Magnetostructural Dynamics with the Extended Broken Symmetry Formalism: Antiferromagnetic [2Fe-2S] Complexes

Experimental and computational study of the exchange interaction between the V(III) centers in the vanadium-cyclal dimer

[V2(HCyclal)2] is prepared by controlled oxidation of vanadium nanoparticles at 50 °C in toluene. The V(0) nanoparticles are synthesized in THF by reduction of VCl3 with lithium naphthalenide. They exhibit very small particle sizes of 1.2 ± 0.2 nm and a high reactivity (e.g. with air or water). By reaction of V(0) nanoparticles with the azacrown ether H4Cyclal, [V2(HCyclal)2] is obtained with deep green crystals and high yield. The title compound exhibits a V(III) dimer (V⋯V: 304.1(1) pm) with two deprotonated [HCyclal]3- ligands as anions. V(0) nanoparticles as well as the sole coordination of V(III) by a crown ether as the ligand and nitrogen as sole coordinating atom are shown for the first time. Magnetic measurements and computational results point to antiferromagnetic coupling within the V(III) couple, establishing an antiferromagnetic spin S = 1 dimer with the magnetic susceptibility determined by the thermal population of the total spin ranging from ST = 0 to ST = 2.

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.

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.

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.

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 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.

Magnetostructural Dynamics from Hubbard-U Corrected Spin-Projection: [2Fe-2S] Complex in Ferredoxin

A Hubbard-corrected spin-projected two-determinant approach, EBS+Uscf, is introduced to treat low-spin ground states of antiferromagnetically coupled transition metal complexes. In addition to providing access to total energies, forces, and ab initio simulations, it allows one to readily compute Heisenberg's exchange coupling J(t) on the fly. By studying the binuclear [2Fe-2S] cofactor in a metalloprotein, Anabaena Fd, within this consistent nonempirical procedure in combination with a QM/MM framework, it is illustrated that spin-projection, self-interaction corrections, thermal fluctuations, and protein matrix shifts are crucial in obtaining ⟨J⟩ close to the experiment.

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.

Binuclear ruthenium complexes of a neutral radical bridging ligand. A new "spin" on mixed valency

The electronic structures of (LX)2Ru(Vd)Ru(LX)2 complexes (Vd = 1,5-diisopropyl-3-(4,6-dimethyl-2-pyrimidinyl)-6-oxoverdazyl radical; LX = acac (acetylacetonate) or hfac (hexafluoroacetylacetonate)) in multiple charge states have been investigated experimentally and computationally. The main focus was to probe the consequences of the interplay between the ruthenium ions and the redox-active verdazyl ligand for possible mixed-valent behavior. Cyclic voltammetry studies reveal one reversible reduction and one reversible oxidation process for both complexes; in addition the acac-based derivative possesses a second reversible oxidation. Analysis of a collection of experimental (X-ray structures, EPR, electronic spectra) and computational (TD-DFT (PCM)) data reveal that the ruthenium ancillary ligands (acac vs hfac) have dramatic consequences for the electronic structures of the complexes in all charge states studied. In the hfac series, the neutral complex is best regarded as a binuclear Ru(II) species bridged by a neutral radical ligand. Reduction to give the anionic complex takes place on the verdazyl ligand, whereas oxidation to the cation (a closed shell species) is shared between Vd and ruthenium. For the acac-based complexes, the neutral species is most accurately represented as a Ru(II)/Ru(III) mixed valent complex containing a bridging verdazyl anion, though some bis(Ru(II))-neutral radical character remains. The monocation complex contains a significant contribution from a "broken symmetry" singlet diradical structure, best represented as a bis-Ru(III) system with an anionic ligand, with significant spin coupling of the two Ru(III) centers via the Vd(-1) ligand (calculated J = -218 cm(-1)). The dication, a spin doublet, consists of two Ru(III) ions linked (and antiferromagnetically coupled) to the neutral radical ligand. Computed net σ- and π-back-donation, spin densities, and orbital populations are provided. Time dependent DFT is used to predict the optical spectra and assign experimental data.

Revealing the magnetostructural dynamics of [2Fe-2S] ferredoxins from reduced-dimensionality analysis of antiferromagnetic exchange coupling fluctuations

Metalloproteins are biomolecular hybrids composed of an "inorganic core" embedded in a "bioorganic matrix". Cofactors typically contain transition metal clusters with complex electronic structure whereas the protein host undergoes dynamics on many length and time scales. This renders computational studies of spectroscopic properties challenging, in particular, when magnetic interactions are involved. In the present study we introduce a simplified description of the antiferromagnetic exchange coupling J in reduced dimensionality which allows one to study magnetostructural dynamics of [2Fe-2S] type iron-sulfur proteins in their oxidized form by molecular dynamics. It is demonstrated that parametrization in terms of a 2D J-surface faithfully reproduces the rigorous results both in vacuo and in Anabaena ferredoxin. In particular, we present a parametrization which relies on a spin-projected density functional approach based on two Kohn-Sham determinants corrected for self-interaction via a self-consistent linear-response Hubbard-U technique. This yields an average J for Anabaena Fd in close agreement with experimental in vitro results without any specific adjustment or fitting. The analytical J-surface can be used for [2Fe-2S] proteins in their oxidized form in general and the idea can be extended to other metalloproteins as well as to other spectroscopic properties.