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

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.

Assessment of the applicability of DFT methods to [Cp*Rh]-catalyzed hydrogen evolution processes

The present computational study provides a benchmark of density functional theory (DFT) methods in describing hydrogen evolution processes catalyzed by [Cp*Rh]-containing organometallic complexes. A test set was composed of 26 elementary reactions featuring chemical transformations and bonding situations essential for the field, including the emerging concept of non-innocent Cp* behavior. Reference values were obtained from a highly accurate 3/4 complete basis set and 6/7 complete PNO space extrapolated DLPNO-CCSD(T) energies. The performance of lower-level extrapolation procedures was also assessed. We considered 84 density functionals (DF) (including 13 generalized gradient approximations (GGA), nine meta-GGAs, 33 hybrids, and 29 double-hybrids) and three composite methods (HF-3c, PBEh-3c, and r2SCAN-3c), combined with different types of dispersion corrections (D3(0), D3BJ, D4, and VV10). The most accurate approach is the PBE0-DH-D3BJ (MAD of 1.36 kcal mol-1) followed by TPSS0-D3BJ (MAD of 1.60 kcal mol-1). Low-cost r2SCAN-3c composite provides a less accurate but much faster alternative (MAD of 2.39 kcal mol-1). The widely used Minnesota-family M06-L, M06, and M06-2X DFs should be avoided (MADs of 3.70, 3.94, and 4.01 kcal mol-1, respectively).

Coordination Induced Spin State Transition Switches the Reactivity of Nickel (II) Porphyrin in Hydrogen Evolution Reaction

Electron spin plays a critical role in chemical processes, particularly in reactions involving metal complexes with unpaired electrons. However, more definitive state-to-state experiments are needed to better elucidate the role of electronic spin. Herein, we chose nickel (II) 5,10,15,20-tetrakis(pentafluorophenyl) porphyrin 1 as a catalyst, which allows switching from a low spin to a high spin state of Ni (II) center through an axial pyridine coordination, for electrocatalytic hydrogen evolution reaction (HER). When pyridine is present, we observed β-hydrogenation of porphyrin through electron transfer followed by proton transfer. In contrast, hydrogen evolution mainly occurs via the concerted proton-coupling electron transfer without pyridine coordination. Similar distinct spin-dependent selectivity was also observed in chemical reduction of 1 by CoCp2 with subsequent addition of pyridinium p-toluenesulfonate. Computational calculations using density functional theory demonstrated that the transition from low spin to high spin state enriches the ligand's electron density after one-electron reduction, leading to preferential protonation of β-periphery rather than meso-position or metal center.

Influence of graphene on the electronic and magnetic properties of an iron(III) porphyrin chloride complex

Although iron-based single atom catalysts are regarded as a promising alternative to precious metal catalysts, their precise electronic structures during catalysis still pose challenges for computational descriptions. A particularly urgent issue to be addressed is the influence of the environment on the electronic structure, and how to describe this accurately using computational methods. Here, we study an iron porphyrin chloride complex adsorbed on a graphene sheet using density functional theory calculations to probe how much the electronic structure is influenced by the presence of a graphene layer. Our results indicate that weak interactions due to van der Waals forces dominate between the porphyrin complex and graphene, and only a small amount of charge is transferred between the two entities. Furthermore, the interplay of the ligand field environment, strong p-d hybridization, and correlation effects within the complex are strongly involved in determining the spin state of the iron ion. By bridging molecular chemistry and solid state physics, this study provides first steps towards a joint analysis of the properties of iron-based catalysts from first principles.

Improving Predictions of Spin-Crossover Complex Properties through DFT Calculations with a Local Hybrid Functional

We conducted a study on the performance of the local hybrid exchange-correlation functional PBE0r for a set of 95 experimentally characterized iron spin-crossover (SCO) complexes [Vennelakanti, V.; J. Chem. Phys. 2023, 159, 024120]. The PBE0r functional is a variant of PBE0 where the exchange correction is restricted to on-site terms formulated on the basis of local orbitals. We determine the free parameters of the PBE0r functional against the experimental data and other hybrid functionals. With a Hartree-Fock (HF) exchange factor of 4%, the PBE0r functional accurately reproduces the electronic and free-energy trends predicted in prior DFT studies for these 95 complexes by using the B3LYP functional. Larger values of HF exchange stabilize high-spin states. The PBE0r-predicted bond lengths tend to exceed the experimental bond lengths, although bond lengths are less sensitive to HF exchange than in global hybrids. The predicted SCO transition temperatures T1/2 from PBE0r correlate moderately with the experimental transition temperatures, showing a slight improvement compared to the previous modB3LYP-predicted T1/2. This study suggests that the PBE0r functional is computationally cost-effective and offers the possibility of simulating larger complexes with accuracy comparable to global hybrid functionals, provided the HF-exchange parameter is carefully optimized.

