Magnetic coupling constants and vibrational frequencies by extended broken symmetry approach with hybrid functionals

Comparative Density Functional Theory Study of Magnetic Exchange Couplings in Dinuclear Transition-Metal Complexes

Multicenter transition-metal complexes (MCTMs) with magnetically interacting ions have been proposed as components for information-processing devices and storage units. For any practical application of MCTMs as magnetic units, it is crucial to characterize their magnetic behavior, and in particular, the isotropic magnetic exchange coupling, J, between its magnetic centers. Due to the large size of typical MCTMs, density functional theory is the only practical electronic structure method for evaluating the J coupling. Here, we assess the accuracy of different density functional approximations for predicting the magnetic couplings of eight dinuclear transition-metal complexes, including five dimanganese, two dicopper, and one divanadium with known reliable experimental J couplings spanning from ferromagnetic to strong antiferromagnetic. The density functionals considered include global hybrid functionals which mix semilocal density functional approximations and exact exchange with a fixed admixing parameter, six local hybrid functionals where the admixing parameters are extended to be spatially dependent, the SCAN and r2SCAN meta-generalized gradient approximations (GGAs), and two widely used GGAs. We found that global hybrids tested in this work have a tendency to over-correct the error in magnetic coupling parameters from the Perdew-Burke-Ernzerhof (PBE) GGA as seen for manganese complexes. The performance of local hybrid density functionals shows no improvement in terms of bias and is scattered without a clear trend, suggesting that more efforts are needed for the extension from global to local hybrid density functionals for this particular property. The SCAN and r2SCAN meta-GGAs are found to perform as well as benchmark global hybrids on most tested complexes. We further analyze the charge density redistribution of meta-GGAs as well as global and local hybrid density functionals with respect to that of PBE, in connection to the self-interaction error or delocalization error.

Study of Self-Interaction Errors in Density Functional Calculations of Magnetic Exchange Coupling Constants Using Three Self-Interaction Correction Methods

We examine the role of self-interaction error (SIE) removal on the evaluation of magnetic exchange coupling constants. In particular, we analyze the effect of scaling down the self-interaction correction (SIC) for three nonempirical density functional approximations (DFAs) namely, the local spin density approximation, the Perdew-Burke-Ernzerhof generalized gradient approximation, and the recent SCAN family of meta-GGA functionals. To this end, we employ three one-electron SIC methods: Perdew-Zunger SIC [Perdew, J. P.; Zunger, A. Phys. Rev. B, 1981, 23, 5048.], the orbitalwise scaled SIC method [Vydrov, O. A. et al. J. Chem. Phys. 2006, 124, 094108.], and the recent local scaling method [Zope, R. R. et al. J. Chem. Phys. 2019, 151, 214108.]. We compute the magnetic exchange coupling constants using the spin projection and nonprojection approaches for sets of molecules composed of dinuclear and polynuclear H···He models, organic radical molecules, and chlorocuprate and compare these results against accurate theories and experiment. Our results show that for the systems that mainly consist of single-electron regions, PZSIC performs well, but for more complex organic systems and the chlorocuprates, an overcorrecting tendency of PZSIC combined with the DFAs utilized in this work is more pronounced, and in such cases, LSIC with kinetic energy density ratio performs better than PZSIC. Analysis of the results in terms of SIC corrections to the density and to the total energy shows that both density and energy correction are required to obtain an improved prediction of magnetic exchange couplings.

