Magnetic Coupling in a Tris-hydroxo-Bridged Chromium Dimer Occurs through Ligand Mediated Superexchange in Conjunction with Through-Space Coupling

Physics and Chemistry of Two-Dimensional Triangulene-Based Lattices

ConspectusTriangulene (TRI) and its heterotriangulene (HT) derivatives are planar, triangle-shaped molecules that, via suitable coupling reactions, can form extended organic two-dimensional (2D) crystal (O2DC) structures. While TRI is a diradical, HTs are either closed-shell molecules or monoradicals which can be stabilized in their cationic form.Triangulene-based O2DCs have a characteristic honeycomb-kagome lattice. This structure gives rise to four characteristic electronic bands: two of them form Dirac points, while the other two are flat and sandwich the Dirac bands. Functionalization and heteroatoms are suitable means to engineer this band structure. Heteroatoms like boron and nitrogen shift the Fermi level upward and downward, respectively, while bridging groups and functionalized triangulene edges can introduce a dispersion to the flat bands.The stable backbone architecture makes 2D HT-polymers ideal for photoelectrochemical applications: (i) bridge functionalization can tune the band gap and maximize absorption, (ii) the choice of the center atom (B or N) controls the band occupation and shifts the Fermi level with respect to vacuum, allowing in some cases for overpotential-free photon-driven surface reactions, and (iii) the large surface area allows for a high flux of educts and products.The spin polarization in TRI and in open-shell HTs is maintained when linking them to dimers or extended frameworks with direct coupling or more elaborate bridging groups (acetylene, diacetylene, and phenyl). The dimers have a high spin-polarization energy and some of them are strongly magnetically coupled, resulting in stable high-spin or broken-symmetry (BS) low-spin systems. As O2DCs, some systems become antiferromagnetic Mott insulators with large band gaps, while others show Stoner ferromagnetism, maintaining the characteristic honeycomb-kagome bands but shifting the opposite spin-polarized bands to different energies. For O2DCs based on aza- and boratriangulene (monoradicals as building blocks), the Fermi level is shifted to a spin-polarized Dirac point, and the systems have a Curie temperature of about 250 K. For half-filled (all-carbon) systems, the Ovchinnikov rule or, equivalently, Lieb's theorem, is sufficient to predict the magnetic ordering of the systems, while the non-half-filled systems (i.e., those with heteroatoms) obey the more involved Goodenough-Kanamori rule to interpret the magnetism on the grounds of fundamental electronic interactions.There remain challenges in experiment and in theory to advance the field of triangulene-based O2DCs: Coupling reactions beyond surface chemistry have to be developed to allow for highly ordered, extended crystals. Multilayer structures, which are unexplored to date, will be inevitable in alternative synthesis approaches. The predictive power of density-functional theory (DFT) within state-of-the-art functionals is limited for the description of magnetic couplings in these systems due to the apparent multireference character and the large spatial extension of the spin centers.

Automatic State Interaction with Large Localized Active Spaces for Multimetallic Systems

The localized active space self-consistent field method factorizes a complete active space wave function into an antisymmetrized product of localized active space wave function fragments. Correlation between fragments is then reintroduced through localized active space state interaction (LASSI), in which the Hamiltonian is diagonalized in a model space of LAS states. However, the optimal procedure for defining the LAS fragments and LASSI model space is unknown. We here present an automated framework to explore systematically convergent sets of model spaces, which we call LASSI[r, q]. This method requires the user to select only r, the number of electron hops from one fragment to another, and q, the number of fragment basis functions per Hilbert space, which converges to CASCI in the limit of r, q → ∞. Numerical tests of this method on the trimetal oxo-centered complexes [Fe(III)Al(III)Fe(II)(μ3-O)(HCOO)6] and [Fe(III)2Fe(II)(μ3-O)(HCOO)6] show efficient convergence to the CASCI limit with 4-10 orders of magnitude fewer states than CASCI.

Voltage-induced modulation of the magnetic exchange in binuclear Fe(III) complex deposited on Au(111) surface

We present a complete computational study devoted to the deposition of a magnetic binuclear complex on a metallic surface, aimed to obtain insight into the interaction of magnetically coupled complexes with their supporting substrates, as well as their response to external electrical stimuli applied through a surface-molecule-STM molecular junction-like architecture. Our results not only show that the deposition is favorable in two of the four studied orientations, but also, that the magnetic coupling is only slightly perturbed once the complex is adsorbed. We observe that the effects of the applied bias voltage on the magnetic coupling strongly depend on the molecule orientation with respect to the surface and the voltage polarity. Further analysis shows that this behavior is attributable to the stabilization/destabilization of the d-type singly occupied orbitals of the iron centers, reinforced by the strong local electric fields and induced charge densities only present in certain orientations of the deposited molecule and applied voltage polarity.

