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

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

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

Mechanistic Principles of Hydrogen Evolution in the Membrane-Bound Hydrogenase

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

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

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

Gaseous inhibition of the transsulfuration pathway by cystathionine β-synthase

The transsulfuration pathway plays a key role in mammals for maintaining the balance between cysteine and homocysteine, whose concentrations are critical in several biochemical processes. Human cystathionine β-synthase is a heme-containing, pyridoxal 5'-phosphate (PLP)-dependent enzyme found in this pathway. The heme group does not participate directly in catalysis, but has a regulatory function, whereby CO or NO binding inhibits the PLP-dependent reactions. In this study, we explore the detailed structural changes responsible for inhibition using quantum chemical calculations to validate the experimentally observed bonding patterns associated with heme CO and NO binding and molecular dynamics simulations to explore the medium-range structural changes triggered by gas binding and propagating to the PLP active site, which is more than 20 Å distant from the heme group. Our results support a previously proposed mechanical signaling model, whereby the cysteine decoordination associated with gas ligand binding leads to breaking of a hydrogen bond with an arginine residue on a neighbouring helix. In turn, this leads to a shift in position of the helix, and hence also of the PLP cofactor, ultimately disrupting a key hydrogen bond that stabilizes the PLP in its catalytically active form.

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.

Ab initio calculation of magnetic anisotropy and thermal spin transition in the variable temperature crystal conformations of [Co(terpy)2]2

The structure-property correlation of [Co(terpy)2]2+, which shows a spin crossover at 270 K, has been computationally investigated based on its variable temperature crystal structures. Among the employed DFT functionals, only the re-parametrized hybrid B3LYP* functional could describe the correct spin transition temperature. Explicit consideration of metal-ligand sigma bonding with dynamic electron correlation is found to be necessary for an accurate determination of the SCO temperature with multi-reference calculations. The metal-ligand axial bond distances are found to be the most significant internal coordinates in deciding SCO. A small structural change along the axial distance causes a change in the t2g orbital splitting pattern and a reorientation of the magnetization axes at the SCO temperature. The complex shows an unusual triaxial magnetic anisotropy, with an easy axis of magnetization developing at higher temperatures. The strong coupling of low-frequency wagging motion of the two terpyridine ligands with the spin states of the complex provides an effective pathway for the relaxation of magnetization, resulting in a small magnetic anisotropy barrier.

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.

Correlating Structure with Spectroscopy in Ascorbate Peroxidase Compound II

Structural and spectroscopic investigations of compound II in ascorbate peroxidase (APX) have yielded conflicting conclusions regarding the protonation state of the crucial Fe(IV) intermediate. Neutron diffraction and crystallographic data support an iron(IV)-hydroxo formulation, whereas Mössbauer, X-ray absorption (XAS), and nuclear resonance vibrational spectroscopy (NRVS) studies appear consistent with an iron(IV)-oxo species. Here we examine APX with spectroscopy-oriented QM/MM calculations and extensive exploration of the conformational space for both possible formulations of compound II. We establish that irrespective of variations in the orientation of a vicinal arginine residue and potential reorganization of proximal water molecules and hydrogen bonding, the Fe-O distances for the oxo and hydroxo forms consistently fall within distinct, narrow, and nonoverlapping ranges. The accuracy of geometric parameters is validated by coupled-cluster calculations with the domain-based local pair natural orbital approach, DLPNO-CCSD(T). QM/MM calculations of spectroscopic properties are conducted for all structural variants, encompassing Mössbauer, optical, X-ray absorption, and X-ray emission spectroscopies and NRVS. All spectroscopic observations can be assigned uniquely to an Fe(IV)═O form. A terminal hydroxy group cannot be reconciled with the spectroscopic data. Under no conditions can the Fe(IV)═O distance be sufficiently elongated to approach the crystallographically reported Fe-O distance. The latter is consistent only with a hydroxo species, either Fe(IV) or Fe(III). Our findings strongly support the Fe(IV)═O formulation of APX-II and highlight unresolved discrepancies in the nature of samples used across different experimental studies.

Rebound or Cage Escape? The Role of the Rebound Barrier for the Reactivity of Non-Heme High-Valent FeIV =O Species

Owing to their high reactivity and selectivity, variations in the spin ground state and a range of possible pathways, high-valent FeIV =O species are popular models with potential bioinspired applications. An interesting example of a structure-reactivity pattern is the detailed study with five nonheme amine-pyridine pentadentate ligand FeIV =O species, including N4py: [(L1 )FeIV =O]2+ (1), bntpen: [(L2 )FeIV =O]2+ (2), py2 tacn: [(L3 )FeIV =O]2+ (3), and two isomeric bispidine derivatives: [(L4 )FeIV =O]2+ (4) and [(L5 )FeIV =O]2+ (5). In this set, the order of increasing reactivity in the hydroxylation of cyclohexane differs from that with cyclohexadiene as substrate. A comprehensive DFT, ab initio CASSCF/NEVPT2 and DLPNO-CCSD(T) study is presented to untangle the observed patterns. These are well reproduced when both activation barriers for the C-H abstraction and the OH rebound are taken into account. An MO, NBO and deformation energy analysis reveals the importance of π(pyr) → π*xz (FeIII -OH) electron donation for weakening the FeIII -OH bond and thus reducing the rebound barrier. This requires that pyridine rings are oriented perpendicularly to the FeIII -OH bond and this is a subtle but crucial point in ligand design for non-heme iron alkane hydroxylation.