State-of-the-art local correlation methods enable affordable gold standard quantum chemistry for up to hundreds of atoms

In this feature, we review the current capabilities of local electron correlation methods up to the coupled cluster model with single, double, and perturbative triple excitations [CCSD(T)], which is a gold standard in quantum chemistry. The main computational aspects of the local method types are assessed from the perspective of applications, but the focus is kept on how to achieve chemical accuracy (i.e., <1 kcal mol-1 uncertainty), as well as on the broad scope of chemical problems made accessible. The performance of state-of-the-art methods is also compared, including the most employed DLPNO and, in particular, our local natural orbital (LNO) CCSD(T) approach. The high accuracy and efficiency of the LNO method makes chemically accurate CCSD(T) computations accessible for molecules of hundreds of atoms with resources affordable to a broad computational community (days on a single CPU and 10-100 GB of memory). Recent developments in LNO-CCSD(T) enable systematic convergence and robust error estimates even for systems of complicated electronic structure or larger size (up to 1000 atoms). The predictive power of current local CCSD(T) methods, usually at about 1-2 order of magnitude higher cost than hybrid density functional theory (DFT), has become outstanding on the palette of computational chemistry applicable for molecules of practical interest. We also review more than 50 LNO-based and other advanced local-CCSD(T) applications for realistic, large systems across molecular interactions as well as main group, transition metal, bio-, and surface chemistry. The examples show that properly executed local-CCSD(T) can contribute to binding, reaction equilibrium, rate constants, etc. which are able to match measurements within the error estimates. These applications demonstrate that modern, open-access, and broadly affordable local methods, such as LNO-CCSD(T), already enable predictive computations and atomistic insight for complicated, real-life molecular processes in realistic environments.

Automated prediction of ground state spin for transition metal complexes

Exploiting crystallographic data repositories for large-scale quantum chemical computations requires the rapid and accurate extraction of the molecular structure, charge and spin from the crystallographic information file. Here, we develop a general approach to assign the ground state spin of transition metal complexes, in complement to our previous efforts on determining metal oxidation states and bond order within the cell2mol software. Starting from a database of 31k transition metal complexes extracted from the Cambridge Structural Database with cell2mol, we construct the TM-GSspin dataset, which contains 2063 mononuclear first row transition metal complexes and their computed ground state spins. TM-GSspin is highly diverse in terms of metals, metal oxidation states, coordination geometries, and coordination sphere compositions. Based on TM-GSspin, we identify correlations between structural and electronic features of the complexes and their ground state spins to develop a rule-based spin state assignment model. Leveraging this knowledge, we construct interpretable descriptors and build a statistical model achieving 98% cross-validated accuracy in predicting the ground state spin across the board. Our approach provides a practical way to determine the ground state spin of transition metal complexes directly from crystal structures without additional computations, thus enabling the automated use of crystallographic data for large-scale computations involving transition metal complexes.

Benchmarking Periodic Density Functional Theory Calculations for Spin-State Energies in Spin-Crossover Systems

Spin energetics is one of the biggest challenges associated with energy calculations for electronic structure methods. The energy differences of the spin states in spin-crossover compounds are very small, making them one of the most difficult systems to calculate. Few methods provide accurate results for calculating these energy differences. In addition, studies have usually focused on calculating energetics of single molecules, while spin-crossover properties are usually experimentally studied in the solid phase. In this paper, we have used periodic boundary conditions employing methods based on density functional theory to calculate the high- and low-spin energy differences for a test case of 20 extended systems. Compounds with different metals and ligands have been selected, and the results indicate that a semiquantitative description of the energy differences can be obtained with the combination of geometry optimization using the PBE functional including many-body dispersion approach and the use of meta-GGA functionals, such as r2SCAN but especially KTBM24, for the energy calculation. Other hybrid functionals, such as TPSSh, give generally good results, but the calculation of the exact exchange with periodic boundary conditions involves a huge increase in computer time and computational resources. It makes the proposed nonhybrid functional approach (KTBM24//PBE+MB) a great advantage for the study of periodic systems.

Predicting spin states of iron porphyrins with DFT methods including crystal packing effects and thermodynamic corrections

Accurate computational treatment of spin states for transition metal complexes, exemplified by iron porphyrins, lies at the heart of quantum bioinorganic chemistry, but at the same time represents a great challenge for approximate density functional theory (DFT) methods, which are predominantly used. Here, the accuracy of DFT methods for spin-state splittings in iron porphyrin is assessed by probing the ability to correctly predict the ground states for six FeIII or FeII complexes experimentally characterized in solid state. For each case, molecular and periodic DFT calculations are employed to quantify the effect of porphyrin side substituents and the crystal packing effect (CPE) on the spin-state splitting. It is proposed to partition the total CPE into additive components, the direct and structural one, the importance of which is shown to significantly vary from case to case. By knowing the substituent effect, the CPE, and the Gibbs free energy thermodynamic correction from calculations, one can employ the experimental ground-state information in order to derive a quantitative constraint on the electronic energy difference for a simplified (porphin) model of the experimentally characterized metalloporphyrin. The constraints derived in such a way-in the form of single or double inequalities-are used to assess the accuracy of dispersion-corrected DFT methods for 6 spin-state splittings of [FeIII(P)(2-MeIm)2]+, [FeIII(P)(2-MeIm)]+, [FeII(P)(THF)2] and [FeII(P)] models (where P is porphin, 2-MeIm is 2-methylimidazole, THF is tetrahydrofuran). These data constitute the new benchmark set of spin states for crystalline iron porphyrins (SSCIP6). The highest accuracy is obtained in the case of double-hybrid functionals (B2PLYP-D3, DSD-PBEB95-D3), whereas hybrid functionals, especially those with reduced admixture of the exact exchange (B3LYP*-D3, TPSSh-D3), are found to considerably overstabilize the intermediate spin state, leading to incorrect ground-state prediction in FeIII porphyrins. The present approach, which can be generalized to other transition metal complexes, is not only useful in method benchmarking, but also sheds light on the interpretations of experimental data for metalloporphyrins, which are important models to understand the electronic properties of heme proteins.

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.

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.

