Performance of CASPT2 and DFT for Relative Spin-State Energetics of Heme Models

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.

Improving the Accuracy in the Prediction of Transition-Metal Spin-State Energetics Using a Robust Variation-Based Approach: Density Functional Theory, CASPT2 and MC-PDFT Applied to the Case Study of Tris-Diimine Fe(II) Complexes

Designing ligands for transition metal complexes with a specified low-spin, high-spin or spin-crossover behavior is challenging. A major advance was recently made by Phan et al. [J. Am. Chem. Soc. 2017, 139, 6437-6447] who showed that the spin state of a homoleptic tris-diimine Fe(II) complex can be predicted from the N-N distance in the free diimine. They could thus predict the change in magnetic behavior on passing from the complexes of 2,2'-bipyridine (bpy), 2,2'-biimidazole (bim) and 2,2'-bis-2-imidazoline (bimz) ligands to those obtained with the modified analogs 4,5-diazafluoren-9-one (dafo), 1,1'-(α,α'-o-xylyl)-2,2'-bisimidazole (xbim) and 2,3,5,6,8,9-hexahydrodiimidazo[1,2-a:2', 1'-c]pyrazine (etbimz), respectively. Theoretically, the challenge lies in the accurate determination of the HS-LS zero-point energy difference ΔEHL°. The issue can be circumvented by using a variation-based approach, wherein ΔEHL° is not directly evaluated but obtained from the estimate of its variation Δ(ΔEHL°) in series of related systems, which include one whose ΔEHL° is accurately known [Phys. Chem. Chem. Phys. 2013, 15, 3752-3763; J. Phys. Chem. A 2022, 126, 6221-6235]. In this study, density functional theory (DFT), second-order multireference perturbation theory in its CASPT2 formulation, multiconfigurational pair DFT (MC-PDFT) and its hybrid formulation (HMC-PDFT) have been applied to the determination of Δ(ΔEHL°) in the pairs of complexes ( [ F e ( b p y ) 3 ] 2 + , [ F e ( d a f o ) 3 ] 2 + ) , ( [ F e ( b i m ) 3 ] 2 + , [ F e ( x b i m ) 3 ] 2 + ) and ( [ F e ( b i m z ) 3 ] 2 + , [ F e ( e t b i m z ) 3 ] 2 + ) . In DFT, we used several semilocal functionals and their global hybrids, as well as their D2, D3, D3BJ and D4 dispersion-corrected forms; and in MC-PDFT, different translated and fully translated functionals. The results are consistent with one another and in very good agreement with experiments. They show small to vanishing dependence on key details of the methods used: namely, the exact-exchange contribution to global hybrids; the ionization potential-electron affinity shift and basis sets used in the CASPT2 calculations; and the active spaces employed for the CASSCF wave functions used in the MC-PDFT and HMC-PDFT calculations. Insights into the change in the spin-state energetics accompanying the ligand exchanges were gained through a complexation energy analysis. Using the accurate CCSD(T) estimate of the HS-LS adiabatic energy difference in [ F e ( N C H ) 6 ] 2 + [J. Chem. Theory Comput. 2012, 8, 4216-4231], the Δ(ΔEHL°)-approach has been applied to the determination of ΔEHL° in the diimine complexes. The CASPT2 and DFT-D2 methods only give results in agreement with experiments. This suggests for the other methods a limitation in their treatment of dispersion which prevents them from accurately describing the spin-state energetics change accompanying the passing from [ F e ( N C H ) 6 ] 2 + with the tetragonal arrangement of its nitrile ligands to the tris-diimine complexes with the trigonal packing of their bulkier ligands.

Mechanism of Peroxynitrite Interaction with Ferric M. tuberculosis Nitrobindin: A Computational Study

Nitrobindins (Nbs) are all-β-barrel heme proteins present along the evolutionary ladder. They display a highly solvent-exposed ferric heme group with the iron atom being coordinated by the proximal His residue and a water molecule at the distal position. Ferric nitrobindins (Nb(III)) play a role in the conversion of toxic peroxynitrite (ONOO-) to harmless nitrate, with the value of the second-order rate constant being similar to those of most heme proteins. The value of the second-order rate constant of Nbs increases as the pH decreases; this suggests that Nb(III) preferentially reacts with peroxynitrous acid (ONOOH), although ONOO- is more nucleophilic. In this work, we shed light on the molecular basis of the ONOO- and ONOOH reactivity of ferric Mycobacterium tuberculosis Nb (Mt-Nb(III)) by dissecting the ligand migration toward the active site, the water molecule release, and the ligand binding process by computer simulations. Classical molecular dynamics simulations were performed by employing a steered molecular dynamics approach and the Jarzynski equality to obtain ligand migration free energy profiles for both ONOO- and ONOOH. Our results indicate that ONOO- and ONOOH migration is almost unhindered, consistent with the exposed metal center of Mt-Nb(III). To further analyze the ligand binding process, we computed potential energy profiles for the displacement of the Fe(III)-coordinated water molecule using a hybrid QM/MM scheme at the DFT level and a nudged elastic band approach. These results indicate that ONOO- exhibits a much larger barrier for ligand displacement than ONOOH, suggesting that water displacement is assisted by protonation of the leaving group by the incoming ONOOH.