Analysis of the Geometric and Electronic Structure of Spin-Coupled Iron-Sulfur Dimers with Broken-Symmetry DFT: Implications for FeMoco

The open-shell electronic structure of iron–sulfur clusters presents considerable challenges to quantum chemistry, with the complex iron–molybdenum cofactor (FeMoco) of nitrogenase representing perhaps the ultimate challenge for either wavefunction or density functional theory. While broken-symmetry density functional theory has seen some success in describing the electronic structure of such cofactors, there is a large exchange–correlation functional dependence in calculations that is not fully understood. In this work, we present a geometric benchmarking test set, FeMoD11, of synthetic spin-coupled Fe–Fe and Mo–Fe dimers, with relevance to the molecular and electronic structure of the Mo-nitrogenase FeMo cofactor. The reference data consists of high-resolution crystal structures of metal dimer compounds in different oxidation states. Multiple density functionals are tested on their ability to reproduce the local geometry, specifically the Fe–Fe/Mo–Fe distance, for both antiferromagnetically coupled and ferromagnetically coupled dimers via the broken-symmetry approach. The metal–metal distance is revealed not only to be highly sensitive to the amount of exact exchange in the functional but also to the specific exchange and correlation functionals. For the antiferromagnetically coupled dimers, the calculated metal–metal distance correlates well with the covalency of the bridging metal–ligand bonds, as revealed via the corresponding orbital analysis, Hirshfeld S/Fe charges, and Fe–S Mayer bond order. Superexchange via bridging ligands is expected to be the dominant interaction in these dimers, and our results suggest that functionals that predict accurate Fe–Fe and Mo–Fe distances describe the overall metal–ligand covalency more accurately and in turn the superexchange of these systems. The best performing density functionals of the 16 tested for the FeMoD11 test set are revealed to be either the nonhybrid functionals r²SCAN and B97-D3 or hybrid functionals with 10–15% exact exchange: TPSSh and B3LYP*. These same four functionals are furthermore found to reproduce the high-resolution X-ray structure of FeMoco well according to quantum mechanics/molecular mechanics (QM/MM) calculations. Almost all nonhybrid functionals systematically underestimate Fe–Fe and Mo–Fe distances (with r²SCAN and B97-D3 being the sole exceptions), while hybrid functionals with >15% exact exchange (including range-separated hybrid functionals) overestimate them. The results overall suggest r²SCAN, B97-D3, TPSSh, and B3LYP* as accurate density functionals for describing the electronic structure of iron–sulfur clusters in general, including the complex FeMoco cluster of nitrogenase.

Auxiliary Atomic Relay Center Facilitates Enhanced Magnetic Couplings in Blatter's Radical

The recent accomplishments in obtaining strong ferromagnetic exchange interactions in organic diradicals have made the field quite fascinating and even more promising toward its technological applications. In this context, herein, we report a unique combination of remarkably strong ferromagnetic exchange interactions coupled with molecular rigidity, utilizing superstable Blatter's radical as a spin source. The planar analogues of the parent Blatter's radical obtained by annulation with a chalcogen coupled to nitronyl nitroxide (NN) are investigated using density functional theory along with the wave function-based multiconfigurational self-consistent field methods, for example, complete active space self-consistent field (CASSCF)-N-electron valence state perturbation theory (NEVPT2). The calculations reveal phenomenal modulation in exchange couplings upon annulation such that remarkably strong ferromagnetic interactions are realized especially for a certain class of the Blatter-NN diradicals. The modulation of spin-spin interactions is rationalized by variation in spin density distribution and molecular torsional angles. We demonstrate that annulation in OMMs opens an additional coupling pathway via auxiliary X-atom acting as the atomic relay center which strongly manipulates the magnitude of exchange couplings.

Spin-symmetrised structures and vibrational frequencies of iron-sulfur clusters

Calculations of relaxed geometries of multi-centre transition metal compounds are routinely carried out using Broken Symmetry Density Functional Theory. The resulting low-spin open shell electronic state is described by one single Slater determinant and is affected by spin contamination. To alleviate this symmetry breaking, the Extended Broken Symmetry (EBS) approach can be applied to complexes with an arbitrary number of local high-spin metal ions. The actual symmetry is therefore reconstructed through minimization of an effective Hamiltonian leading to a relaxed geometry consistent with the magnetic couplings. In the present work we extend the approach already introduced by [Chu et al., J. Chem. Theory Comput., 2017, 13, 4675] to the calculation of vibrational frequencies. As prototypes we have considered the iron-sulfur clusters Fe₂S₂Cl₄^2- and Fe₄S₄Cl₄. We have compared the results obtained for different spin states (high spin, broken symmetry and extended broken symmetry) and by using different DFT functionals (B3LYP, OPBE, BP, M06 and B2PLYP) and a post-HF method (SCS-MP2). The data have shown that for specific vibrational modes the EBS technique produces shifts up to 40 cm^-1 with respect to the routinely used Broken Symmetry approach, indicating that the use of a consistent spin-symmetrised state is a crucial ingredient for an accurate description of vibrational properties, as certified by the comparison with the experimental data for the Fe₂S₂Cl₄^2- cluster.