Exploring CPS-Extrapolated DLPNO-CCSD(T1) Reference Values for Benchmarking DFT Methods on Enzymatically Catalyzed Reactions

Domain-based local pair natural orbital coupled-cluster singles doubles with perturbative triples [DLPNO-CCSD(T)] is regularly used to calculate reliable benchmark reference values at a computational cost significantly lower than that of canonical CCSD(T). Recent work has shown that even greater accuracy can be obtained at only a small additional cost through extrapolation to the complete PNO space (CPS) limit. Herein, we test two levels of CPS extrapolation, CPS(5,6), which approximates the accuracy of standard TightPNO, and CPS(6,7), which surpasses it, as benchmark values to test density functional approximations (DFAs) on a small set of organic and transition-metal-dependent enzyme active site models. Between the different reference levels of theory, there are changes in the magnitudes of the absolute deviations for all functionals, but these are small and there is minimal impact on the relative rankings of the tested DFAs. The differences are more significant for the metalloenzymes than the organic enzymes, so we repeat the tests on our entire ENZYMES22 set of organic enzyme active site models [Wappett, D. A.; Goerigk, L. J. Phys. Chem. A 2019, 123, 7057-7074] to confirm that using the CPS extrapolations for the reference values has negligible impact on the benchmarking outcomes. This means that we can particularly recommend CPS(5,6) as an alternative to standard TightPNO settings for calculating reference values, increasing the applicability of DLPNO-CCSD(T) in benchmarking reaction energies and barrier heights of larger models of organic enzymes. DLPNO-CCSD(T1)/CPS(6,7) energies for ENZYMES22 are finally presented as updated reference values for the set, reflecting the recent improvements in the method.

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

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

Conformational energies of reference organic molecules: benchmarking of common efficient computational methods against coupled cluster theory

We selected 145 reference organic molecules that include model fragments used in computer-aided drug design. We calculated 158 conformational energies and barriers using force fields, with wide applicability in commercial and free softwares and extensive application on the calculation of conformational energies of organic molecules, e.g. the UFF and DREIDING force fields, the Allinger's force fields MM3-96, MM3-00, MM4-8, the MM2-91 clones MMX and MM+, the MMFF94 force field, MM4, ab initio Hartree-Fock (HF) theory with different basis sets, the standard density functional theory B3LYP, the second-order post-HF MP2 theory and the Domain-based Local Pair Natural Orbital Coupled Cluster DLPNO-CCSD(T) theory, with the latter used for accurate reference values. The data set of the organic molecules includes hydrocarbons, haloalkanes, conjugated compounds, and oxygen-, nitrogen-, phosphorus- and sulphur-containing compounds. We reviewed in detail the conformational aspects of these model organic molecules providing the current understanding of the steric and electronic factors that determine the stability of low energy conformers and the literature including previous experimental observations and calculated findings. While progress on the computer hardware allows the calculations of thousands of conformations for later use in drug design projects, this study is an update from previous classical studies that used, as reference values, experimental ones using a variety of methods and different environments. The lowest mean error against the DLPNO-CCSD(T) reference was calculated for MP2 (0.35 kcal mol-1), followed by B3LYP (0.69 kcal mol-1) and the HF theories (0.81-1.0 kcal mol-1). As regards the force fields, the lowest errors were observed for the Allinger's force fields MM3-00 (1.28 kcal mol-1), ΜΜ3-96 (1.40 kcal mol-1) and the Halgren's MMFF94 force field (1.30 kcal mol-1) and then for the MM2-91 clones MMX (1.77 kcal mol-1) and MM+ (2.01 kcal mol-1) and MM4 (2.05 kcal mol-1). The DREIDING (3.63 kcal mol-1) and UFF (3.77 kcal mol-1) force fields have the lowest performance. These model organic molecules we used are often present as fragments in drug-like molecules. The values calculated using DLPNO-CCSD(T) make up a valuable data set for further comparisons and for improved force field parameterization.

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.

Natural Abiotic Iron-Oxido-Mediated Formation of C1 and C2 Compounds from Environmentally Important Methyl-Substituted Substrates

Organic and inorganic volatile compounds containing one carbon atom (C1), such as carbon dioxide, methane, methanol, formaldehyde, carbon monoxide, and chloromethane, are ubiquitous in the environment, are key components in global carbon cycling, play an important role in atmospheric physics and chemistry, e.g., as greenhouse gases, destroy stratospheric and tropospheric ozone, and control the atmospheric oxidation capacity. Up to now, most C1 compounds in the environment were associated with complex metabolic and enzymatic pathways in organisms or to combustion processes of organic matter. We now present compelling evidence that many C1 and C2 compounds have a common origin in methyl groups of methyl-substituted substrates that are cleaved by the iron oxide-mediated formation of methyl radicals. This scenario is derived from experiments with a mechanistically well-studied bispidine-iron-oxido complex as oxidant and dimethyl sulfoxide as the environmentally relevant model substrate and is supported by computational modeling based on density functional theory and ab initio quantum-chemical studies. The exhaustive experimental model studies, also involving extensive isotope labeling, are complemented with the substitution of the bispidine model system by environmentally relevant iron oxides and, finally, a collection of soils with varying iron and organic matter contents. The combination of all data suggests that the iron oxide-mediated formation of methyl radicals from methyl-substituted substrates is a common abiotic source for widespread C1 and C2 compounds in the environment.