Can Iron-Porphyrins Behave as Single-Molecule Magnets?

The study of magnetic properties, especially the magnetic anisotropy of iron-porphyrin complexes employing multiconfigurational methods, is quite challenging due to many strongly correlated electrons in nearly degenerate orbitals. However, a prerequisite for observing the magnetic anisotropy and slow magnetization relaxation, the zero-field splitting parameter, D, was experimentally observed decades ago for halide-based axially ligated penta-coordinate Fe(III)-porphyrins. In these complexes, the signs of D were reported mostly as positive; in a few cases, inconclusive signs of the D parameter were also mentioned. However, no ab initio calculations have been reported to shed light on this. Deciphering the electronic structure of these penta-coordinated complexes employing the complete active space self-consistent field method and N-electron valence second-order perturbation theory, we confirm the positive D values. However, a negative D value is highly desired to observe the single-molecule magnet properties without an external magnetic field, which we observed in the Fe(II)-porphyrin complexes with axial imidazole ligands instead of halide ligands. The detailed analysis of the multireference wave functions unravels the role of axial ligands in determining the sign and magnitude of the D parameters.

Reaction pathways, proton transfer, and proton pumping in ba3 class cytochrome c oxidase: perspectives from DFT quantum chemistry and molecular dynamics

After drawing comparisons between the reaction pathways of cytochrome c oxidase (CcO, Complex 4) and the preceding complex cytochrome bc1 (Complex 3), both being proton pumping complexes along the electron transport chain, we provide an analysis of the reaction pathways in bacterial ba3 class CcO, comparing spectroscopic results and kinetics observations with results from DFT calculations. For an important arc of the catalytic cycle in CcO, we can trace the energy pathways for the chemical protons and show how these pathways drive proton pumping of the vectorial protons. We then explore the proton loading network above the Fe heme a3-CuB catalytic center, showing how protons are loaded in and then released by combining DFT-based reaction energies with molecular dynamics simulations over states of that cycle. We also propose some additional reaction pathways for the chemical and vector protons based on our recent work with spectroscopic support.

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

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

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

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

Metastable Charged Dimers in Organometallic Species: A Look into Hydrogen Bonding between Metallocene Derivatives

Multiply charged complexes bound by noncovalent interactions have been previously described in the literature, although they were mostly focused on organic and main group inorganic systems. In this work, we show that similar complexes can also be found for organometallic systems containing transition metals and deepen in the reasons behind the existence of these species. We have studied the structures, binding energies, and dissociation profiles in the gas phase of a series of charged hydrogen-bonded dimers of metallocene (Ru, Co, Rh, and Mn) derivatives isoelectronic with the ferrocene dimer. Our results indicate that the carboxylic acid-containing dimers are more strongly bonded and present larger barriers to dissociation than the amide ones and that the cationic complexes tend to be more stable than the anionic ones. Additionally, we describe for the first time the symmetric proton transfer that can occur while in the metastable phase. Finally, we use a density-based energy decomposition analysis to shine light on the nature of the interaction between the dimers.

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.

Reliably assessing the electronic structure of cytochrome P450 on today's classical computers and tomorrow's quantum computers

Chemical simulation is one of the most promising applications for future quantum computers. It is thought that quantum computers may enable accurate simulation for complex molecules that are otherwise impossible to simulate classically; that is, it displays quantum advantage. To better understand quantum advantage in chemical simulation, we explore what quantum and classical resources are required to simulate a series of pharmaceutically relevant molecules. Using classical methods, we show that reliable classical simulation of these molecules requires significant resources and therefore is a promising candidate for quantum simulation. We estimate the quantum resources, both in overall simulation time and the size. The insights from this study pave the way for future quantum simulation of complex molecules.

