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.

Modeling size and edge functionalization of MXene-based quantum dots and their effect on electronic and magnetic properties

In the last six years, the synthesis of MXene-based quantum dots (MXQDs) has gained widespread attention. Due to the quantum confinement effect, it is possible to significantly improve their properties compared to 2D counterparts, such as higher chemical stability and better electronic and optical properties. However, despite the growing interest in their properties, much remains unexplored. One of the biggest challenges is to study in more detail the structure of quantum dots, in particular, their edge functionalization and its effect on their properties. In this paper, the structural stability and electronic and magnetic properties of Ti2CO2 QDs based on different lateral dimensions and edge functionalization (-O, -F, and -OH) are investigated using density functional theory. The study shows that the energy gap of Ti2CO2-O QDs decreases with increasing lateral size for both nonmagnetic (spin-unpolarized, close shell) and magnetic (spin-polarized, open shell) cases. Furthermore, the magnetic behavior of quantum dots was revealed by shrinking from 2D Ti2CO2 to 0D Ti2CO2 QDs with lateral dimensions below 1.4 nm. The binding energy confirms the stability of all three types of edge functionalization, while the most stable structure was observed under fully saturated edge oxygenation. Moreover, it was also found that the spin density distribution and the energy gap of Ti2CO2-X QDs (X = O, F, and OH) are both dependent on the type of atom saturation. Size and edge confinement modeling has been demonstrated to be an effective tool for tuning the electronic and magnetic properties of MXQDs. Moreover, the observed enhanced spin polarization together with tunable magnetic properties makes the ultrafine Ti2CO2-X QDs promising candidates for spintronic applications.

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.

Spin state and magnetic coupling in polynuclear Ni(II) complexes from density functional theory: is there an optimal amount of Fock exchange?

Reliable prediction of the ground-state spin and magnetic coupling constants in transition-metal complexes is a well-known challenge for density functional theory (DFT). One popular strategy for addressing this long-standing issue involves the modification of the fraction of Fock exchange in a hybrid functional. Here we explore the viability of this approach using three polynuclear metal-organic complexes based on a Ni4O4 cubane motif, having different ground state spin values (S = 0, 2, 4) owing to the use of different ligands. We systematically search for an optimum fraction of Fock exchange, across various global, range-separated, and double hybrid functionals. We find that for all functionals tested, at best there only exists a very narrow range of Fock exchange fractions which results in a correct prediction of the ground-state spin for all three complexes. The useful range is functional dependent, but general trends can be identified. Typically, at least two similar systems must be used in order to determine both an upper and lower limit of the optimal range. This is likely owing to conflicting demands of minimizing delocalization errors, which typically requires a higher percentage of Fock exchange, and addressing static correlation, which typically requires a lower one. Furthermore, we find that within the optimal range of Fock exchange, the sign and relative magnitude of Ni-Ni magnetic coupling constants are reasonably well reproduced, but there is still room for quantitative improvement in the prediction. Thus, the prediction of spin state and magnetic coupling in polynuclear complexes remains an ongoing challenge for DFT.

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.

Mean Value Ensemble Hubbard-U Correction for Spin-Crossover Molecules

High-throughput searches for spin-crossover molecules require Hubbard-U corrections to common density functional exchange-correlation (XC) approximations. However, the U_eff values obtained from linear response or based on previous studies overcorrect the spin-crossover energies. We demonstrate that employing a linearly mixed ensemble average spin state as the reference configuration for the linear response calculation of U_eff resolves this issue. Validation on a commonly used set of spin-crossover complexes shows that these ensemble U_eff values consistently are smaller than those calculated directly on a pure spin state, irrespective of whether that be low- or high-spin. Adiabatic crossover energies using this methodology for a generalized gradient approximation XC functional are closer to the expected target energy range than with conventional U_eff values. Based on the observation that the U_eff correction is similar for different complexes that share transition metals with the same oxidation state, we devise a set of recommended averaged U_eff values for high-throughput calculations.

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.

Iron-Catalyzed Olefin Metathesis: Recent Theoretical and Experimental Advances

The "metathesis reaction" is a straightforward and often metal-catalyzed chemical reaction that transforms two hydrocarbon molecules to two new hydrocarbons by exchange of molecular fragments. Alkane, alkene and alkyne metathesis have become an important tool in synthetic chemistry and have provided access to complex organic structures. Since the discovery of industrial olefin metathesis in the 1960s, many modifications have been reported; thus, increasing scope and improving reaction selectivity. Olefin metathesis catalysts based on high-valent group six elements or Ru(IV) have been developed and improved through ligand modifications. In addition, significant effort was invested to realize olefin metathesis with a non-toxic, bio-compatible and one of the most abundant elements in the earth's crust; namely, iron. First evidences suggest that low-valent Fe(II) complexes are active in olefin metathesis. Although the latter has not been unambiguously established, this review summarizes the key advances in the field and aims to guide through the challenges.