Water-Mediated Charge Transfer and Electron Localization in a Co3Fe2 Cyanide-Bridged Trigonal Bipyramidal Complex

The effects of temperature and chemical environment on a pentanuclear cyanide-bridged, trigonal bipyramidal molecular paramagnet have been investigated. Using element- and oxidation state-specific near-ambient pressure X-ray photoemission spectroscopy (NAP-XPS) to probe charge transfer and second order, nonlinear vibrational spectroscopy, which is sensitive to symmetry changes based on charge (de)localization coupled with DFT, a detailed picture of environmental effects on charge-transfer-induced spin transitions is presented. The molecular cluster, Co3Fe2(tmphen)6(μ-CN)6(t-CN)6, abbrev. Co3Fe2, shows changes in electronic behavior depending on the chemical environment. NAP-XPS shows that temperature changes induce a metal-to-metal charge transfer (MMCT) in Co3Fe2 between a Co and Fe center, while cycling between ultrahigh vacuum and 2 mbar of water at constant temperature causes oxidation state changes not fully captured by the MMCT picture. Sum frequency generation vibrational spectroscopy (SFG-VS) probes the role of the cyanide ligand, which controls the electron (de)localization via the superexchange coupling. Spectral shifts and intensity changes indicate a change from a charge delocalized, Robin-Day class II/III high spin state to a charge-localized, class I low spin state consistent with DFT. In the presence of a H-bonding solvent, the complex adopts a localized electronic structure, while removal of the solvent delocalizes the charges and drives an MMCT. This change in Robin-Day classification of the complex as a function of chemical environment results in reversible switching of the dipole moment, analogous to molecular multiferroics. These results illustrate the important role of the chemical environment and solvation on underlying charge and spin transitions in this and related complexes.

Multiconfiguration Pair-Density Functional Theory for Vertical Excitation Energies in Actinide Molecules

Modeling actinides with electronic structure theories is challenging because these systems present a strong ligand field and metal-ligand covalency. We systematically investigate the effectiveness of pair-density functional theory (PDFT) for the calculation of vertical excitation energies in An(III), [AnIIICl6]3-, and [AnVIO2]2+ (An = U, Np, Pu, and Am). We compare the performance of PDFT, hybrid PDFT, and multistate PDFT with traditional active-space methods followed by perturbation theory, like multistate CASPT2, and with experimental data. Overall, multistate PDFT gives quantitative agreement with multistate CASPT2 at a significantly reduced computational cost.

Galván I, Alavi A, Aleotti F, Aquilante F, Autschbach J, Avagliano D, Baiardi A, Bao JJ, Battaglia S, Birnoschi L, Blanco-González A, Bokarev SI, Broer R, Cacciari R, Calio PB, Carlson RK, Carvalho Couto R, Cerdán L, Chibotaru LF, Chilton NF, Church JR, Conti I, Coriani S, Cuéllar-Zuquin J, Daoud RE, Dattani N, Decleva P, de Graaf C, Delcey M, De Vico L, Dobrautz W, Dong SS, Feng R, Ferré N, Filatov(Gulak) M, Gagliardi L, Garavelli M, González L, Guan Y, Guo M, Hennefarth MR, Hermes MR, Hoyer CE, Huix-Rotllant M, Jaiswal VK, Kaiser A, Kaliakin DS, Khamesian M, King DS, Kochetov V, Krośnicki M, Kumaar AA, Larsson ED, Lehtola S, Lepetit MB, Lischka H, López Ríos P, Lundberg M, Ma D, Mai S, Marquetand P, Merritt ICD, Montorsi F, Mörchen M, Nenov A, Nguyen VHA, Nishimoto Y, Oakley MS, Olivucci M, Oppel M, Padula D, Pandharkar R, Phung QM, Plasser F, Raggi G, Rebolini E, Reiher M, Rivalta I, Roca-Sanjuán D, Romig T, Safari AA, Sánchez-Mansilla A, Sand AM, Schapiro I, Scott TR, Segarra-Martí J, Segatta F, Sergentu DC, Sharma P, Shepard R, Shu Y, Staab JK, Straatsma TP, Sørensen LK, Tenorio BNC, Truhlar DG, Ungur L, Vacher M, Veryazov V, Voß TA, Weser O, Wu D, Yang X, Yarkony D, Zhou C, Zobel JP, Lindh R. The OpenMolcas Web: A Community-Driven Approach to Advancing Computational Chemistry