An accurate assessment of how quantum computers can be used for chemical simulation, especially their potential computational advantages, provides important context on how to deploy these future devices. To perform this assessment reliably, quantum resource estimates must be coupled with classical computations attempting to answer relevant chemical questions and to define the classical algorithms simulation frontier. Herein, we explore the quantum computation and classical computation resources required to assess the electronic structure of cytochrome P450 enzymes (CYPs) and thus define a classical–quantum advantage boundary. This is accomplished by analyzing the convergence of density matrix renormalization group plus n-electron valence state perturbation theory (DMRG+NEVPT2) and coupled-cluster singles doubles with noniterative triples [CCSD(T)] calculations for spin gaps in models of the CYP catalytic cycle that indicate multireference character. The quantum resources required to perform phase estimation using qubitized quantum walks are calculated for the same systems. Compilation into the surface code provides runtime estimates to compare directly to DMRG runtimes and to evaluate potential quantum advantage. Both classical and quantum resource estimates suggest that simulation of CYP models at scales large enough to balance dynamic and multiconfigurational electron correlation has the potential to be a quantum advantage problem and emphasizes the important interplay between classical computations and quantum algorithms development for chemical simulation.

The many-body electronic interactions of Fe(II)–porphyrin

Fe(II)–porphyrin complexes exhibit a diverse range of electronic interactions between the metal and macrocycle. Herein, the incremental full configuration interaction method is applied to the entire space of valence orbitals of a Fe(II)–porphyrin model using a modest basis set. A novel visualization framework is proposed to analyze individual many-body contributions to the correlation energy, providing detailed maps of this complex’s highly correlated electronic structure. This technique is used to parse the numerous interactions of two low-lying triplet states (3A2g and 3Eg) and to show that strong metal d–d and macrocycle π–π orbital interactions preferentially stabilize the 3A2g state. d–π interactions, on the other hand, preferentially stabilize the 3Eg state and primarily appear when correlating six electrons at a time. Ultimately, the Fe(II)–porphyrin model’s full set of 88 valence electrons are correlated in 275 orbitals, showing the interactions up to the 4-body level, which covers the great majority of correlations in this system.

On the Sensitivity to Density-Functional Approximations for CO Binding Energies of Single-Atom Catalysts in Nitrogen-Doped Graphene

Density functional theory (DFT) methods are the working horse in screening new catalytic materials. They are widely used to predict trends in binding energies, which are then used to compare the activity of different materials. The binding strength of CO is an important descriptor to the CO₂ reduction catalytic activity of the single transition metal atoms embedded on nitrogen-doped graphene (TM/NG). In this work, however, we show that CO binding strengths in different TM/NG has very different sensitivity to DFT methods. Specifically, Fe/NG CO binding energy changes dramatically with the percentage of exact exchange in the functional; Co/NG does less so, while Ni/NG nearly has no change. Such varying behaviors is a direct result of different local spin configurations, similar to the performance of DFT methods for metal porphyrin complexes. Therefore, caution should be exercised when using DFT binding energies for quantitative predictions in TM/NG single atom catalysis.

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

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

Nonheme Iron Imido Complexes Bearing a Non-Innocent Ligand: A Synthetic Chameleon Species in Oxidation Reactions

High-valent iron-imido complexes can perform C-H activation and sulfimidation reactions, but are far less studied than the more ubiquitous iron-oxo species. As case studies, we have looked at a recently published iron(V)-imido ligand π-cation radical complex, which is formally an iron(VI)-imido complex [Fe^V (NTs)(TAML^+. )] (1; NTs=tosylimido), and an iron(V)-imido complex [Fe^V (NTs)(TAML)]^- (2). Using a theoretical approach, we found that they have multiple energetically close-lying electromers, sometimes even without changing spin states, reminiscent of the so-called Compound I in Cytochrome P450. When studying their reactivity theoretically, it is indeed found that their electronic structures may change to perform efficient oxidations, emulating the multi-spin state reactivity in Fe^IV O systems. This is actually in contrast to the known [Fe^V (O)(TAML)]^- species (3), where the reactions occur only on the ground spin state. We also looked into the whole reaction pathway for the C-H bond activation of 1,4-cyclohexadiene by these intermediates to reproduce the experimentally observed products, including steps that usually attract no interest (neither theoretically nor experimentally) due to their non-rate-limiting status and fast reactivity. A new "clustering non-rebound mechanism" is presented for this C-H activation reaction.