The role of the intermediate triplet state in iron-catalyzed multi-state C-H activation

Efficient activation and functionalization of the C-H bond under mild conditions are of a great interest in chemical synthesis. We investigate the previously proposed spin-accelerated activation of the C(sp²)-H bond by a Fe(II)-based catalyst to clarify the role of the intermediate triplet state in the reaction mechanism. High-level electronic structure calculations on a small model of a catalytic system utilizing the coupled cluster with the single, double, and perturbative triple excitations [CCSD(T)] are used to select the density functional for the full-size model. Our analysis indicates that the previously proposed two-state quintet-singlet reaction pathway is unlikely to be efficient due to a very weak spin-orbit coupling between these two spin states. We propose a more favorable multi-state quintet-triplet-singlet reaction pathway and discuss the importance of the intermediate triplet state. This triplet state facilitates a spin-accelerated reaction mechanism by strongly coupling to both quintet and singlet states. Our calculations show that the C-H bond activation through the proposed quintet-triplet-singlet reaction pathway is more thermodynamically favorable than the single-state quintet and two-state singlet-quintet mechanisms.

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.

EPR Spectroscopy of Cu(II) Complexes: Prediction of g-Tensors Using Double-Hybrid Density Functional Theory

Computational electron paramagnetic resonance (EPR) spectroscopy is an important field of applied quantum chemistry that contributes greatly to connecting spectroscopic observations with the fundamental description of electronic structure for open-shell molecules. However, not all EPR parameters can be predicted accurately and reliably for all chemical systems. Among transition metal ions, Cu(II) centers in inorganic chemistry and biology, and their associated EPR properties such as hyperfine coupling and g-tensors, pose exceptional difficulties for all levels of quantum chemistry. In the present work, we approach the problem of Cu(II) g-tensor calculations using double-hybrid density functional theory (DHDFT). Using a reference set of 18 structurally and spectroscopically characterized Cu(II) complexes, we evaluate a wide range of modern double-hybrid density functionals (DHDFs) that have not been applied previously to this problem. Our results suggest that the current generation of DHDFs consistently and systematically outperform other computational approaches. The B2GP-PLYP and PBE0-DH functionals are singled out as the best DHDFs on average for the prediction of Cu(II) g-tensors. The performance of the different functionals is discussed and suggestions are made for practical applications and future methodological developments.

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.

On the Use of Normalized Metrics for Density Sensitivity Analysis in DFT

In the past years, there has been a discussion about how the errors in density functional theory might be related to errors in the self-consistent densities obtained from different density functional approximations. This, in turn, brings up the discussion about the different ways in which we can measure such errors and develop metrics that assess the sensitivity of calculated energies to changes in the density. It is important to realize that there cannot be a unique metric in order to look at this density sensitivity, simultaneously needing size-extensive and size-intensive metrics. In this study, we report two metrics that are widely applicable to any density functional approximation. We also show how they can be used to classify different chemical systems of interest with respect to their sensitivity to small variations in the density.

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.

Exploring human porphobilinogen synthase metalloprotein by quantum biochemistry and evolutionary methods

Previous studies have shown the porphobilinogen synthase (PBGS) zinc-binding mechanism and its conservation among the living cells. However, the precise molecular interaction of zinc with the active center of the enzyme is unknown. In particular, quantum chemistry techniques within the density functional theory (DFT) framework have been the key methodology to describe metalloproteins, when one is looking for a compromise between accuracy and computational feasibility. Considering this, we used DFT-based models within the molecular fractionation with conjugate caps scheme to evaluate the binding energy features of zinc interacting with the human PBGS. Besides, phylogenetic and clustering analyses were successfully employed in extracting useful information from protein sequences to identify groups of conserved residues that build the ions-binding site. Our results also report a conservative assessment of the relevant amino acids, as well as the benchmark analysis of the calculation models used. The most relevant intermolecular interactions in Zn2+-PBGS are due to the amino acids CYS0122, CYS0124, CYS0132, ASP0169, SER0168, ARG0221, HIS0131, ASP0120, GLY0133, VAL0121, ARG0209, and ARG0174. Among these residues, we highlighted ASP0120, GLY0133, HIS0131, SER0168, and ARG0209 by co-occurring in all clusters generated by unsupervised clustering analysis. On the other hand, the triple cysteines at 2.5 Å from zinc (CYS0122, CYS0124, and CYS0132) have the highest energy attraction and are absent in the taxa Viridiplantae, Sar, Rhodophyta, and some Bacteria. Additionally, the performance of the DFT-based models shows that the processing time-dependence is more associated with the choice of the basis set than the exchange-correlation functional.

