Orbital Entanglement Analysis of Exchange-Coupled Systems

Magnetic Interactions in a [Co(II)3Er(III)(OR)4] Model Cubane through Forefront Multiconfigurational Methods

Strong electron correlation effects are one of the major challenges in modern quantum chemistry. Polynuclear transition metal clusters are peculiar examples of systems featuring such forms of electron correlation. Multireference strategies, often based on but not limited to the concept of complete active space, are adopted to accurately account for strong electron correlation and to resolve their complex electronic structures. However, transition metal clusters already containing four magnetic centers with multiple unpaired electrons make conventional active space based strategies prohibitively expensive, due to their unfavorable scaling with the size of the active space. In this work, forefront techniques, such as density matrix renormalization group (DMRG), full configuration interaction quantum Monte Carlo (FCIQMC), and multiconfiguration pair-density functional theory (MCPDFT), are employed to overcome the computational limitation of conventional multireference approaches and to accurately investigate the magnetic interactions taking place in a [Co(II)₃Er(III)(OR)₄] (chemical formula [Co(II)₃Er(III)(hmp)₄(μ₂-OAc)₂(OH)₃(H₂O)], hmp = 2-(hydroxymethyl)-pyridine) model cubane water oxidation catalyst. Complete active spaces with up to 56 electrons in 56 orbitals have been constructed for the seven energetically lowest different spin states. Relative energies, local spin, and spin-spin correlation values are reported and provide crucial insights on the spin interactions for this model system, pivotal in the rationalization of the catalytic activity of this system in the water-splitting reaction. A ferromagnetic ground state is found with a very small, ∼50 cm^-1, highest-to-lowest spin gap. Moreover, for the energetically lowest states, S = 3-6, the three Co(II) sites exhibit parallel aligned spins, and for the lower states, S = 0-2, two Co(II) sites retain strong parallel spin alignment.

The Chromium Dimer: Closing a Chapter of Quantum Chemistry

The complex electronic structure and unusual potential energy curve of the chromium dimer have fascinated scientists for decades, with agreement between theory and experiment so far elusive. Here, we present a new ab initio simulation of the potential energy curve and vibrational spectrum that significantly improves on all earlier estimates. Our data support a shift in earlier experimental assignments of a cluster of vibrational frequencies by one quantum number. The new vibrational assignment yields an experimentally derived potential energy curve in quantitative agreement with theory across all bond lengths and across all measured frequencies. By solving this long-standing problem, our results raise the possibility of quantitative quantum chemical modeling of transition metal clusters with spectroscopic accuracy.

The Apparently Unreactive Substrate Facilitates the Electron Transfer for Dioxygen Activation in Rieske Dioxygenases

Rieske dioxygenases belong to the non‐heme iron family of oxygenases and catalyze important cis‐dihydroxylation as well as O‐/N‐dealkylation and oxidative cyclization reactions for a wide range of substrates. The lack of substrate coordination at the non‐heme ferrous iron center, however, makes it particularly challenging to delineate the role of the substrate for productive O2 activation. Here, we studied the role of the substrate in the key elementary reaction leading to O2 activation from a theoretical perspective by systematically considering (i) the 6‐coordinate to 5‐coordinate conversion of the non‐heme Fe^II upon abstraction of a water ligand, (ii) binding of O2 , and (iii) transfer of an electron from the Rieske cluster. We systematically evaluated the spin‐state‐dependent reaction energies and structural effects at the active site for all combinations of the three elementary processes in the presence and absence of substrate using naphthalene dioxygenase as a prototypical Rieske dioxygenase. We find that reaction energies for the generation of a coordination vacancy at the non‐heme FeII center through thermoneutral H₂O reorientation and exothermic O2 binding prior to Rieske cluster oxidation are largely insensitive to the presence of naphthalene and do not lead to formation of any of the known reactive Fe‐oxygen species. By contrast, the role of the substrate becomes evident after Rieske cluster oxidation and exclusively for the 6‐coordinate non‐heme FeII sites in that the additional electron is found at the substrate instead of at the iron and oxygen atoms. Our results imply an allosteric control of the substrate on Rieske dioxygenase reactivity to happen prior to changes at the non‐heme FeII in agreement with a strategy that avoids unproductive O2 activation.