The redox potential of a heme cofactor in Nitrosomonas europaea cytochrome c peroxidase: a polarizable QM/MM study

Redox reactions are crucial to biological processes that protect organisms against oxidative stress. Metalloenzymes, such as peroxidases which reduce excess reactive oxygen species into water, play a key role in detoxification mechanisms. Here we present the results of a polarizable QM/MM study of the reduction potential of the electron transfer heme in the cytochrome c peroxidase of Nitrosomonas europaea. We have found that environment polarization does not substantially affect the computed value of the redox potential. Particular attention has been given to analyzing the role of electrostatic interactions within the protein environment and the solvent on tuning the redox potential of the heme co-factor. We have found that the electrostatic interactions predominantly explain the fluctuations of the vertical ionization/attachment energies of the heme for the sampled configurations, and that the long range electrostatic interactions (up to 40 Å) contribute substantially to the absolute values of the vertical energy gaps.

Controlling the spin-crossover behavior of the [Cr(indenyl)2] family via ligand functionalization

In this work, a computational method to study the spin-crossover (SCO) behavior of the [Cr(n-Meindenyl)2] family was carried out. Using the TPSSh/Def2TZVP method with the GD3BJ dispersion correction scheme, we computed the thermochemistry and transition temperatures (T1/2) for all members of this family, which are in excellent agreement with the available experimental data. Moreover, the computed data allow us to build a model that describes the effect of functionalizing the indenyl ligand in different positions on the spin-state energy gap and transition temperature. Our results show that the C4 and C7 positions of the indenyl ligand have a greater effect on tuning the SCO properties of such complexes. The model quantitatively reproduces the DFT calculations, thus providing a powerful tool to analyze and predict the SCO properties in any member of the [Cr(n-Meindenyl)2] family.

The importance of exact exchange-A methodological investigation of NO reduction in heme-copper oxidases

Significant improvements of the density functional theory (DFT) methodology during the past few decades have made DFT calculations a powerful tool in studies of enzymatic reaction mechanisms. For metalloenzymes, however, there are still concerns about the reliability in the DFT-results. Therefore, a systematic study is performed where the fraction of exact exchange in a hybrid DFT functional is used as a parameter. By varying this parameter, a set of different but related functionals are obtained. The various functionals are applied to one of the reactions occurring in the enzyme family heme-copper oxidases, the reduction of nitric oxide (NO) to nitrous oxide (N₂O) and water. The results show that, even though certain parts of the calculated energetics exhibit large variations, the qualitative pictures of the reaction mechanisms are quite stable. Furthermore, it is found that the functional with 15% exact exchange (B3LYP^*) gives the best agreement with experimental data for the particular reactions studied. An important aspect of the procedure used is that the computational results are carefully combined with a few more general experimental data to obtain a complete description of the entire catalytic cycle of the reactions studied.

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.

Activation of O2 and NO in heme-copper oxidases - mechanistic insights from computational modelling

Heme-copper oxidases are transmembrane enzymes involved in aerobic and anaerobic respiration. The largest subgroup contains the cytochrome c oxidases (CcO), which reduce molecular oxygen to water. A significant part of the free energy released in this exergonic process is conserved as an electrochemical gradient across the membrane, via two processes, electrogenic chemistry and proton pumping. A deviant subgroup is the cytochrome c dependent NO reductases (cNOR), which reduce nitric oxide to nitrous oxide and water. This is also an exergonic reaction, but in this case none of the released free energy is conserved. Computational studies applying hybrid density functional theory to cluster models of the bimetallic active sites in the heme-copper oxidases are reviewed. To obtain a reliable description of the reaction mechanisms, energy profiles of the entire catalytic cycles, including the reduction steps have to be constructed. This requires a careful combination of computational results with certain experimental data. Computational studies have elucidated mechanistic details of the chemical parts of the reactions, involving cleavage and formation of covalent bonds, which have not been obtainable from pure experimental investigations. Important insights regarding the mechanisms of energy conservation have also been gained. The computational studies show that the reduction potentials of the active site cofactors in the CcOs are large enough to afford electrogenic chemistry and proton pumping, i.e. efficient energy conservation. These results solve a conflict between different types of experimental data. A mechanism for the proton pumping, involving a specific and crucial role for the active site tyrosine, conserved in all CcOs, is suggested. For the cNORs, the calculations show that the low reduction potentials of the active site cofactors are optimized for fast elimination of the toxic NO molecules. At the same time, the low reduction potentials lead to endergonic reduction steps with high barriers. To prevent even higher barriers, which would lead to a too slow reaction, when the electrochemical gradient across the membrane is present, the chemistry must occur in a non-electrogenic manner. This explains why there is no energy conservation in cNOR.