Zeolites at the Molecular Level: What Can Be Learned from Molecular Modeling

This review puts the development of molecular modeling methods in the context of their applications to zeolitic active sites. We attempt to highlight the utmost necessity of close cooperation between theory and experiment, resulting both in advances in computational methods and in progress in experimental techniques.

On the mechanism of the formation of alkyl-Ni(I) catalysts

The formation of catalytically active alkyl-Ni(i) complexes by comproportionation of diorgano-Ni(ii) precursors and Ni(0) species proceeds easily through triplet states by alkyl ligand exchange. The process involves inversion of the configuration at the carbon that is transferred.

Electronic and geometric structure effects on one-electron oxidation of first-row transition metals in the same ligand framework

Developing new transition metal catalysts requires understanding of how both metal and ligand properties determine reactivity. Since metal complexes bearing ligands of the Py5 family (2,6-bis-[(2-pyridyl)methyl]pyridine) have been employed in many fields in the past 20 years, we set out here to understand their redox properties by studying a series of base metal ions (M = Mn, Fe, Co, and Ni) within the Py5OH (pyridine-2,6-diylbis[di-(pyridin-2-yl)methanol]) variant. Both reduced (M^II) and the one-electron oxidized (M^III) species were carefully characterized using a combination of X-ray crystallography, X-ray absorption spectroscopy, cyclic voltammetry, and density-functional theory calculations. The observed metal-ligand interactions and electrochemical properties do not always follow consistent trends along the periodic table. We demonstrate that this observation cannot be explained by only considering orbital and geometric relaxation, and that spin multiplicity changes needed to be included into the DFT calculations to reproduce and understand these trends. In addition, exchange reactions of the sixth ligand coordinated to the metal, were analysed. Finally, by including published data of the extensively characterised Py5OMe (pyridine-2,6-diylbis[di-(pyridin-2-yl)methoxymethane])complexes, the special characteristics of the less common Py5OH ligand were extracted. This comparison highlights the non-innocent effect of the distal OH functionalization on the geometry, and consequently on the electronic structure of the metal complexes. Together, this gives a complete analysis of metal and ligand degrees of freedom for these base metal complexes, while also providing general insights into how to control electrochemical processes of transition metal complexes.

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

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

A Theoretical Assessment of Spin and Charge States in Binuclear Cobalt-Ruthenium Complexes: Implications for a Creutz-Taube Model Ion Separated by a C60-Derivative Bridging Ligand

We investigate the spin-state energetics and the role of ionic charges in the electronic configuration of binuclear complexes of the form [(NH₃)₅Co(py)-X-(py)Ru(NH₃)₅]^q+. In these compounds with q = 4-6, py = pyridine, and X = C≡C and C₆₀, the Co-Ru distance varies from ∼1.4 to ∼2.1 nm. We carry out a systematic electronic structure calculation using different exchange-correlation (xc) approaches within spin-density functional theory, which are largely employed to investigate the properties of a variety of coordination complexes. To evaluate the effects of spin states and type of spacer in the bridging ligand on the valence tautomerism between Co^2+/3+ and Ru^2+/3+, we examine in more detail the case of Creutz-Taube-type ions [(NH₃)₅Co(py)-X-(py)Ru(NH₃)₅]⁵⁺. Our analysis shows that the stabilization of low- and high-spin states critically depends on the total charge of the complex, type of X-bridged ligand, and employed xc approach to calculate the electron spin density. Importantly, the C₆₀-bridged group may result in a blockage of the valence tautomerism of the Creutz-Taube complex, inducing bistable charge configurations. Overall, our results also show that an adiabatic description in terms of the frontier molecular spin-orbitals for analyzing the distinct spin-charge states of these complexes may dramatically depend on the density-functional description.