The developments of the open-source OpenMolcas chemistry software environment since spring 2020 are described, with a focus on novel functionalities accessible in the stable branch of the package or via interfaces with other packages. These developments span a wide range of topics in computational chemistry and are presented in thematic sections: electronic structure theory, electronic spectroscopy simulations, analytic gradients and molecular structure optimizations, ab initio molecular dynamics, and other new features. This report offers an overview of the chemical phenomena and processes OpenMolcas can address, while showing that OpenMolcas is an attractive platform for state-of-the-art atomistic computer simulations.

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.

Natural orbitals and two-particle correlators as tools for the analysis of effective exchange couplings in solids

Using generalizations of spin-averaged natural orbitals and two-particle charge correlators for solids, we investigate the electronic structure of antiferromagnetic transition-metal oxides with a fully self-consistent, imaginary-time GW method. Our findings disagree with the Goodenough-Kanamori (GK) rules that are commonly used for the qualitative interpretation of such solids. First, we found a strong dependence of the natural orbital occupancies on momenta, contradicting GK assumptions. Second, along the momentum path, the character of natural orbitals changes. In particular, the contributions of oxygen 2s orbitals are important, which has not been considered in the GK rules. To analyze the influence of the electronic correlation on the values of effective exchange coupling constants, we use both natural orbitals and two-particle correlators and show that electronic screening modulates the degree of superexchange by stabilizing the charge-transfer contributions, which greatly affects these coupling constants. Finally, we give a set of predictions and recommendations regarding the use of density functional, Green's function, and wave-function methods for evaluating effective magnetic couplings in molecules and solids.

Synthesis and Properties of Ba6Fe2Te3S7, with an Fe Dimer in a Magnetic Singlet State

A new quaternary sulfide telluride, Ba6Fe2Te3S7, was synthesized by a solid-state reaction, and its crystal structure is novel. X-ray diffraction data on powder and single crystals reveal an orthorhombic lattice with a = 9.7543(3) Å, b = 18.2766(6) Å, and c = 12.0549(4) Å, and the noncentrosymmetric space group Cmc21 (No. 36). The properties of the compound were studied by magnetic susceptibility investigations, specific heat measurements, Mössbauer spectroscopy, and density functional theory calculations. Assuming Ba2+ and, as verified by the Mössbauer spectra, Fe3+, the charge balance requires the presence of a polytelluride, suggested to be a straight-chain [Te34-] polyanion. Further, the crystal structure contains [Fe2S7]8- dimers of two vertex-sharing tetrahedra, with a nearly linear Fe-S-Fe atom arrangement. The dimer exhibits antiferromagnetic coupling, with a coupling constant J = -10.5 meV (H = -2JS1S2) and S = 5/2, resulting in a spin singlet ground state. The interdimer magnetic interaction is so weak that the magnetic dimers can be treated as individuals.

Predicting Magnetic Coupling and Spin-Polarization Energy in Triangulene Analogues

Triangulene and its analogue metal-free magnetic systems have garnered increasing attention since their discovery. Predicting the magnetic coupling and spin-polarization energy with quantitative accuracy is beyond the predictive power of today's density functional theory (DFT) due to their intrinsic multireference character. Herein, we create a benchmark dataset of 25 magnetic systems with nonlocal spin densities, including the triangulene monomer, dimer, and their analogues. We calculate the magnetic coupling (J) and spin-polarization energy (ΔE_spin) of these systems using complete active space self-consistent field (CASSCF) and coupled-cluster methods as high-quality reference values. This reference data is then used to benchmark 22 DFT functionals commonly used in material science. Our results show that, while some functionals consistently correctly predict the qualitative character of the ground state, achieving quantitative accuracy with small relative errors is currently not feasible. PBE0, M06-2X, and MN15 are predicting the correct electronic ground state for all systems investigated here and also have the lowest mean absolute error for predicting both ΔE_spin (0.34, 0.32, and 0.31 eV) and J (11.74, 12.66, and 10.64 meV). They may therefore also serve as starting points for higher-level methods such as the GW or the random phase approximation. As other functionals fail for the prediction of the ground state, they cannot be recommended for metal-free magnetic systems.

