFe complex, acting as a Lewis acid, while the added buffer, acting as Lewis base, abstracts protons to form a hydride complex. Furthermore, the higher basicity in the proton abstraction reaction characterises reaction more exergonic and lowers the reaction barrier. In addition, in the proton abstraction by the water molecule, the reaction barrier was lowered when anion such as Cl⁻ is in the vicinity of the water. Understanding the chemical species that contribute to the catalytic process in cooperation with the metal catalyst at the atomic level should help to maximise the function of the catalyst.

Hydrogen is a clean fuel alternative to fossil fuels, and it is vital to develop catalysts for its efficient activation and production.

Interactions between large molecules pose a puzzle for reference quantum mechanical methods

Quantum-mechanical methods are used for understanding molecular interactions throughout the natural sciences. Quantum diffusion Monte Carlo (DMC) and coupled cluster with single, double, and perturbative triple excitations [CCSD(T)] are state-of-the-art trusted wavefunction methods that have been shown to yield accurate interaction energies for small organic molecules. These methods provide valuable reference information for widely-used semi-empirical and machine learning potentials, especially where experimental information is scarce. However, agreement for systems beyond small molecules is a crucial remaining milestone for cementing the benchmark accuracy of these methods. We show that CCSD(T) and DMC interaction energies are not consistent for a set of polarizable supramolecules. Whilst there is agreement for some of the complexes, in a few key systems disagreements of up to 8 kcal mol^-1 remain. These findings thus indicate that more caution is required when aiming at reproducible non-covalent interactions between extended molecules.

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.

Putting Density Functional Theory to the Test in Machine-Learning-Accelerated Materials Discovery

Accelerated discovery with machine learning (ML) has begun to provide the advances in efficiency needed to overcome the combinatorial challenge of computational materials design. Nevertheless, ML-accelerated discovery both inherits the biases of training data derived from density functional theory (DFT) and leads to many attempted calculations that are doomed to fail. Many compelling functional materials and catalytic processes involve strained chemical bonds, open-shell radicals and diradicals, or metal-organic bonds to open-shell transition-metal centers. Although promising targets, these materials present unique challenges for electronic structure methods and combinatorial challenges for their discovery. In this Perspective, we describe the advances needed in accuracy, efficiency, and approach beyond what is typical in conventional DFT-based ML workflows. These challenges have begun to be addressed through ML models trained to predict the results of multiple methods or the differences between them, enabling quantitative sensitivity analysis. For DFT to be trusted for a given data point in a high-throughput screen, it must pass a series of tests. ML models that predict the likelihood of calculation success and detect the presence of strong correlation will enable rapid diagnoses and adaptation strategies. These "decision engines" represent the first steps toward autonomous workflows that avoid the need for expert determination of the robustness of DFT-based materials discoveries.

Improved Spin-State Energy Differences of Fe(II) Molecular and Crystalline Complexes via the Hubbard U-Corrected Density

We recently showed that the DFT+U approach with a linear-response U yields adiabatic energy differences biased toward high spin [Mariano et al. J. Chem. Theory Comput. 2020, 16, 6755-6762]. Such bias is removed here by employing a density-corrected DFT approach where the PBE functional is evaluated on the Hubbard U-corrected density. The adiabatic energy differences of six Fe(II) molecular complexes computed using this approach, named PBE[U] here, are in excellent agreement with coupled cluster-corrected CASPT2 values for both weak- and strong-field ligands resulting in a mean absolute error (MAE) of 0.44 eV, smaller than that of the recently proposed Hartree-Fock density-corrected DFT (1.22 eV) and any other tested functional, including the best performer TPSSh (0.49 eV). We take advantage of the computational efficiency of this approach and compute the adiabatic energy differences of five molecular crystals using PBE[U] with periodic boundary conditions. The results show, again, an excellent agreement (MAE = 0.07 eV) with experimentally extracted values and a superior performance compared with the best performers M06-L (MAE = 0.08 eV) and TPSSh (MAE = 0.31 eV) computed on molecular fragments.