Energy decomposition analysis based on broken symmetry unrestricted density functional theory

In this paper, the generalized Kohn-Sham energy decomposition analysis (GKS-EDA) scheme is extended to molecular interactions in open shell singlet states, which is a challenge for many popular EDA methods due to the multireference character. Based on broken symmetry (BS) unrestricted density functional theory with a spin projection approximation, the extension scheme, named GKS-EDA(BS) in this paper, divides the total interaction energy into electrostatic, exchange-repulsion, polarization, correlation, and dispersion terms. Test examples include the pancake bond in the phenalenyl dimer, the ligand interactions in the Fe(ii)-porphyrin complexes, and the radical interactions in dehydrogenated guanine-cytosine base pairs and show that GKS-EDA(BS) is a practical EDA tool for open shell singlet systems.

Assessment of Double-Hybrid Density Functional Theory for Magnetic Exchange Coupling in Manganese Complexes

Molecular systems containing magnetically interacting (exchange-coupled) manganese ions are important in catalysis, biomimetic chemistry, and molecular magnetism. The reliable prediction of exchange coupling constants with quantum chemical methods is key for tracing the relationships between structure and magnetic properties in these systems. Density functional theory (DFT) in the broken-symmetry approach has been employed extensively for this purpose and hybrid functionals with moderate levels of Hartree–Fock exchange admixture have often been shown to perform adequately. Double-hybrid density functionals that introduce a second-order perturbational contribution to the Kohn–Sham energy are generally regarded as a superior approach for most molecular properties, but their performance remains unexplored for exchange-coupled manganese systems. An assessment of various double-hybrid functionals for the prediction of exchange coupling constants is presented here using a set of experimentally characterized dinuclear manganese complexes that cover a wide range of exchange coupling situations. Double-hybrid functionals perform more uniformly compared to conventional DFT methods, but they fail to deliver improved accuracy or reliability in the prediction of exchange coupling constants. Reparametrized double-hybrid density functionals (DHDFs) perform no better, and most often worse, than the original B2-PLYP double-hybrid method. All DHDFs are surpassed by the hybrid-meta-generalized gradient approximation (GGA) TPSSh functional. Possible directions for future methodological developments are discussed.

Spin, Valence, and Structural Isomerism in the S3 State of the Oxygen-Evolving Complex of Photosystem II as a Manifestation of Multimetallic Cooperativity

Photosynthetic water oxidation is catalyzed by a Mn₄CaO₅-cluster in photosystem II through an S-state cycle. Understanding the roles of heterogeneity in each S-state, as identified recently by the EPR spectroscopy, is very important to gain a complete description of the catalytic mechanism. We performed herein hybrid DFT calculations within the broken-symmetry formalism and associated analyses of Heisenberg spin models to study the electronic and spin structures of various isomeric structural motifs (hydroxo-oxo, oxyl-oxo, peroxo, and superoxo species) in the S₃ state. Our extensive study reveals several factors that affect the spin ground state: (1) (formal) Mn oxidation state; (2) metal-ligand covalency; (3) coordination geometry; and (4) structural change of the Mn cluster induced by alternations in Mn···Mn distances. Some combination of these effects could selectively stabilize/destabilize some spin states. We found that the high spin state ( S_total = 6) of the oxyl-oxo species can be causative for catalytic function, which manifests through mixing of the metal-ligand character in magnetic orbitals at relatively short O5···O6 distances (<2.0 Å) and long Mn_A···O5 distances (>2.0 Å). These results will serve as a basis to conceptually identify and rationalize the physicochemical synergisms that can be evoked by the unique "distorted chair" topology of the cluster through cooperative Jahn-Teller effects on multimetallic centers.