Revisiting magnetic exchange couplings in heterodinuclear complexes through the decomposition method in KS-DFT

Providing tools to understand the physical mechanisms governing magnetic properties in transition metal-based compounds is still of great interest. Here, the magnetic exchange coupling in a series of heterodinuclear complexes is investigated by means of the decomposition method. This work presents the first application of the decomposition method to systems where magnetic centres may bear more than one unpaired electron. By decomposing the coupling into three physical contributions (direct exchange, kinetic exchange, and spin polarisation), we provide numerical arguments to confirm or infirm the rationalisation allowed by the conceptual analysis of the magnetic d orbitals. We also take advantage of the recently proposed generalisation of the method [David et al., J. Chem. Theory Comput., 2023, 19, 157] to get more insights into the underlying mechanisms by disentangling the coupling between centres into its electron-electron interactions.

Consistent Evaluation of Magnetic Exchange Couplings in Multicenter Compounds in KS-DFT: The Recomposition Method

The use of broken-symmetry calculations in Kohn-Sham density functional theory has offered an affordable route to study magnetic exchange couplings in transition-metal-based compounds. However, computing this property in compounds exhibiting several couplings is still challenging and especially due to the difficulties to overcome the well-known problem of spin contamination. Here, we present a new and general method to compute magnetic exchange couplings in systems featuring several spin sites. To provide a consistent spin decontamination of J values, our strategy exploits the decomposition method of the magnetic exchange coupling proposed by Coulaud et al. and generalizes our previous work on diradical compounds where the overall magnetic exchange coupling is defined as the sum of its three main and properly extracted physical contributions (direct exchange, kinetic exchange, and spin polarization). In this aim, the generalized extraction of all contributions is presented to systems with multiple spin sites bearing one unpaired electron. This is done by proposing a new paradigm to treat the kinetic exchange contribution, which proceeds through monorelaxations of the magnetic orbitals. This method, so-called the recomposition method, is applied to a compound featuring three Cu(II) ions with a linear arrangement and to a recently synthesized complex containing a Cu₄O₄ cubane unit presenting an unusual magnetic behavior.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Localized Active Space-State Interaction: a Multireference Method for Chemical Insight

Multireference electronic structure methods, like the complete active space (CAS) self-consistent field model, have long been used to characterize chemically interesting processes. Important work has been done in recent years to develop modifications having a lower computational cost than CAS, but typically these methods offer no more chemical insight than that from the CAS solution being approximated. In this paper, we present the localized active space-state interaction (LASSI) method that can be used not only to lower the intrinsic cost of the multireference calculation but also to improve interpretability. The localized active space (LAS) approach utilizes the local nature of the electron-electron correlation to express a composite wave function as an antisymmetrized product of unentangled wave functions in local active subspaces. LASSI then uses these LAS states as a basis from which to express complete molecular wave functions. This not only makes the molecular wave function more compact but also permits flexibility in choosing those states to be included in the basis. Such selective inclusion of states translates to the selective inclusion of specific types of interactions, thereby allowing a quantitative analysis of these interactions. We demonstrate the use of LASSI to study charge migration and spin-flip excitations in multireference organic molecules. We also compute the J coupling parameter for a bimetallic compound using various LAS bases to construct the Hamiltonian to provide insights into the coupling mechanism.

Computational design of magnetic molecules and their environment using quantum chemistry, machine learning and multiscale simulations

Having served as a playground for fundamental studies on the physics of d and f electrons for almost a century, magnetic molecules are now becoming increasingly important for technological applications, such as magnetic resonance, data storage, spintronics and quantum information. All of these applications require the preservation and control of spins in time, an ability hampered by the interaction with the environment, namely with other spins, conduction electrons, molecular vibrations and electromagnetic fields. Thus, the design of a novel magnetic molecule with tailored properties is a formidable task, which does not only concern its electronic structures but also calls for a deep understanding of the interaction among all the degrees of freedom at play. This Review describes how state-of-the-art ab initio computational methods, combined with data-driven approaches to materials modelling, can be integrated into a fully multiscale strategy capable of defining design rules for magnetic molecules.

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.