Predicting valence tautomerism in diverse cobalt-dioxolene complexes: elucidation of the role of ligands and solvent

The ability of molecular switches to reversibly interconvert between different forms promises potential applications at the scale of single molecules up to bulk materials. One type of molecular switch comprises cobalt-dioxolene compounds that exhibit thermally-induced valence tautomerism (VT) interconversions between low spin Co(iii)-catecholate (LS-CoIII-cat) and high spin Co(ii)-semiquinonate (HS-CoII-sq) forms. Two families of these compounds have been investigated for decades but have generally been considered separately: neutral [Co(diox)(sq)(N2L)] and cationic [Co(diox)(N4L)]+ complexes (diox = generic dioxolene, N2L/N4L = bidentate/tetradentate N-donor ancillary ligand). Computational identification of promising new candidate compounds prior to experimental exploration is beneficial for environmental and cost considerations but requires a thorough understanding of the underlying thermochemical parameters that influence the switching. Herein, we report a robust approach for the analysis of both cobalt-dioxolene families, which involved a quantitative density functional theory-based study benchmarked with reliable quasi-experimental references. The best-performing M06L-D4/def2-TZVPP level of theory has subsequently been verified by the synthesis and experimental investigation of three new complexes, two of which exhibit thermally-induced VT, while the third remains in the LS-CoIII-cat form across all temperatures, in agreement with prediction. Valence tautomerism in solution is markedly solvent-dependent, but the origin of this has not been definitively established. We have extended our computational approach to elucidate the correlation of VT transition temperature with solvent stabilisation energy and change in dipole moment. This new understanding may inform the development of VT compounds for applications in soft materials including films, gels, and polymers.

Modeling Fe(II) Complexes Using Neural Networks

We report a Fe(II) data set of more than 23000 conformers in both low-spin (LS) and high-spin (HS) states. This data set was generated to develop a neural network model that is capable of predicting the energy and the energy splitting as a function of the conformation of a Fe(II) organometallic complex. In order to achieve this, we propose a type of scaled electronic embedding to cover the long-range interactions implicitly in our neural network describing the Fe(II) organometallic complexes. For the total energy prediction, the lowest MAE is 0.037 eV, while the lowest MAE of the splitting energy is 0.030 eV. Compared to baseline models, which only incorporate short-range interactions, our scaled electronic embeddings improve the accuracy by over 70% for the prediction of the total energy and the splitting energy. With regard to semiempirical methods, our proposed models reduce the MAE, with respect to these methods, by 2 orders of magnitude.

FeN4 Environments upon Reduction: A Computational Analysis of Spin States, Spectroscopic Properties, and Active Species

FeN4 motifs, found, for instance, in bioinorganic chemistry as heme-type cofactors, play a crucial role in man-made FeNC catalysts for the oxygen reduction reaction. Such single-atom catalysts are a potential alternative to platinum-based catalysts in fuel cells. Since FeNC catalysts are prepared via pyrolysis, the resulting materials are amorphous and contain side phases and impurities. Therefore, the geometric and electronic nature of the catalytically active FeN4 site remains to be clarified. To further understand the behavior of FeN4 centers in electrochemistry and their expected spectroscopic behavior upon reduction, we investigate two FeN4 environments (pyrrolic and pyridinic). These are represented by the model complexes [Fe(TPP)Cl] and [Fe(phen2N2)Cl], where TPP = tetraphenylporphyrin and phen = 1,10-phenanthroline. We predict their Mössbauer, UV-vis, and NRV spectral data using density functional theory as windows into their electronic structure differences. By varying the axial ligand, we further show how well small chemical changes in both complexes can be discerned. We find that the differences in ligand field strength in pyrrolic and pyridinic coordination result in different spin ground states, which in turn leads to distinct Mössbauer spectroscopic properties. As a result, pyrrolic nitrogen donors with a weaker ligand field are predicted to show more pronounced spectroscopic differences under in situ and operando conditions, while pyridinic nitrogen donors are expected to show less pronounced spectroscopic changes upon reduction and/or ligand loss. We therefore suggest that a weaker ligand field leads to better detectability of catalytic intermediates in in situ and operando experiments.

Multi-d-Occupancy as an Alternative Definition for the Double d-Shell Effect

Despite the prevalence of first-row transition metal-containing compounds in virtually all areas of chemistry, the accurate modeling of these systems is a known challenge for the theoretical chemistry community. Such a challenge is shown in a myriad of facets; among them are difficulties in defining ground-state multiplicities, disagreement in the results from methods considered highly accurate, and convergence problems in calculations for excited states. These problems cause a scarcity of reliable theoretical data for transition metal-containing systems. In this work, we explore the double d-shell effect that plagues and makes the application of multireference methods to this type of system difficult. We propose an alternative definition for this effect based on the mixing among d-occupancy configurations or the multi-d-occupancy character of the wave function. Moreover, we present a protocol able to include this effect in multireference calculations using an active space smaller than that usually used in the literature. A molybdenum-copper model system and its copper subsystem are used as example study cases, in particular, the molybdenum-copper charge transfer of the former and the electron affinity of the latter. We have shown that our alternative definition can be used to analyze their reference wave functions qualitatively. Based on this qualitative description, it is possible to optimize an active space without a second d-shell able to obtain relative energies accurately. Seeing the double d-shell effect through the lens of a multi-d-occupancy character, it is possible to correctly describe the wave function, improve the accuracy of the relative energies, and reduce the computational cost of multireference calculations. That way, we believe that this alternative definition has the potential to improve the modeling of first-row transition metal-containing compounds both for their ground and excited electronic structures.