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

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

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

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

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

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

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

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

Comparison of the Performance of Density Functional Methods for the Description of Spin States and Binding Energies of Porphyrins

This work analyzes the performance of 250 electronic structure theory methods (including 240 density functional approximations) for the description of spin states and the binding properties of iron, manganese, and cobalt porphyrins. The assessment employs the Por21 database of high-level computational data (CASPT2 reference energies taken from the literature). Results show that current approximations fail to achieve the "chemical accuracy" target of 1.0 kcal/mol by a long margin. The best-performing methods achieve a mean unsigned error (MUE) <15.0 kcal/mol, but the errors are at least twice as large for most methods. Semilocal functionals and global hybrid functionals with a low percentage of exact exchange are found to be the least problematic for spin states and binding energies, in agreement with the general knowledge in transition metal computational chemistry. Approximations with high percentages of exact exchange (including range-separated and double-hybrid functionals) can lead to catastrophic failures. More modern approximations usually perform better than older functionals. An accurate statistical analysis of the results also casts doubts on some of the reference energies calculated using multireference methods. Suggestions and general guidelines for users are provided in the conclusions. These results hopefully stimulate advances for both the wave function and the density functional side of electronic structure calculations.

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

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

Exploring the Accuracy Limits of PNO-Based Local Coupled-Cluster Calculations for Transition-Metal Complexes

While the domain-based local pair natural orbital coupled-cluster method with singles, doubles, and perturbative triples (DLPNO-CCSD(T)) has proven instrumental for computing energies and properties of large and complex systems accurately, calculations on first-row transition metals with a complex electronic structure remain challenging. In this work, we identify and address the two main error sources that influence the DLPNO-CCSD(T) accuracy in this context, namely, (i) correlation effects from the 3s and 3p semicore orbitals and (ii) dynamic correlation-induced orbital relaxation (DCIOR) effects that are not described by the local MP2 guess. We present a computational strategy that allows us to completely eliminate the DLPNO error associated with semicore correlation effects, while increasing, at the same time, the efficiency of the method. As regards the DCIOR effects, we introduce a diagnostic for estimating the deviation between DLPNO-CCSD(T) and canonical CCSD(T) for systems with significant orbital relaxation.

Ionization energies of metallocenes: a coupled cluster study of cobaltocene

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

Advances in Parallel Heat Bath Configuration Interaction

Heat-bath configuration interaction (HCI) is a deterministic method that approaches the full CI limit at greatly reduced computational cost. In this work, computational improvements to the HCI algorithm are introduced targeting speed, parallel efficiency, and memory requirements. The new implementation introduces a hash function to distribute determinants and takes advantage of MPI and OpenMP for parallelism allowing for a (22e,168o) active space to be studied, which explicitly includes 2.39 × 10⁷ variational determinants and 8.95 × 10¹⁰ perturbative determinants. Benchmarks show up to 86% parallel efficiency of the perturbative step on 32 nodes (4096 cores) and a total efficiency of 74%. The new HCI implementation is benchmarked for accuracy against prior results and applied to study the triplet-quintet gap in the challenging [FeO(NH₃)₅]²⁺ complex.

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

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

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.

Molecular properties and tautomeric equilibria of isolated flavins

Flavins are employed as redox cofactors and chromophores in a plethora of flavoenzymes. Their versatility is an outcome of intrinsic molecular properties of the isoalloxazine ring modulated by the protein scaffold and surrounding solvent. Thus, an investigation of isolated flavins with high-level electronic-structure methods and with error assessment of the calculated properties will contribute to building better models of flavin reactivity. Here, we benchmarked ground-state properties such as electron affinity, gas-phase basicity, dipole moment, torsion energy, and tautomer stability for lumiflavins in all biologically relevant oxidation and charge states. Overall, multiconfigurational effects are small and chemical accuracy is achieved by coupled-cluster treatments of energetic properties. Augmented basis sets and extrapolations to the complete basis-set limit are necessary for consistent agreement with experimental energetics. Among DFT functionals tested, M06-2X shows the best performance for most properties, except gas-phase basicity, in which M06 and CAM-B3LYP perform better. Moreover, dipole moments of radical flavins show large deviations for all functionals studied. Tautomers with noncanonical protonation states are significantly populated at normal temperatures, adding to the complexity of modeling flavins. These results will guide future computational studies of flavoproteins and flavin chemistry by indicating the limitations of electronic-structure methodologies and the contributions of multiple tautomeric states.