The Effect of Geometry, Spin, and Orbital Optimization in Achieving Accurate, Correlated Results for Iron-Sulfur Cubanes

Iron-sulfur clusters comprise an important functional motif in the catalytic centers of biological systems, capable of enabling important chemical transformations at ambient conditions. This remarkable capability derives from a notoriously complex electronic structure that is characterized by a high density of states that is sensitive to geometric changes. The spectral sensitivity to subtle geometric changes has received little attention from correlated, large active space calculations, owing partly to the exceptional computational complexity for treating these large and correlated systems accurately. To provide insight into this aspect, we report the first Complete Active Space Self Consistent Field (CASSCF) calculations for different geometries of the [Fe(II/III)₄S₄(SMe)₄]^-2 clusters using two complementary, correlated solvers: spin-pure Adaptive Sampling Configuration Interaction (ASCI) and Density Matrix Renormalization Group (DMRG). We find that the previously established picture of a double-exchange driven magnetic structure, with minute energy gaps (<1 mHa) between consecutive spin states, has a weak dependence on the underlying geometry. However, the spin gap between the singlet and the spin state 2S + 1 = 19, corresponding to a maximal number of Fe-d electrons being unpaired and of parallel spin, is strongly geometry dependent, changing by a factor of 3 upon slight deformations that are still within biologically relevant parameters. The CASSCF orbital optimization procedure, using active spaces as large as 86 electrons in 52 orbitals, was found to reduce this gap compared to typical mean-field orbital approaches. Our results show the need for performing large active space calculations to unveil the challenging electronic structure of these complex catalytic centers and should serve as accurate starting points for fully correlated treatments upon inclusion of dynamical correlation outside the active space.

Electronic structure of strongly correlated systems: recent developments in multiconfiguration pair-density functional theory and multiconfiguration nonclassical-energy functional theory

Strong electron correlation plays an important role in transition-metal and heavy-metal chemistry, magnetic molecules, bond breaking, biradicals, excited states, and many functional materials, but it provides a significant challenge for modern electronic structure theory. The treatment of strongly correlated systems usually requires a multireference method to adequately describe spin densities and near-degeneracy correlation. However, quantitative computation of dynamic correlation with multireference wave functions is often difficult or impractical. Multiconfiguration pair-density functional theory (MC-PDFT) provides a way to blend multiconfiguration wave function theory and density functional theory to quantitatively treat both near-degeneracy correlation and dynamic correlation in strongly correlated systems; it is more affordable than multireference perturbation theory, multireference configuration interaction, or multireference coupled cluster theory and more accurate for many properties than Kohn–Sham density functional theory. This perspective article provides a brief introduction to strongly correlated systems and previously reviewed progress on MC-PDFT followed by a discussion of several recent developments and applications of MC-PDFT and related methods, including localized-active-space MC-PDFT, generalized active-space MC-PDFT, density-matrix-renormalization-group MC-PDFT, hybrid MC-PDFT, multistate MC-PDFT, spin–orbit coupling, analytic gradients, and dipole moments. We also review the more recently introduced multiconfiguration nonclassical-energy functional theory (MC-NEFT), which is like MC-PDFT but allows for other ingredients in the nonclassical-energy functional. We discuss two new kinds of MC-NEFT methods, namely multiconfiguration density coherence functional theory and machine-learned functionals.

This feature article overviews recent work on active spaces, matrix product reference states, treatment of quasidegeneracy, hybrid theory, density-coherence functionals, machine-learned functionals, spin–orbit coupling, gradients, and dipole moments.

Quantum-classical hybrid algorithm for the simulation of all-electron correlation

While chemical systems containing hundreds to thousands of electrons remain beyond the reach of quantum devices, hybrid quantum-classical algorithms present a promising pathway toward a quantum advantage. Hybrid algorithms treat the exponentially scaling part of the calculation-the static correlation-on the quantum computer and the non-exponentially scaling part-the dynamic correlation-on the classical computer. While a variety of algorithms have been proposed, the dependence of many methods on the total wave function limits the development of easy-to-use classical post-processing implementations. Here, we present a novel combination of quantum and classical algorithms, which computes the all-electron energy of a strongly correlated molecular system on the classical computer from the 2-electron reduced density matrix (2-RDM) evaluated on the quantum device. Significantly, we circumvent the wave function in the all-electron calculations by using density matrix methods that only require input of the statically correlated 2-RDM. Although the algorithm is completely general, we test it with two classical density matrix methods, the anti-Hermitian contracted Schrödinger equation (ACSE) and multiconfiguration pair-density functional theories, using the recently developed quantum ACSE method for simulating the statically correlated 2-RDM. We obtain experimental accuracy for the relative energies of all three benzyne isomers and thereby demonstrate the ability of the developed algorithm to achieve chemically relevant and accurate results on noisy intermediate-scale quantum devices.