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.

Unprecedented interplay of antiferro- and ferromagnetic exchange interactions through intermolecular hydrogen bonds in mononuclear Cu(ii) complexes

A nontrivial case of a combination of ferro- and antiferromagnetic exchange interactions in mononuclear copper(ii) complexes, determined by HB networks, is reported.

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.

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

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

Is density functional theory accurate for lytic polysaccharide monooxygenase enzymes?

The lytic polysaccharide monooxygenase (LPMO) enzymes boost polysaccharide depolymerization through oxidative chemistry, which has fueled the hope for more energy-efficient production of biofuel. We have recently proposed a mechanism for the oxidation of the polysaccharide substrate (E. D. Hedegård and U. Ryde, Chem. Sci., 2018, 9, 3866-3880). In this mechanism, intermediates with superoxide, oxyl, as well as hydroxyl (i.e. [CuO2]+, [CuO]+ and [CuOH]2+) cores were involved. These complexes can have both singlet and triplet spin states, and both spin-states may be important for how LPMOs function during catalytic turnover. Previous calculations on LPMOs have exclusively been based on density functional theory (DFT). However, different DFT functionals are known to display large differences for spin-state splittings in transition-metal complexes, and this has also been an issue for LPMOs. In this paper, we study the accuracy of DFT for spin-state splittings in superoxide, oxyl, and hydroxyl intermediates involved in LPMO turnover. As reference we employ multiconfigurational perturbation theory (CASPT2).

Spin-inversion mechanisms in O2 binding to a model heme compound: A perspective from nonadiabatic wave packet calculations

Spin-inversion dynamics in O₂ binding to a model heme complex, which consisted of Fe(II)-porphyrin and imidazole, were studied using nonadiabatic wave packet dynamics calculations. We considered three active nuclear degrees of freedom in the dynamics, including the motions along the Fe-O distance, Fe-O-O angle, and Fe out-of-plane distance. Spin-free potential energy surfaces for the singlet, triplet, quintet, and septet states were developed using density functional theory calculations, and spin-orbit coupling elements were obtained from CASSCF-level electronic structure calculations. The spin-inversion mainly occurred between the singlet state and one of the triplet states due to large spin-orbit couplings and the contributions of other states were extremely small. The present quantum dynamics calculations suggested that the narrow crossing region model plays a dominant role in the O₂ binding dynamics. In addition, the one-dimensional Landau-Zener model underestimated the nonadiabatic transition probability.

Role of the Two Metals in the Active Sites of Heme Copper Oxidases-A Study of NO Reduction in cbb3 Cytochrome c Oxidase

The superfamily of heme copper oxidases reduces molecular oxygen or nitric oxide, and the active sites comprise a high-spin heme group (a₃ or b₃) and a non-heme metal (Cu_B or Fe_B). The cbb₃ C family of cytochrome c oxidases, with the high-spin heme b₃ and Cu_B in the active site, is a subfamily of the heme copper oxidases that can reduce both molecular oxygen, which is the main substrate, and nitric oxide. The mechanism for NO reduction in cbb₃ oxidase is studied here using hybrid density functional theory and compared to other cytochrome c oxidases (A and B families), with a high-spin heme a₃ and Cu_B in the active site, and to cytochrome c dependent NO reductase, with a high-spin heme b₃ and a non-heme Fe_B in the active site. It is found that the reaction mechanism and the detailed reaction energetics of the cbb₃ oxidases are not similar to those of cytochrome c dependent NO reductase, which has the same type of high-spin heme group but a different non-heme metal. This is in contrast to earlier expectations. Instead, the NO reduction mechanism in cbb₃ oxidases is very similar to that in the other cytochrome c oxidases, with the same non-heme metal, Cu_B, and is independent of the type of high-spin heme group. The conclusion is that the type of non-heme metal (Cu_B or Fe_B) in the active site of the heme copper oxidases is more important for the reaction mechanisms than the type of high-spin heme, at least for the NO reduction reaction. The reason is that the proton-coupled reduction potentials of the active site cofactors determine the energetics for the NO reduction reaction, and they depend to a larger extent on the non-heme metal. Observed differences in NO reduction reactivity among the various cytochrome c oxidases may be explained by differences outside the BNC, affecting the rate of proton transfer, rather than in the BNC itself.