Meeting the Challenge of Magnetic Coupling in a Triply-Bridged Chromium Dimer: Complementary Broken-Symmetry Density Functional Theory and Multireference Density Matrix Renormalization Group Perspectives

Face-sharing octahedral dinuclear Cr(III) compounds with d³–d³ electronic configurations represent nontrivial examples of electronic complexity, posing particular challenges for theoretical and computational studies. A tris-hydroxy-bridged Cr(III)–Cr(III) system has proven to be a richly rewarding target for studies of magnetism and electron paramagnetic resonance spectroscopy. It was also reported to be a peculiarly difficult system to treat with density functional theory (DFT). In this work the magnetic coupling problem for this dimer is approached with broken-symmetry (BS)-DFT and multireference calculations that utilize the density matrix renormalization group (DMRG) to handle full-valence active spaces. BS-DFT is shown to recover the correct ordering and energy spacing of Heisenberg spin states if used in conjunction with appropriate spin projection procedures, albeit with pronounced functional sensitivity. The contrasting conclusions of previous studies are traced to incorrect inclusion of electronically excited configurations. Analysis of the direct and differential overlap of corresponding orbital pairs from the BS-DFT solution indicates that metal–metal through-space interaction is the dominant contributor to antiferromagnetic coupling. At the DFT level a procedure that utilizes pseudopotential substitution is demonstrated that allows evaluation of the direct exchange vs superexchange contributions. A complementary description is obtained with DMRG-SCF calculations that enable state-averaged CASSCF calculations with both metal and bridge orbitals in the active space. A localized orbital subspace analysis supports the DFT conclusions that in contrast to doubly bridged isoelectronic analogues, antiferromagnetic coupling in the chromium dimer arises primarily from direct metal–metal interaction but is significantly enhanced by ligand-mediated superexchange.

Exchange Coupling Interactions from the Density Matrix Renormalization Group and N-Electron Valence Perturbation Theory: Application to a Biomimetic Mixed-Valence Manganese Complex

The accurate description of magnetic level energetics in oligonuclear exchange-coupled transition-metal complexes remains a formidable challenge for quantum chemistry. The density matrix renormalization group (DMRG) brings such systems for the first time easily within reach of multireference wave function methods by enabling the use of unprecedentedly large active spaces. But does this guarantee systematic improvement in predictive ability and, if so, under which conditions? We identify operational parameters in the use of DMRG using as a test system an experimentally characterized mixed-valence bis-μ-oxo/μ-acetato Mn(III,IV) dimer, a model for the oxygen-evolving complex of photosystem II. A complete active space of all metal 3d and bridge 2p orbitals proved to be the smallest meaningful starting point; this is readily accessible with DMRG and greatly improves on the unrealistic metal-only configuration interaction or complete active space self-consistent field (CASSCF) values. Orbital optimization is critical for stabilizing the antiferromagnetic state, while a state-averaged approach over all spin states involved is required to avoid artificial deviations from isotropic behavior that are associated with state-specific calculations. Selective inclusion of localized orbital subspaces enables probing the relative contributions of different ligands and distinct superexchange pathways. Overall, however, full-valence DMRG-CASSCF calculations fall short of providing a quantitative description of the exchange coupling owing to insufficient recovery of dynamic correlation. Quantitatively accurate results can be achieved through a DMRG implementation of second order N-electron valence perturbation theory (NEVPT2) in conjunction with a full-valence metal and ligand active space. Perspectives for future applications of DMRG-CASSCF/NEVPT2 to exchange coupling in oligonuclear clusters are discussed.