Spin-state dependence of exchange-correlation holes

Applications of density-functional theory (DFT) in computational chemistry rely on an approximate exchange-correlation (xc) functional. However, existing approximations can fail dramatically for open-shell molecules, in particular for transition-metal complexes or radicals. Most importantly, predicting energy differences between different spin-states with approximate exchange-correlation functionals remains extremely challenging. Formally, it is known that the exact xc functional should be spin-state dependent, but none of the available approximations feature such an explicit spin-state dependence [C. R. Jacob and M. Reiher, Int. J. Quantum Chem., 2012, 112, 3661-3684]. Thus, to find novel approximations for the xc functional for open-shell systems, the development of spin-state dependent xc functionals appears to be a promising avenue. Here, we set out to shed light on the spin-state dependence of the xc functional by investigating the underlying xc holes, which we extract from configuration interaction calculations for model systems. We analyze the similarities and differences between the xc holes of the lowest-energy singlet and triplet states of the dihydrogen molecule, the helium atom, and the lithium dimer. To shed further light on the spin-state dependence of these xc holes we also discuss exact conditions that can be derived from the spin structure of the reduced two-electron density matrix. Altogether, our results suggest several possible routes towards the construction of explicitly spin-state dependent approximations for the xc functional.

A Review of Density Functional Models for the Description of Fe(II) Spin-Crossover Complexes

Spin-crossover (SCO) materials have for more than 30 years stood out for their vast application potential in memory, sensing and display devices. To reach magnetic multistability conditions, the high-spin (HS) and low-spin (LS) states have to be carefully balanced by ligand field stabilization and spin-pairing energies. Both effects could be effectively modelled by electronic structure theory, if the description would be accurate enough to describe these concurrent influences to within a few kJ/mol. Such a milestone would allow for the in silico-driven development of SCO complexes. However, so far, the ab initio simulation of such systems has been dominated by general gradient approximation density functional calculations. The latter can only provide the right answer for the wrong reasons, given that the LS states are grossly over-stabilized. In this contribution, we explore different venues for the parameterization of hybrid functionals. A fitting set is provided on the basis of explicitly correlated coupled cluster calculations, with single- and multi-dimensional fitting approaches being tested to selected classes of hybrid functionals (hybrid, range-separated, and local hybrid). Promising agreement to benchmark data is found for a rescaled PBE0 hybrid functional and a local version thereof, with a discussion of different atomic exchange factors.

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

In this work, a detailed study of spin-state splittings in three spin crossover model compounds with DLPNO-CCSD(T) is presented. The performance in comparison to canonical CCSD(T) is assessed in detail. It was found that spin-state splittings with chemical accuracy, compared to the canonical results, are achieved when the full iterative triples (T1) scheme and TightPNO settings are applied and relativistic effects are taken into account. Having established the level of accuracy that can be reached relative to the canonical results, we have undertaken a detailed basis set study in the second part of the study. The slow convergence of the results of correlated calculations with respect to basis set extension is particularly acute for spin-state splittings for reasons discussed in detail in this Article. In fact, for some of the studied systems, 5Z basis sets are necessary in order to come close to the basis set limit that is estimated here by basis set extrapolation. Finally, the results of the present work are compared to available literature. In general, acceptable agreement with previous CCSD(T) results is found, although notable deviations stemming from differences in methodology and basis sets are noted. It is noted that the published CASPT2 numbers are far away from the extrapolated CCSD(T) numbers. In addition, dynamic quantum Monte Carlo results differ by several tens of kcal/mol from the CCSD(T) numbers. A comparison to DFT results produced with a range of popular density functionals shows the expected scattering of results and showcases the difficulty of applying DFT to spin-state energies.

Transition Metal Spin-State Energetics by MC-PDFT with High Local Exchange

The energetics of the spin states of transition metal complexes have been explored with a variety of electronic structure methods, but the calculations require a compromise between accuracy and affordability. In this work, the spin splittings of several iron complexes are studied with multiconfiguration pair-density functional theory (MC-PDFT). The results are compared to previously published results obtained by complete active space second-order perturbation theory (CASPT2) and CASPT2 with coupled-cluster semicore correlation (CASPT2/CC). In contrast to CASPT2's systematic overstabilization of high-spin states with respect to the CASPT2/CC reference, MC-PDFT with the tPBE on-top functional understabilizes high-spin states. This systematic understabilization is largely corrected by revising the exchange and correlation contributions to the on-top functional using the high local-exchange approximation (tPBE-HLE). Moreover, tPBE-HLE correctly predicts the spin of the ground state in most cases, while CASPT2 incorrectly predicts high-spin ground states in all cases. This is encouraging for practical work because tPBE and tPBE-HLE are faster than CASPT2 by a factor of 50 even in a moderately sized example.