Spin Splitting Energy of Transition Metals: A New, More Affordable Wave Function Benchmark Method and Its Use to Test Density Functional Theory

Accurately predicting the spin splitting energy of chemical species is important for understanding their reactivity and magnetic properties, but it is very challenging, especially for molecules containing transition metals. One impediment to progress is the scarcity of accurate benchmark data. Here we report a set of calculations designed to yield reliable benchmarks for simple transition-metal complexes that can be used to test density functional methods that are affordable for large systems of more practical interest. Various wave function methods are tested against experiment for Fe²⁺, Fe³⁺, and Co³⁺, including CASSCF, CASPT2, CASPT3, MRCISD, MRCISD+Q, ACPF, AQCC, CCSD(T), and CASPT2/CCSD(T) and also a new method called CASPT2.5, which is performed by taking the average of the CASPT2 and CASPT3 energies. We find that MRCISD+Q, ACPF, and AQCC require smaller active spaces for good accuracy than are required by CASPT2 and CASPT3, and this aspect may be important for calculations on larger molecules; here we find that CASPT2.5 extrapolated to a complete basis set is the most suitable method-in terms of computational cost and in terms of accuracy on monatomic systems-and therefore we chose this method for molecular benchmarks. Then Kohn-Sham density functional calculations with 60 exchange-correlation functionals are tested for FeF₂, FeCl₂, and CoF₂. We find that MN15-L, M06-SX, and revM06 have very good agreement with CASPT2.5 benchmarks in terms of both the spin splitting energy and the optimized geometry for each spin state. In addition, we recommend def2-TZVP as the most suitable basis set to perform density functional calculations for molecular spin splitting energies; extra polarization functions in the basis set do not help to increase the accuracy of the spin splitting energy in KS calculations.

Seeing Is Believing: Experimental Spin States from Machine Learning Model Structure Predictions

Determination of ground-state spins of open-shell transition-metal complexes is critical to understanding catalytic and materials properties but also challenging with approximate electronic structure methods. As an alternative approach, we demonstrate how structure alone can be used to guide assignment of ground-state spin from experimentally determined crystal structures of transition-metal complexes. We first identify the limits of distance-based heuristics from distributions of metal–ligand bond lengths of over 2000 unique mononuclear Fe(II)/Fe(III) transition-metal complexes. To overcome these limits, we employ artificial neural networks (ANNs) to predict spin-state-dependent metal–ligand bond lengths and classify experimental ground-state spins based on agreement of experimental structures with the ANN predictions. Although the ANN is trained on hybrid density functional theory data, we exploit the method-insensitivity of geometric properties to enable assignment of ground states for the majority (ca. 80–90%) of structures. We demonstrate the utility of the ANN by data-mining the literature for spin-crossover (SCO) complexes, which have experimentally observed temperature-dependent geometric structure changes, by correctly assigning almost all (>95%) spin states in the 46 Fe(II) SCO complex set. This approach represents a promising complement to more conventional energy-based spin-state assignment from electronic structure theory at the low cost of a machine learning model.

A multireference coupled-electron pair approximation combined with complete-active space perturbation theory in local pair-natural orbital framework

The Complete-Active Space Second-order Perturbation Theory (CASPT2) has been one of the most widely-used methods for reliably calculating electronic structures of multireference systems. Because of its lowest level treatment of dynamic correlation, it has a high computational feasibility; however, its accuracy in some cases falls short of needs. Here, as a simple yet higher-order alternative, we introduce a hybrid theory of the CASPT2 and a multireference variant of the Coupled-Electron Pair Approximation (CEPA), which is a class of high level correlation theory. A central feature of our theory (CEPT2) is to use the two underlying theories for describing different divisions of correlation components based on the full internal contraction framework. The external components, which usually give a major contribution to the dynamic correlation, are intensively described using the CEPA Ansatz, while the rests are treated at the CASPT2 level. Furthermore, to drastically reduce the computational demands, we have incorporated the pair-natural orbital (PNO) method into our multireference implementations. This development, thus, requires highly complex derivations and coding, while it has been largely facilitated with an automatic expression and code generation technique. To highlight the accuracy of the CEPT2 approach and to assess the errors caused by the PNO truncation, benchmark calculations are shown on small- to medium-size molecules, illustrating the high accuracy of the present CEPT2 model. By tightening the truncation thresholds, the PNO-CEPT2 energy converges toward the canonical counterpart and is more accurate than that of PNO-CASPT2 as long as the same truncation thresholds are used.

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.

Large-scale comparison of 3d and 4d transition metal complexes illuminates the reduced effect of exchange on second-row spin-state energetics

The origin of distinct 3d vs. 4d transition metal complex sensitivity to exchange is explored over a large data set.