Linear-Scaling Local Natural Orbital CCSD(T) Approach for Open-Shell Systems: Algorithms, Benchmarks, and Large-Scale Applications

The extension of the highly optimized local natural orbital (LNO) coupled cluster (CC) with single-, double-, and perturbative triple excitations [LNO-CCSD(T)] method is presented for high-spin open-shell molecules based on restricted open-shell references. The techniques enabling the outstanding efficiency of the closed-shell LNO-CCSD(T) variant are adopted, including the iteration- and redundancy-free second-order Møller-Plesset and (T) formulations as well as the integral-direct, memory- and disk use-economic, and OpenMP-parallel algorithms. For large molecules, the efficiency of our open-shell LNO-CCSD(T) method approaches that of its closed-shell parent method due to the application of restricted orbital sets for demanding integral transformations and a novel approximation for higher-order long-range spin-polarization effects. The accuracy of open-shell LNO-CCSD(T) is extensively tested for radicals and reactions thereof, ionization processes, as well as spin-state splittings, and transition-metal compounds. At the size range where the canonical CCSD(T) reference is accessible (up to 20-30 atoms), the average open-shell LNO-CCSD(T) correlation energies are found to be 99.9 to 99.95% accurate, which translates into average absolute deviations of a few tenths of kcal/mol in the investigated energy differences already with the default settings. For more extensive molecules, the local errors may grow, but they can be estimated and decreased via affordable systematic convergence studies. This enables the accurate modeling of large systems with complex electronic structures, as illustrated on open-shell organic radicals and transition-metal complexes of up to 179 atoms as well as on challenging biochemical systems, including up to 601 atoms and 11,000 basis functions. While the protein models involve difficulties for local approximations, such as the spin states of a bounded iron ion or an extremely delocalized singly occupied orbital, the corresponding single-node LNO-CCSD(T) computations were feasible in a matter of days with 10s to 100 GB of memory use. Therefore, the new LNO-CCSD(T) implementation enables highly accurate computations for open-shell systems of unprecedented size and complexity with widely accessible hardware.

Local Oxidation States in {FeNO}6-8 Porphyrins: Insights from DMRG/CASSCF-CASPT2 Calculations

A first DMRG/CASSCF-CASPT2 study of a series of paradigmatic {FeNO}6, {FeNO}7, and {FeNO}8 heme-nitrosyl complexes has led to substantial new insight as well as uncovered key shortcomings of the DFT approach. By virtue of its balanced treatment of static and dynamic correlation, the calculations have provided some of the most authoritative information available to date on the energetics of low- versus high-spin states of different classes of heme-nitrosyl complexes. Thus, the calculations indicate low doublet-quartet gaps of 1-4 kcal/mol for {FeNO}7 complexes and high singlet-triplet gaps of ≳20 kcal/mol for both {FeNO}6 and {FeNO}8 complexes. In contrast, DFT calculations yield widely divergent spin state gaps as a function of the exchange-correlation functional. DMRG-CASSCF calculations also help calibrate DFT spin densities for {FeNO}7 complexes, pointing to those obtained from classic pure functionals as the most accurate. The general picture appears to be that nearly all the spin density of Fe[P](NO) is localized on the Fe, while the axial ligand imidazole (ImH) in Fe[P](NO)(ImH) pushes a part of the spin density onto the NO moiety. An analysis of the DMRG-CASSCF wave function in terms of localized orbitals and of the resulting configuration state functions in terms of resonance forms with varying NO(π*) occupancies has allowed us to address the longstanding question of local oxidation states in heme-nitrosyl complexes. The analysis indicates NO(neutral) resonance forms [i.e., Fe(II)-NO0 and Fe(III)-NO0] as the major contributors to both {FeNO}6 and {FeNO}7 complexes. This finding is at variance with the common formulation of {FeNO}6 hemes as Fe(II)-NO+ species but is consonant with an Fe L-edge XAS analysis by Solomon and co-workers. For the {FeNO}8 complex {Fe[P](NO)}-, our analysis suggests a resonance hybrid description: Fe(I)-NO0 ↔ Fe(II)-NO-, in agreement with earlier DFT studies. Vibrational analyses of the compounds studied indicate an imperfect but fair correlation between the NO stretching frequency and NO(π*) occupancy, highlighting the usefulness of vibrational data as a preliminary indicator of the NO oxidation state.

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.

Artificial Neural Network-Based Density Functional Approach for Adiabatic Energy Differences in Transition Metal Complexes

During the past decades, approximate Kohn-Sham density functional theory schemes have garnered many successes in computational chemistry and physics, yet the performance in the prediction of spin state energetics is often unsatisfactory. By means of a machine learning approach, an enhanced exchange and correlation functional is developed to describe adiabatic energy differences in transition metal complexes. The functional is based on the computationally efficient revision of the regularized, strongly constrained, and appropriately normed functional and improved by an artificial neural network correction trained over a small data set of electronic densities, atomization energies, and/or spin state energetics. The training process, performed using a bioinspired nongradient-based approach adapted for this work from the particle swarm optimization, is analyzed and discussed extensively. The resulting machine learned meta-generalized gradient approximation functional is shown to outperform most known density functionals in the prediction of adiabatic energy differences for a diverse set of transition metal complexes with varying local coordinations and metal choices.

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.