Effects of Static Correlation between Spin Centers in Multicenter Transition Metal Complexes

Multicenter transition metal complexes are the key moieties of many processes in chemistry, biochemistry, and materials science such as in the active sites of enzymes, molecular catalysts, and biological electron carriers. Their electronic structure, often characterized by high-spin-polarized metal sites, is a challenge for theoretical chemists because of their high degree of dynamical and static correlation. Static correlation is necessary both for the appropriate description of the metal-ligand bonding and for a correct description of the multideterminant character arising from the magnetic interactions between spin centers. Density functional theory (DFT) is usually applied using a single-determinant broken-symmetry state that is lacking the correct spin symmetry when the ground state has total low-spin character. To alleviate this drawback, we use the extended broken-symmetry (EBS) approach to derive approximate ground-state energies and, for the first time, forces for the correctly symmetric ground state of an arbitrary number of spin centers within the framework of the Heisenberg-Dirac-van Vleck Hamiltonian. Remarkably, the proposed procedure supplies relaxed geometries that are fully consistent with the calculated J-coupling constants. We apply the method to investigate the relaxed geometrical structure of the low-spin ground state of iron-sulfur clusters with two, three, and four iron centers. We observed significant differences in both geometrical parameters and coupling constant J between the symmetrized ground state, the high-spin, and the broken-symmetry optimized structures. These changes are often comparable with the differences observed by using different functionals, and the use of EBS always improves the description of the studied systems. It will be therefore important to include it in any DFT attempt to quantitatively describe multicenter transition metal complexes in the future.

Distinguishing the Protonation State of the Histidine Ligand to the Oxidized Iron-Sulfur Cluster from the MitoNEET Family of Proteins

The iron-sulfur cluster located in the recently discovered human mitoNEET protein (and related proteins) is structurally similar to the more well-known ferredoxin and Rieske clusters. Although its biological function is uncertain, the iron-sulfur cluster in mitoNEET has been proposed to undergo proton-coupled electron transfer involving the histidine ligand to the cluster. The cluster is also released from the protein at low pH. This contribution reports density functional calculations to model the structures, vibrations, and Heisenberg coupling constants (J) for high-spin (HS), broken symmetry (BS) singlet, and extended broken symmetry (EBS) singlet states of the oxidized iron-sulfur cluster from mitoNEET. This work suggests that J values or ¹⁵ N isotopic frequency shifts may provide methods for determining experimentally whether the histidine ligand to the oxidized iron-sulfur cluster in human mitoNEET and mitoNEET-related proteins is protonated or deprotonated.

Distinguishing Protonation States of Histidine Ligands to the Oxidized Rieske Iron-Sulfur Cluster through (15) N Vibrational Frequency Shifts

The Rieske [2Fe-2S] cluster is a vital component of many oxidoreductases, including mitochondrial cytochrome bc1; its chloroplast equivalent, cytochrome b6f; one class of dioxygenases; and arsenite oxidase. The Rieske cluster acts as an electron shuttle and its reduction is believed to couple with protonation of one of the cluster's His ligands. In cytochromes bc1 and b6f, for example, the Rieske cluster acts as the first electron acceptor in a modified Q cycle. The protonation states of the cluster's His ligands determine its ability to accept a proton and possibly an electron through a hydrogen bond to the electron carrier, ubiquinol. Experimental determination of the protonation states of a Rieske cluster's two His ligands by NMR spectroscopy is difficult, due to the close proximity of the two paramagnetic iron atoms of the cluster. Therefore, this work reports density functional calculations and proposes that difference vibrational spectroscopy with (15) N isotopic substitution may be used to assign the protonation states of the His ligands of the oxidized Rieske [2Fe-2S] complex.

Aqueous Fe2S2 cluster: structure, magnetic coupling, and hydration behaviour from Hubbard U density functional theory

We present a DFT + U investigation of the all-ferrous Fe2S2 cluster in aqueous solution. We determine the value of U by tuning the geometry of the cluster in the gas-phase to that obtained by the highly accurate CCSD(T) method. When the optimised value of U is employed for the aqueous Fe2S2 cluster (Fe2S2(aq)), the resulting geometry agrees well with the X-ray diffraction structure, while the magnetic coupling is in line with the estimate from Mössbauer data. Molecular dynamics trajectories predict Fe2S2(aq) to be stable in water, regardless of the introduction of U. However, significant differences arise in the geometry, hydration, and exchange constant of the solvated clusters.