Accurate computed spin-state energetics for Co(iii) complexes: implications for modelling homogeneous catalysis

DLPNO-CCSD(T) calculations provide accurate spin state energetics for a range of Co(iii) complexes and so represent a promising approach to modelling homogeneous catalysis based on Co(iii) species.

The mechanism for oxygen reduction in the C family cbb3 cytochrome c oxidases - Implications for the proton pumping stoichiometry

Cytochrome c oxidases (CcOs) couple the exergonic reduction of molecular oxygen to proton pumping across the membrane in which they are embedded, thereby conserving a significant part of the free energy. The A family CcOs are known to pump four protons per oxygen molecule, while there is no consensus regarding the proton pumping stoichiometry for the C family cbb₃ oxidases. Hybrid density functional theory is used here to investigate the catalytic mechanism for oxygen reduction in cbb₃ oxidases. A surprising result is that the barrier for O O bond cleavage at the mixed valence reduction level seems to be too high compared to the overall reaction rate of the enzyme. It is therefore suggested that the O O bond is cleaved only after the first proton coupled reduction step, and that this reduction step most likely is not coupled to proton pumping. Furthermore, since the cbb₃ oxidases have only one proton channel leading to the active site, it is proposed that the activated E_H intermediate, suggested to be responsible for proton pumping in one of the reduction steps in the A family, cannot be involved in the catalytic cycle for cbb₃, which results in the lack of proton pumping also in the E to R reduction step. In summary, the calculations indicate that only two protons are pumped per oxygen molecule in cbb₃ oxidases. However, more experimental information on this divergent enzyme is needed, e.g. whether the flow of electrons resembles that in the other more well-studied CcO families.

The Role of Vibrational Anharmonicity in the Computational Study of Thermal Spin Crossover

Spin crossover in transition metal complexes can be studied in great detail with computational chemistry. Over the years, the understanding has grown that the relative stability of high-spin (HS) versus low-spin (LS) states is a subtle balance of many factors that all need to be taken into account for a reliable description. Among the different contributions, the zero-point energy (ZPE) and the entropy play key roles. These quantities are usually calculated assuming a harmonic oscillator model for the molecular vibrations. We investigated the impact of including anharmonic corrections on the ZPE and the entropy and indirectly on the critical temperature of spin crossover. As test systems, we used a set of ten Fe(II) complexes and one Fe(III) complex, covering different coordination modes (mono-, bi-, and tri-dentate ligands), decreasing coordination number upon spin crossover, coordination by second- and third-row atoms, and changes in the oxidation state. The results show that the anharmonicity has a measurable effect, but it is in general rather small, and tendencies are not easily recognized. As a conclusion, we put forward that for high precision results, one should be aware of the anharmonic effects, but as long as computational chemistry is still struggling with other larger factors like the influence of the environment and the accurate determination of the electronic energy difference between HS and LS, the anharmonicity of the vibrational modes is a minor concern.

Understanding the Reactivity of Mn-Oxo Porphyrins for Substrate Hydroxylation: Theoretical Predictions and Experimental Evidence Reconciled

The Mn-oxo porphyrin (MnOP) mechanism for substrate hydroxylation is computationally studied with the aim to better understand reactivity in these systems. Theoretical studies suggest Mn(V)OP species to be very reactive intermediates with thermally accessible reaction barriers represented by low-spin/high-spin-crossover occurring in the Mn(V)OP oxidant, and kinetics for selected Mn(V)OP species indeed find high reactivity. On the other hand, MnOP complexes lead to modest yields in hydroxylation reactions of several different substrates, implying low rate constants and high reaction barriers. The resolution of this inconsistency is very important to understand the reactivity of Mn-oxo porphyrins and to improve the catalytic conditions. In this work we use the toluene hydroxylation by the Mn(V)OP(H₂O)⁺ complex as a case study to gain deep insight into the reaction mechanism. Minimum energy crossing point (MECP) results on the H-abstraction process from toluene indicate a first crossover from a singlet to a triplet spin state of the Mn(V)OP(H₂O)⁺ species with a thermally accessible barrier, followed by a very facile H-abstraction by the triplet complex. Issues concerning (i) the validation of the level of the density functional theory employed (BP86) to describe the singlet-triplet energy gap in the Mn(V)OP(H₂O)⁺ system versus highly accurate DMRG-CASPT2/CC calculations, and (ii) the influence of the axial ligand (X = none, Cl^-, CH₃CN, OH^-, and O^2-) on MnOP reactivity, which models the different experimental conditions, are addressed. The ligand trans influence mainly controls the reactivity through the singlet-triplet energy gap modulation, with the porphyrin ruffling distortion also finely tuning it. Finally, a stepwise model for the H-abstraction process is proposed which allows a direct comparison between the calculated and experimentally measured Gibbs free activation energy barriers ( Zhang et al. J. Am. Chem. Soc. 2005 , 127 , 6573 - 6582 ). The low yields in catalysis are shown not to be due to low reactivity of Mn(V).