Mechanism and origins of cobalt-catalyzed ligand-controlled regiodivergent C-H functionalization of aldehydes with enynes

The influence of the P-M-P bite angle in diphosphine ligands on selectivity has been observed in various catalytic reactions. A better understanding of the ligand bite angle concept is important for the rational design of efficient catalytic systems. In the present work, the mechanism of cobalt-catalyzed C-H functionalization of aldehydes with enynes and how the diphosphine ligands alter regioselectivity were investigated by density functional theory (DFT) calculations. The catalytic cycle is initiated by the oxidative cyclization of enynes rather than the oxidative addition of aldehydes. Regioselectivity arises from competing σ-bond metathesis and migratory insertion steps, in which the steric effects of diphosphine ligands are the dominant factors influencing the activation barriers. The calculations indicate that σ-bond metathesis is more challenging and its feasibility is highly dependent on the ligand bite angle. The improved mechanistic understanding will enable further design of transition-metal-catalyzed selective cyclization reactions.

Toward Benchmark-Quality Ab Initio Predictions for 3d Transition Metal Electrocatalysts: A Comparison of CCSD(T) and ph-AFQMC

Generating accurate ab initio ionization energies for transition metal complexes is an important step toward the accurate computational description of their electrocatalytic reactions. Benchmark-quality data is required for testing existing theoretical methods and developing new ones but is complicated to obtain for many transition metal compounds due to the potential presence of both strong dynamical and static electron correlation. In this regime, it is questionable whether the so-called gold standard, coupled cluster with singles, doubles, and perturbative triples (CCSD(T)), provides the desired level of accuracy─roughly 1-3 kcal/mol. In this work, we compiled a test set of 28 3d metal-containing molecules relevant to homogeneous electrocatalysis (termed 3dTMV) and computed their vertical ionization energies (ionization potentials) with CCSD(T) and phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) in the def2-SVP basis set. A substantial effort has been made to converge away the phaseless bias in the ph-AFQMC reference values. We assess a wide variety of multireference diagnostics and find that spin-symmetry breaking of the CCSD wave function and the PBE0 density functional correlate well with our analysis of multiconfigurational wave functions. We propose quantitative criteria based on symmetry breaking to delineate correlation regimes inside of which appropriately performed CCSD(T) can produce mean absolute deviations from the ph-AFQMC reference values of roughly 2 kcal/mol or less and outside of which CCSD(T) is expected to fail. We also present a preliminary assessment of density functional theory (DFT) functionals on the 3dTMV set.

Unraveling the Spin-State Energetics of FeN4 Complexes with Ab Initio Methods

A systematic analysis was conducted to explore the spin-state energetics of a series of 19 FeN4 complexes. The performance of a large number of multireference methods was assessed, highlighting the significant challenges associated with accurately describing the spin-state energetics of FeN4 complexes. Most multireference methods were found to be susceptible to errors originating from the reference CASSCF wavefunction, leading to an overstabilization of high-spin states. Nonetheless, a few multireference methods, namely, CASPT2/CC, DSRG-MRPT3, and LDSRG(2), demonstrated promising performance compared to the benchmark CCSD(T) method. Furthermore, our study revealed that FeN4 complexes having a quintet ground state are exceedingly rare. Accordingly, only one specific model (Fe(L2)) and one synthesized complex (Fe(OTBP)) have the quintet ground state among the studied complexes. This scarcity of quintet FeN4 complexes highlights the unique nature of these systems and raises intriguing questions regarding the factors influencing spin states, such as the size of the macrocycle cavity, the introduction of substituents, or the induction of out-of-plane deformation.

Insights into the deviation from piecewise linearity in transition metal complexes from supervised machine learning models

Virtual high-throughput screening (VHTS) and machine learning (ML) with density functional theory (DFT) suffer from inaccuracies from the underlying density functional approximation (DFA). Many of these inaccuracies can be traced to the lack of derivative discontinuity that leads to a curvature in the energy with electron addition or removal. Over a dataset of nearly one thousand transition metal complexes typical of VHTS applications, we computed and analyzed the average curvature (i.e., deviation from piecewise linearity) for 23 density functional approximations spanning multiple rungs of "Jacob's ladder". While we observe the expected dependence of the curvatures on Hartree-Fock exchange, we note limited correlation of curvature values between different rungs of "Jacob's ladder". We train ML models (i.e., artificial neural networks or ANNs) to predict the curvature and the associated frontier orbital energies for each of these 23 functionals and then interpret differences in curvature among the different DFAs through analysis of the ML models. Notably, we observe spin to play a much more important role in determining the curvature of range-separated and double hybrids in comparison to semi-local functionals, explaining why curvature values are weakly correlated between these and other families of functionals. Over a space of 187.2k hypothetical compounds, we use our ANNs to pinpoint DFAs for which representative transition metal complexes have near-zero curvature with low uncertainty, demonstrating an approach to accelerate screening of complexes with targeted optical gaps.

Mechanism of Nitrogen-Carbon Bond Formation from Iron(IV) Disilylhydrazido Intermediates during N2 Reduction

We recently reported a reaction sequence that activates C-H bonds in simple arenes as well as the N-N triple bond in N₂, delivering the aryl group to N₂ to form a new N-C bond (Nature 2020, 584, 221). This enables the transformation of abundant feedstocks (arenes and N₂) into N-containing organic compounds. The key N-C bond forming step occurs upon partial silylation of N₂. However, the pathway through which reduction, silylation, and migration occurred was unknown. Here, we describe synthetic, structural, magnetic, spectroscopic, kinetic, and computational studies that elucidate the steps of this transformation. N₂ must be silylated twice at the distal N atom before aryl migration can occur, and sequential silyl radical and silyl cation addition is a kinetically competent pathway to a formally iron(IV)-NN(SiMe₃)₂ intermediate that can be isolated at low temperature. Kinetic studies show its first-order conversion to the migrated product, and DFT calculations indicate a concerted transition state for migration. The electronic structure of the formally iron(IV) intermediate is examined using DFT and CASSCF calculations, which reveal contributions from iron(II) and iron(III) resonance forms with oxidized NNSi₂ ligands. The depletion of electron density from the Fe-coordinated N atom makes it electrophilic enough to accept the incoming aryl group. This new pathway for the N-C bond formation offers a method for functionalizing N₂ using organometallic chemistry.