Magnetic exchange coupling in Cu dimers studied with modern multireference methods and broken-symmetry coupled cluster theory

Spin-state energetics of exchange-coupled copper complexes pose a persistent challenge for applied quantum chemistry. Here, we provide a comprehensive comparison of all available theoretical approaches to the problem of exchange coupling in two antiferromagnetically coupled bis-μ-hydroxo Cu(II) dimers. The evaluated methods include multireference methods based on the density matrix renormalization group (DMRG), multireference methods that incorporate dynamic electron correlation either perturbatively, such as the N-electron valence state perturbation theory, or variationally, such as the difference-dedicated configuration interaction. In addition, we contrast the multireference results with those obtained using broken-symmetry approaches that utilize either density functional theory or, as demonstrated here for the first time in such systems, a local implementation of coupled cluster theory. The results show that the spin-state energetics of these copper dimers are dominated by dynamic electron correlation and represent an impossible challenge for multireference methods that rely on brute-force expansion of the active space to recover correlation energy. Therefore, DMRG-based methods even at the limit of their applicability cannot describe quantitatively the antiferromagnetic exchange coupling in these dimers, in contrast to dinuclear complexes of earlier transition metal ions. The convergence of the broken-symmetry coupled cluster approach is studied and shown to be a limiting factor for the practical application of the method. The advantages and disadvantages of all approaches are discussed, and recommendations are made for future developments.

Modern multireference methods and their application in transition metal chemistry

Transition metal chemistry is a challenging playground for quantum chemical methods owing to the simultaneous presence of static and dynamic electron correlation effects in many systems. Wavefunction based multireference (MR) methods constitute a physically sound and systematically improvable Ansatz to deal with this complexity but suffer from some conceptual difficulties and high computational costs. The latter problem partially arises from the unfavorable scaling of the Full Configuration Interaction (Full-CI) problem which in the majority of MR methods is solved for a subset of the molecular orbital space, the so-called active space. In the last years multiple methods such as modern variants of selected CI, Full-CI Quantum Monte Carlo (FCIQMC) and the density matrix renormalization group (DMRG) have been developed that solve the Full-CI problem approximately for a fraction of the computational cost required by conventional techniques thereby significantly extending the range of applicability of modern MR methods. This perspective review outlines recent advancements in the field of MR electronic structure methods together with the resulting chances and challenges for theoretical studies in the field of transition metal chemistry. In light of its emerging importance a special focus is put on the selection of adequate active spaces and the concomitant development of numerous selection aides in recent years.

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.

Accurate singlet-triplet gaps in biradicals via the spin averaged anti-Hermitian contracted Schrödinger equation

The accurate description of biradical systems, and in particular the resolution of their singlet-triplet gaps, has long posed a major challenge to the development of electronic structure theories. Biradicaloid singlet ground states are often marked by strong correlation and, hence, may not be accurately treated by mainstream, single-reference methods such as density functional theory or coupled cluster theory. The anti-Hermitian contracted Schrödinger equation (ACSE), whose fundamental quantity is the two-electron reduced density matrix rather than the N-electron wave function, has previously been shown to account for both dynamic and strong correlations when seeded with a strongly correlated guess from a complete active space (CAS) calculation. Here, we develop a spin-averaged implementation of the ACSE, allowing it to treat higher multiplicity states from the CAS input without additional state preparation. We apply the spin-averaged ACSE to calculate the singlet-triplet gaps in a set of small main group biradicaloids, as well as the organic four-electron biradicals trimethylenemethane and cyclobutadiene, and naphthalene, benchmarking the results against other state-of-the-art methods reported in the literature.

Beyond the Classical Contributions to Exchange Coupling in Binuclear Transition Metal Complexes

Complexes with two or more magnetically coupled metal ions have attracted considerable attention as catalysts of many vital processes, single-molecule magnets, or spin-crossover compounds. Elucidation of their electronic structures is essential for understanding their catalytic and magnetic properties. Here, we provide an unprecedented insight into exchange-coupling mechanisms between the magnetic centers in six prototypical bis-μ-oxo bimetallic M₂O₂ complexes, including two biologically relevant models of non-heme iron enzymes. Employing multiconfigurational/multireference methods and related orbital entanglement analysis, we revealed the essential and counterintuitive role of predominantly unoccupied valence metal d orbitals in their strong antiferromagnetic coupling. We found that the participation of these orbitals is twofold. First, they enhance the superexchange between the singly occupied d orbitals. Second, they become substantially occupied and thus directly magnetically active, which we perceive as a new mechanism of the exchange interaction between the magnetic transition metal centers.