Active Site Midpoint Potentials in Different Cytochrome c Oxidase Families: A Computational Comparison

Cytochrome c oxidase (C cO) is the terminal enzyme in the respiratory electron transport chain, reducing molecular oxygen to water. The binuclear active site in C cO comprises a high-spin heme associated with a Cu_B complex and a redox active tyrosine. The electron transport in the respiratory chain is driven by increasing midpoint potentials of the involved cofactors, resulting in a release of free energy, which is stored by coupling the electron transfer to proton translocation across a membrane, building up an electrochemical gradient. In this context, the midpoint potentials of the active site cofactors in the C cOs are of special interest, since they determine the driving forces for the individual oxygen reduction steps and thereby affect the efficiency of the proton pumping. It has been difficult to obtain useful information on some of these midpoint potentials from experiments. However, since each of the reduction steps in the catalytic cycle of oxygen reduction to water corresponds to the formation of an O-H bond, they can be calculated with a reasonably high accuracy using quantum chemical methods. From the calculated O-H bond strengths, the proton-coupled midpoint potentials of the active site cofactors can be estimated. Using models representing the different families of C cO's (A, B, and C), the calculations give midpoint potentials that should be relevant during catalytic turnover. The calculations also suggest possible explanations for why some experimentally measured potentials deviate significantly from the calculated ones, i.e., for Cu_B in all oxidase families, and for heme b₃ in the C family.

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.

Limits of Coupled-Cluster Calculations for Non-Heme Iron Complexes

In a large variety of studies, the coupled-cluster method with singles, doubles, and perturbative triples (CCSD(T)) is used as a reference for benchmarking the performance of density functional theory (DFT) functionals. In the case of open-shell species, this theory can be applied in different forms depending on the restricted or unrestricted treatment of spin. In this study, we show that these different approaches can produce results which deviate by ∼5 kcal/mol for different species on the potential energy surfaces. This was demonstrated for a simple model of the C-H activation carried out by non-heme iron enzymes. Assessing the limits of CCSD(T) prior to its use as a general benchmark tool is warranted. This was done using higher-order coupled-cluster calculations as well as multiconfigurational second-order perturbation theory (CASPT2), since iron-oxo species present some multireference character. Furthermore, we tested two different implementations of the local coupled-cluster method and compared them to the CCSD(T) results, showing that even though these novel approaches are promising, without further developments they appear not to be suitable for describing two-state reactivity of the system investigated in the current study. Additionally, we implemented and assessed the performance of the hotspot approach for the local unrestricted CCSD(T) scheme which aims at reducing the pair error for systems containing transition metals.

Mechanisms for enzymatic reduction of nitric oxide to nitrous oxide - A comparison between nitric oxide reductase and cytochrome c oxidase

Cytochrome c oxidases (CcO) reduce O₂ to H₂O in the respiratory chain of mitochondria and many aerobic bacteria. In addition, some species of CcO can also reduce NO to N₂O and water while others cannot. Here, the mechanism for NO-reduction in CcO is investigated using quantum mechanical calculations. Comparison is made to the corresponding reaction in a "true" cytochrome c-dependent NO reductase (cNOR). The calculations show that in cNOR, where the reduction potentials are low, the toxic NO molecules are rapidly reduced, while the higher reduction potentials in CcO lead to a slower or even impossible reaction, consistent with experimental observations. In both enzymes the reaction is initiated by addition of two NO molecules to the reduced active site, forming a hyponitrite intermediate. In cNOR, N₂O can then be formed using only the active-site electrons. In contrast, in CcO, one proton-coupled reduction step most likely has to occur before N₂O can be formed, and furthermore, proton transfer is most likely rate-limiting. This can explain why different CcO species with the same heme a₃-Cu active site differ with respect to NO reduction efficiency, since they have a varying number and/or properties of proton channels. Finally, the calculations also indicate that a conserved active site valine plays a role in reducing the rate of NO reduction in CcO.