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.

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).

Redox-active ligands as a challenge for electronic structure methods

To improve the catalytic activity of 3d transition metal catalysts, redox-active ligands are a promising tool. These ligands influence the oxidation state of the metal center as well as the ground spin-state and make the experimental determination of both properties challenging. Therefore, first-principles calculations, in particular employing density functional theory with a proper choice of exchange-correlation (xc) functional, are crucial. Common xc functionals were tested on a simple class of metal complexes: homoleptic, octahedral tris(diimine) iron(II) complexes. The spin-state energy splittings for most of these complexes showed the expected linear dependence on the amount of exact exchange included in the xc functionals. Even though varying redox-activity affects the electronic structure of the complexes considerably, the sensitivity of the spin-state energetics to the exact exchange admixture is surprisingly small. For iron(II) complexes with highly redox-active ligands and for a broad range of ligands in the reduced tris(diimine) iron(I) complexes, self-consistent field convergence to local minima was observed, which differ from the global minimum in the redox state of the ligand. This may also result in convergence to a molecular structure that corresponds to an energetically higher-lying local minimum. One criterion to detect such behavior is a change in the sign of the slope for the dependence of the spin-state energy splittings on the amount of exact exchange. We discuss possible protocols for dealing with such artifacts in cases in which a large number of calculations makes checking by hand unfeasible.

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+.

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.

Accurate computed singlet-triplet energy differences for cobalt systems: implication for two-state reactivity

Accurate singlet-triplet energy differences for cobalt and rhodium complexes were calculated by using several wave function methods, such as MRCISD, CASPT2, CCSD(T) and BCCD(T). Relaxed energy differences were obtained by considering the singlet and triplet complexes, each at the minimum of their potential energy surfaces. Active spaces for multireference calculations were carefully checked to provide accurate results. The considered systems are built by increasing progressively the first coordination sphere around the metal. We included in our set two CpCoX complexes (Cp = cyclopentadienyl, X = alkenyl ligand), which have been suggested as intermediates in cycloaddition reactions. Indeed, cobalt systems have been used for more than a decade as active species in this kind of transformations, for which a two-state reactivity has been proposed. Most of the considered systems display a triplet ground state. However, in the case of a reaction intermediate, while a triplet ground state was predicted on the basis of Density Functional Theory results, our calculations suggest a singlet ground state. This stems from the competition between the exchange term (stabilising the triplet) and the accessibility of an intramolecular coordination (stabilising the singlet). This finding has an impact on the general mechanism of the cycloaddition reaction. Analogous rhodium systems were also studied and, as expected, they have a larger tendency to electron pairing than cobalt species.

Adiabatic models for the quantum dynamics of surface scattering with lattice effects

In this contribution, we review models for the lattice effects in quantum dynamics calculations on surface scattering, which is important to modeling heterogeneous catalysis for achieving an interpretation of experimental measurements. Unlike dynamics models for reactions in the gas phase, those for heterogeneous reactions have to include the effects of the surface. For manageable computational costs in calculations, the effects of static surface (SS) are firstly modeled as this is simply and easily implemented. Then, the SS model has to be improved to include the effects of the flexible surface, that is the lattice effects. To do this, various surface models have been designed where the coordinates of the surface atoms are introduced in the Hamiltonian operator, especially those of the top surface atom. Based on this model Hamiltonian operator, extensive multi-dimension quantum dynamics calculations can be performed to recover the lattice effects. Here, we first review an overview of the techniques in constructing the Hamiltonian operator, which is a sum of the kinetic energy operator (KEO) and potential energy surface (PES). Since the PES containing the coordinates of the surface atoms in a cell is still expensive, the SS model is often accepted. We consider a mathematical model, called the coupled harmonic oscillator (CHO) model, to introduce the concepts of adiabatic and diabatic representations for separating the molecule and surface. Under the adiabatic model, we further introduce the expansion model where the potential function is Taylor expanded around the optimized geometry of the surface. By an expansion model truncated at the first and second order, various coupling surface models between the molecule and surface are derived. Moreover, by further and deeply understanding the adiabatic representation, an effective Hamiltonian operator is obtained by optimizing the total wave function in factorized form. By this factorized form of wave function and effective Hamiltonian operator, the geometry phase of the surface wave function is theoretically found. This theoretical prediction may be measured by carefully designing experiments. Finally, discussions on the adiabatic representation, the PES construction, and possibility of the classical-dynamics solutions are given. Based on these discussions, a simple outlook on the dynamics of photocatalytics is finally given.