Spin modification of iron(ii) complexes via covalent (dative) and dispersion guided non-covalent bonding with N-heterocyclic carbenes: DFT, DLPNO-CCSD(T) and MCSCF studies

Spin crossover from high spin Fe(ii)-phthalocyanine to low or intermediate spin via either dative covalent or non-covalent interaction by just varying the substituent using the same core ligand.

Atom-Economical Ni-Catalyzed Diborylative Cyclization of Enynes: Preparation of Unsymmetrical Diboronates

We report a Ni-catalyzed diborylative cyclization of enynes that affords carbo- and heterocycles containing both alkyl- and alkenylboronates. The reaction is fully atom-economical, shows a broad scope, and employs a powerful and inexpensive catalytic Ni-based system. The reaction mechanism seems to involve activation of the enyne by Ni(0) through oxidative cyclometalation of the enyne prior to diboron reagent activation. An unprecedented dinuclear bis(organometallic) Ni(I) intermediate complex was isolated.

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

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

Cubane-Type [Mo3 S4 M] Clusters with First-Row Groups 4-10 Transition-Metal Halides Supported by C5 Me5 Ligands on Molybdenum

A synthetic protocol was developed for a series of cubane-type [Mo₃ S₄ M] clusters that incorporate halides of first-row transition metals (M) from Groups 4-10. This protocol is based on the anionic cluster platform [Cp*₃ Mo₃ S₄ ]^- ([1]^- ; Cp*=η⁵ -C₅ Me₅ ), which crystallizes when K(18-crown-6) is used as the counter cation. Treatment of in situ-generated [1]^- with such transition-metal halides led to the formation of [Mo₃ S₄ M] clusters, in which the M/halide ratio gradually changes from 1:2 to 1:1.5 and to 1:1, when moving from early to late transition metals. This trend suggests a tendency for early transition metals to tolerate higher oxidation states and adopt larger ionic radii relative to late transition metals. The properties of the [Mo₃ S₄ Fe] cluster 6 a were investigated in detail by using ⁵⁷ Fe Mössbauer spectroscopy and computational methods.

Spin States and Other Ligand-Field States of Aqua Complexes Revisited with Multireference ab Initio Calculations Including Solvation Effects

High-level multireference (CASPT2, NEVPT2) calculations are reported for transition metal aqua complexes with electronic configurations from (3d)¹ to (3d)⁸. We focus on the experimentally evidenced excitation energies to their various ligand-field states, including different spin states. By employing models accounting for both explicit and implicit solvation, we find that solvation effect may contribute up to 0.5 eV to the excitation energies depending on the charge of ion and character of the electronic transition. We further demonstrate that with an adequate choice of the active space and the energetics extrapolated to the complete basis set limit, the presently computed excitation energies are in a good agreement with the experimental data. This allows us to conclusively resolve significant discrepancies reported in earlier theory works [e.g., J. Phys. Chem. C 2014 , 118 , 29196 - 29208 ]. For the benchmark set of 19 spin-forbidden and 24 spin-allowed transitions (for which experimental data are unambiguous), we find the mean absolute error of 0.15 or 0.13 eV and the maximum error of 0.56 or 0.42 eV for CASPT2 or NEVPT2 calculations, respectively. For the particularly challenging sextet-quartet gap for [Fe(H₂O)₆]³⁺, we support our interpretation by additional calculations with multireference configuration interaction (MRCI) and coupled cluster theory up to the CCSDT(Q) level. By underlining a rather subtle interplay between the solvation and correlation effects, the findings of this Article are relevant not only for modeling and interpretation of optical spectra of transition metal complexes but also in further benchmarking of theoretical methods for the challenging problem of spin-state energetics.