Magnetostructural dynamics of Rieske versus ferredoxin iron-sulfur cofactors

The local chemical environment of the [2Fe-2S] cofactor hosted by ferredoxin and Rieske-type proteins is fundamentally different due to the presence of distinct ligands at the two iron centers in the case of Rieske proteins, whereas they are identical in ferredoxins. This renders Rieske [2Fe-2S] cores chemically asymmetric and results in more complex vibrational spectra as compared to ferredoxin. Likewise, one would expect other properties, for instance the dynamics of the magnetic exchange coupling constant J, to be also more complex. Applying ab initio molecular dynamics using our recently introduced spin-constrained two-determinant extended broken symmetry (CEBS) approach to Rieske and ferredoxin model complexes at 300 K, we extract the molecular fluctuations and the resulting magnetostructural cross-correlations involving the antiferromagnetic exchange interaction J(t). This analysis demonstrates that the details of the magnetostructural dynamics are indeed distinctly different for Rieske and ferredoxin cofactors, while the time averages of 〈J〉 are shown to be essentially identical. In particular, the frequency window between about 200 and 350 cm(-1), is a "fingerprint region" that allows one to distinguish chemically asymmetric from symmetric cofactors and thus Rieske proteins from ferredoxins.

The iron-sulfur core in Rieske proteins is not symmetric

At variance with ferredoxins, Rieske-type proteins contain a chemically asymmetric iron-sulfur cluster. Nevertheless, X-ray crystallography apparently finds their [2Fe-2S] cores to be structurally symmetric or very close to symmetric (i.e. the four iron-sulfur bonds in the [2Fe-2S] core are equidistant). Using a combination of advanced density-based approaches, including finite-temperature molecular dynamics to access thermal fluctuations and free-energy profiles, in conjunction with correlated wavefunction-based methods we clearly predict an asymmetric core structure. This reveals a fundamental and intrinsic difference within the iron-sulfur clusters hosted by Rieske proteins and ferredoxins and thus opens up a new dimension for the ongoing efforts in understanding the role of Rieske-type [2Fe-2S] cluster in electron transfer processes that occur in almost all biological systems.

On the magnetic and spectroscopic properties of high-valent Mn3CaO4 cubanes as structural units of natural and artificial water-oxidizing catalysts

The Mn(IV)3CaO4 cubane is a structural motif present in the oxygen-evolving complex (OEC) of photosystem II and in water-oxidizing Mn/Ca layered oxides. This work investigates the magnetic and spectroscopic properties of two recently synthesized complexes and a series of idealized models that incorporate this structural unit. Magnetic interactions, accessible spin states, and (55)Mn isotropic hyperfine couplings are computed with quantum chemical methods and form the basis for structure-property correlations. Additionally, the effects of oxo-bridge protonation and one-electron reduction are examined. The calculated properties are found to be in excellent agreement with available experimental data. It is established that all synthetic and model Mn(IV)3CaO4 cubane complexes have the same high-spin S = (9)/2 ground state. The magnetic coupling conditions under which different ground spin states can be accessed are determined. Substitution of Mn(IV) magnetic centers by diamagnetic ions [e.g., Ge(IV)] allows one to "switch off" specific spin sites in order to examine the magnetic orbitals along individual Mn-Mn exchange pathways, which confirms the predominance of ferromagnetic interactions within the cubane framework. The span of the Heisenberg spin ladder is found to correlate inversely with the number of protonated oxo bridges. Energetic comparisons for protonated models show that the tris-μ-oxo bridge connecting only Mn ions in the cubane has the lowest proton affinity and that the average relaxation energy per additional proton is on the order of 18 kcal·mol(-1), thus making access to ground states other than the high-spin S = (9)/2 state in these cubanes unlikely. The relevance of these cubanes for the OEC and synthetic oxides is discussed.