A combined first- and second-order optimization method for improving convergence of Hartree–Fock and Kohn–Sham calculations

In this work, we investigate the optimization of Hartree–Fock (HF) orbitals with our recently proposed combined first- and second-order (SO-SCI) method, which was originally developed for multi-configuration self-consistent field (MCSCF) and complete active space SCF (CASSCF) calculations. In MCSCF/CASSCF, it unites a second-order optimization of the active orbitals with a Fock-based first-order treatment of the remaining closed-virtual orbital rotations. In the case of the single-determinant wavefunctions, the active space is replaced by a preselected “second-order domain,” and all rotations involving orbitals in this subspace are treated at second-order. The method has been implemented for spin-restricted and spin-unrestricted Hartree–Fock (RHF, UHF), configuration-averaged Hartree–Fock (CAHF), as well as Kohn–Sham (KS) density functional theory (RKS, UKS). For each of these cases, various choices of the second-order domain have been tested, and appropriate defaults are proposed. The performance of the method is demonstrated for several transition metal complexes. It is shown that the SO-SCI optimization provides faster and more robust convergence than the standard SCF procedure but requires, in many cases, even less computation time. In difficult cases, the SO-SCI method not only speeds up convergence but also avoids convergence to saddle-points. Furthermore, it helps to find spin-symmetry broken solutions in the cases of UHF or UKS. In the case of CAHF, convergence can also be significantly improved as compared to a previous SCF implementation. This is particularly important for multi-center cases with two or more equal heavy atoms. The performance is demonstrated for various two-center complexes with different lanthanide atoms.

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.

Comprehensive Theoretical View of the [Cu2O2] Side-on-Peroxo-/Bis-μ-Oxo Equilibria

Coupled binuclear copper (CBC) sites are employed by many metalloenzymes to catalyze a broad set of biochemical transformations. Typically, the CBC catalytic sites are activated by the O2 molecule to form various [Cu2O2] reactive species. This has also inspired synthesis and development of various biomimetic inorganic complexes featuring the CBC core. From theoretical perspective, the [Cu2O2] reactivity often hinges on the side-on-peroxo-dicopper(II) (P) vs. bis-μ-oxo-dicopper(III) (O) isomerism - an equilibrium that has become almost iconic in theoretical bioinorganic chemistry. Herein, we present a comprehensive calibration and evaluation of the performance of various composite computational protocols available in contemporary computational chemistry, involving coupled-cluster and multi-reference (relativistic) wave function methods, popular density functionals and solvation models. Starting with the well-studied reference [Cu2O2(NH3)6]2+ system, we compared the performance of electronic structure methods and discussed the relativistic effects. This allowed us to select several 'calibrated' DFT functionals that can be conveniently employed to study ten experimentally well-characterized [Cu2O2] inorganic systems. We mostly correctly predicted the lowest-energy structures (P vs. O) of the studied systems. In addition, we present calibration of the used electronic structure methods for prediction of the spectroscopic features of the [Cu2O2] core.

BuRNN: Buffer Region Neural Network Approach for Polarizable-Embedding Neural Network/Molecular Mechanics Simulations

Hybrid quantum mechanics/molecular mechanics (QM/MM) simulations have advanced the field of computational chemistry tremendously. However, they require the partitioning of a system into two different regions that are treated at different levels of theory, which can cause artifacts at the interface. Furthermore, they are still limited by high computational costs of quantum chemical calculations. In this work, we develop the buffer region neural network (BuRNN), an alternative approach to existing QM/MM schemes, which introduces a buffer region that experiences full electronic polarization by the inner QM region to minimize artifacts. The interactions between the QM and the buffer region are described by deep neural networks (NNs), which leads to the high computational efficiency of this hybrid NN/MM scheme while retaining quantum chemical accuracy. We demonstrate the BuRNN approach by performing NN/MM simulations of the hexa-aqua iron complex.

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.

Hidden ordered structure in the archetypical Fe(pyrazine)[Pt(CN)4] spin-crossover porous coordination compound

The actual low-spin state of Fe(pyrazine)Pt(CN)4 implies an ordered arrangement of the pyrazine rings, which significantly affects the spin transition.

Catalytic properties of the ferryl ion in the solid state: a computational review

This review summarises the last findings in the emerging field of heterogeneous catalytic oxidation of light alkanes by ferryl species supported on solid-state systems such as the conversion of methane into methanol by FeO-MOF74.

A computational protocol for the calculation of the standard reduction potential of iron complexes: application to Fe2+/3+-Aβ model systems relevant to Alzheimer's disease

Iron complexes play a key role in several biological processes, and they are also related to the development of neurological disorders, such as Alzheimer's and Parkinson's diseases. One of the main properties involved in these processes is the standard reduction potential (SRP) of iron complexes. However, the calculation of this property is challenging, mainly due to problems in the electronic structure description, solvent effects and the thermodynamic cycles used for its calculation. In this work, we proposed a computational protocol for the calculation of SRPs of iron complexes by evaluating a wide range of density functionals for the electronic structure description, two implicit solvent models with varying radii and two thermodynamic cycles. Results show that the M06L density functional in combination with the SMD solvation model and the isodesmic method provides good results compared with SRP experimental values for a set of iron complexes. Finally, this protocol was applied to three Fe^2+/3+-Aβ model systems involved in the development of Alzheimer's disease and the obtained SRP values are in good agreement with those reported previously by means of MP2 calculations.

Iron complexes play a key role in the development of neurological disorders, such as Alzheimer's disease. We provide a computational protocol based on DFT for the calculation of standard reduction potentials of iron complexes relevant to Alzheimer's disease.