Toward Highly Accurate Spin State Energetics in First-Row Transition Metal Complexes: A Combined CASPT2/CC Approach

In previous work on the performance of multiconfigurational second-order perturbation theory (CASPT2) in describing spin state energetics in first-row transition metal systems [ Pierloot et al. J. Chem. Theory Comput. 2017 , 13 , 537 - 553 ], we showed that standard CASPT2 works well for valence correlation but does not describe the metal semicore (3s3p) correlation effects accurately. This failure is partially responsible for the well-known bias toward high-spin states of CASPT2. In this paper, we expand our previous work and show that this bias could be partly removed with a combined CASPT2/CC approach: using high-quality CASPT2 with extensive correlation-consistent basis sets for valence correlation and low-cost CCSD(T) calculations with minimal basis sets for the metal semicore (3s3p) correlation effects. We demonstrate that this approach is efficient by studying the spin state energetics of a series of iron complexes modeling important intermediates in oxidative catalytic processes in chemistry and biochemistry. On the basis of a comparison with bare CCSD(T) results from this and previous work, the average error of the CASPT2/CC approach is estimated at around 2 kcal mol^-1 in favor of high spin states.

Base-Free Iron Hydrosilylene Complexes via an α-Hydride Migration that Induces Spin Pairing

Two new base-free hydrosilylene complexes of iron were synthesized using the novel starting material Cp*( ⁱPr₂MeP)FeMes. These Cp*( ⁱPr₂MeP)Fe(H)SiHR (R = DMP, Trip) complexes are in equilibrium with the corresponding iron silyl complexes, Cp*( ⁱPr₂MeP)FeSiH₂R, which can be trapped and characterized for R = Trip. Unlike the Ru analogues, the Fe silylene complex with R = DMP is observed to undergo an intramolecular C-H activation involving formal addition of a benzylic C-H bond across the Fe-Si bond. This increased activity for bond activations is also observed for reactions with hydrogen, where Fe reacts faster than a Ru analog to form the hydrogenation product, Cp*( ⁱPr₂MeP)H₂FeSiH₂DMP.

Assessment of the ground spin state of iron(i) complexes: insights from DFT predictive models

Factors governing the ground spin state of iron(i) complexes are analyzed by DFT methods. An efficient benchmarking procedure is reported.

Conceptual Insights into DFT Spin-State Energetics of Octahedral Transition-Metal Complexes through a Density Difference Analysis

In this study, an intuitive concept is derived, which explains the characteristic dependence of spin-state energetics on the exact exchange admixture of DFT functionals in the case of octahedral transition metal complexes. The change in electron density distributions upon varying the admixture, c₃ , in the B3LYP functional is analyzed for archetype ionic and covalent systems as well as for the Fe²⁺ ion in an ideal octahedral field. An understanding of how the DFT description of the electronic structure of octahedral complexes changes as a function of c₃ is sought. A systematic spin-state energy analysis of 50 octahedral complexes of various metals and ligands with consistent experimental data is presented, allowing the derivation, in theory, of an optimal c₃ value for each system. The notion that the admixture dependence of DFT spin-state energetics stems from the treatment of nondynamic electrons arising from the mixing of (M-Lz2 )⁰ (dz2 )² and (M-Lx2-y2 )⁰ (dx2-y2 )² configurations into the dominant (M-Lx2-y2 )² (dx2-y2 )⁰ and (M-Lx2-y2 )² (dx2-y2 )⁰ ones in the low(er) spin states is put forward. That is, in the effort to mimic such electron-electron interactions, E_x^LDA overestimates, whereas exact exchange downplays the contribution of this type of electron correlation to the stability of low(er) spin states, leading to the widespread practical observation that the higher the exact exchange admixture, the more stable the high-spin-state configuration.

On the structures, spin states, and optical properties of titanium, platinum, and iron azacalixphyrins: a DFT study

Azacalixphyrins (ACP) constitute a new class of macrocycles isoelectronic and isostructural to porphyrins. Herein, we report the first theoretical investigation of the properties of the ACP macrocycles metallated at their centre by titanium, platinum, and iron ions. We considered both the original phenyl-type ACP and new pyridyl-type forms. Our results indicate that the metallation greatly impacts the global structure of the macrocycle through pseudo Jahn-Teller effects, giving rise to a possible conformational transition between D_2d and S₄ structures. Such an effect could not be found in the metal-free ACPs. In addition, we find that, in contrast to the purely singlet platinum ACPs, and the purely triplet iron ACPs, several spin states are energetically close in the titanium ACPs, especially when weak field ligands are bound in axial positions to the metallic centre. According to TD-DFT calculations, metallation also tunes the optical properties. In particular, the absorption band in the near infrared region undergoes a hypsochromic shift of ca. 100-200 nm when going from the D_2d to the S₄ structures. We quantify how the addition of electroactive ligands in the axial position can increase or tune down these spectral changes. This contribution therefore supports the development of ACP coordination complexes.