A DMRG/CASPT2 Investigation of Metallocorroles: Quantifying Ligand Noninnocence in Archetypal 3d and 4d Element Derivatives

Hybrid density functional theory (B3LYP) and density matrix renormalization group (DMRG) theory have been used to quantitatively compare the degree of ligand noninnocence (corrole radical character) in seven archetypal metallocorroles. The seven complexes, in decreasing order of corrole noninnocent character, are Mn[Cor]Cl > Fe[Cor]Cl > Fe[Cor](NO) > Mo[Cor]Cl2 > Ru[Cor](NO) ≈ Mn[Cor]Ph ≈ Fe[Cor]Ph ≈ 0, where [Cor] refers to the unsubstituted corrolato ligand. DMRG-based second-order perturbation theory calculations have also yielded detailed excited-state energetics data on the compounds, shedding light on periodic trends involving middle transition elements. Thus, whereas the ground state of Fe[Cor](NO) (S = 0) is best described as a locally S = 1/2 {FeNO}7 unit antiferromagnetically coupled to a corrole A' radical, the calculations confirm that Ru[Cor](NO) may be described as simply {RuNO}6-Cor3-, that is, having an innocent corrole macrocycle. Furthermore, whereas the ferromagnetically coupled S = 1{FeNO}7-Cor•2- state of Fe[Cor](NO) is only ∼17.5 kcal/mol higher than the S = 0 ground state, the analogous triplet state of Ru[Cor](NO) is higher by a far larger margin (37.4 kcal/mol) relative to the ground state. In the same vein, Mo[Cor]Cl2 exhibits an adiabatic doublet-quartet gap of 36.1 kcal/mol. The large energy gaps associated with metal-ligand spin coupling in Ru[Cor](NO) and Mo[Cor]Cl2 reflect the much greater covalent character of 4d-π interactions relative to analogous interactions involving 3d orbitals. As far as excited-state energetics is concerned, DMRG-CASPT2 calculations provide moderate validation for hybrid density functional theory (B3LYP) for qualitative purposes, but underscore the possibility of large errors (>10 kcal/mol) in interstate energy differences.

Stereodivergent atom-transfer radical cyclization by engineered cytochromes P450

Naturally occurring enzymes can be a source of unnatural reactivity that can be molded by directed evolution to generate efficient biocatalysts with valuable activities. Owing to the lack of exploitable stereocontrol elements in synthetic systems, steering the absolute and relative stereochemistry of free-radical processes is notoriously difficult in asymmetric catalysis. Inspired by the innate redox properties of first-row transition-metal cofactors, we repurposed cytochromes P450 to catalyze stereoselective atom-transfer radical cyclization. A set of metalloenzymes was engineered to impose substantial stereocontrol over the radical addition step and the halogen rebound step in these unnatural processes, allowing enantio- and diastereodivergent radical catalysis. This evolvable metalloenzyme platform represents a promising solution to tame fleeting radical intermediates for asymmetric catalysis.

Accurate calculation of spin-state energy gaps in Fe(III) spin-crossover systems using density functional methods

Fe(III) complexes are receiving ever-increasing attention as spin crossover (SCO) systems because they are usually air stable, as opposed to Fe(II) complexes, which are prone to oxidation. Here, we present the first systematic study exclusively devoted to assess the accuracy of several exchange-correlation functionals when it comes to predicting the energy gap between the high-spin (S = 5/2) and the low-spin (S = 1/2) states of Fe(III) complexes. Using a dataset of 24 different Fe(III) hexacoordinated complexes, it is demonstrated that the B3LYP* functional is an excellent choice not only for predicting spin-state energy gaps for Fe(III) complexes undergoing spin-transitions but also for discriminating Fe(III) complexes that are either low- or high-spin in the whole range of temperatures. Our benchmark study has led to the identification of a very versatile Fe(III) compound whose SCO properties can be engineered upon changing a single axial ligand. Overall, this work demonstrates that B3LYP* is a reliable functional for screening new spin-crossover systems with tailored properties.

Computational Study of CO2 Reduction Catalyzed by Iron(I) Complex at Different Spin States: Cooperativity of Hydrogen Bonding and Auxiliary Group Effect

To explore alternative approaches to the CO₂ reduction to formate and provide an insight into the spin state effect on the CO₂ reduction, we theoretically designed a kind of low-valence iron(I) model complex, whose doublet, quartet, and sextet states are denoted as ² Fe(I), ⁴ Fe(I), and ⁶ Fe(I), respectively. This complex is featured with an iron(I) center, which bonds to a 1,2-ethanediamine (en) and a 2-hydroxy-biphenyl group. Reaction mechanisms for the CO₂ reduction to formate catalyzed by this iron(I) model complex were explored using density functional theory (DFT) computations. Studies showed that the univalent iron(I) compound can efficiently fix and activate a CO₂ molecule, whereas its oxidized forms with trivalent iron(III) or bivalent iron(II) cannot activate CO₂. For the iron(I) compound, it was found that the lowest spin state ² Fe(I) is the most favorable for the CO₂ reduction as the reactions barriers involving ² Fe(I), ⁴ Fe(I), and ⁶ Fe(I) are 25.6, 37.2, and 35.9 kcal/mol, respectively. Yet, a photosensitizer-free visible-light-mediated high-low spin shift from ⁴ Fe(I) and ⁶ Fe(I) to ² Fe(I) is likely through the reverse intersystem crossing (RIC) because the ⁴ Fe(I) and ⁶ Fe(I) compounds have strong absorption in the visible-light range. Notably, the synergistic interaction between the hydrogen bonding from the auxiliary hydroxyl group in the 2-hydroxy-biphenyl moiety to CO₂ and an intermediate five-membered ring promotes the proton transfer, leading to the formation of the -COOH moiety from CO₂ and the Fe-O bond. With the addition of H₂, one H₂ molecule is split by the Fe-O bond and thus serves as H atom sources for both the CO₂ reduction and the recovery of the auxiliary hydroxyl group. The present theoretical study provides a novel solution for the challenging CO₂ reduction, which calls for further experimental verifications.

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.