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.

Modeling magnetic interactions in high-valent trinuclear [Mn3(IV)O4]4+ complexes through highly compressed multi-configurational wave functions

In this work we apply a quantum chemical framework, recently designed in our laboratories, to rationalize the low-energy electronic spectrum and the magnetic properties of an homo-valent trinuclear [Mn₃^(IV)O₄]⁴⁺ model of the oxygen-evolving center in photosystem II. The method is based on chemically motivated molecular orbital unitary transformations, and the optimization of spin-adapted many-body wave functions, both for ground- and excited-states, in the transformed MO basis. In this basis, the configuration interaction Hamiltonian matrix of exchange-coupled multi-center clusters is extremely sparse and characterized by a unique block diagonal structure. This property leads to highly compressed wave functions (oligo- or single-reference) and crucially enables state-specific optimizations. This work is the first showing that compression and selective targeting of ground- and excited-states wave functions is possible for systems with three magnetic centers that are not exactly half-filled, and that potentially exhibit frustrated spin interactions. The reduced multi-reference character of the wave function greatly simplifies the interpretation of the ground- and excited-state electronic structures, and provides a route for the direct rationalization of magnetic interactions in these compounds, often considered a challenge in polynuclear transition-metal chemistry. In this study, strong electron correlation effects have explicitly been described by conventional and stochastic multiconfigurational methodologies, while dynamic correlation effects have been accounted for by multiconfigurational second order perturbation theory, CASPT2. Ab initio results for the [Mn₃^(IV)O₄]⁴⁺ system have been mapped to a three-site Heisenberg model with two magnetic coupling constants. The magnetic coupling constants and the temperature dependence of the effective magnetic moment predicted by the ab initio calculations are in good agreement with the available experimental data, and confirm the antiferromagnetic interaction among the three magnetic centers, while providing a simple and rigorous description of the noncollinearity of the local spins, that characterize most of the low-energy states for this system.

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.

Bonding and Magnetic Exchange Pathways in Nature's Water-Oxidizing Complex

The nature of the bonding and magnetic exchange pathways of the water-oxidizing complex of photosystem 2 is explored using broken symmetry density functional theory. The electronic structure and superexchange pathways are illustrated and analyzed using corresponding orbitals and intrinsic bond orbitals. These demonstrate a dominating influence on the bonding and magnetic interactions by both the geometrical structure of the Mn₄CaO₅ core complex and the ionic interactions of the oxo bridges with the neighboring Ca²⁺ ion. The demonstrated ionic nature of the Ca²⁺ bonds is proposed to contribute to the stabilization of the oxygen atoms participating in O-O bond formation.

Resolution of Low-Energy States in Spin-Exchange Transition-Metal Clusters: Case Study of Singlet States in [Fe(III)4S4] Cubanes

Polynuclear transition-metal (PNTM) clusters owe their catalytic activity to numerous energetically low-lying spin states and stable oxidation states. The characterization of their electronic structure represents one of the greatest challenges of modern chemistry. We propose a theoretical framework that enables the resolution of targeted electronic states with ease and apply it to two [Fe(III)₄S₄] cubanes. Through direct access to their many-body wave functions, we identify important correlation mechanisms and their interplay with the geometrical distortions observed in these clusters, which are core properties in understanding their catalytic activity. The simulated magnetic coupling constants predicted by our strategy allow us to make qualitative connections between spin interactions and geometrical distortions, demonstrating its predictive power. Moreover, despite its simplicity, the strategy provides magnetic coupling constants in good agreement with the available experimental ones. The complexes are intrinsically frustrated anti-ferromagnets, and the obtained spin structures together with the geometrical distortions represent two possible ways to release spin frustration (spin-driven Jahn–Teller distortion). Our paradigm provides a simple, yet rigorous, route to uncover the electronic structure of PNTM clusters and may be applied to a wide variety of such clusters.

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.

Interpretation of Exchange Interaction through Orbital Entanglement

Recently, the analysis of single-orbital entropy and mutual information has been introduced as a tool for the investigation of contributions to the exchange (J) coupling between open-shell metal ions [Stein et al. J. Phys. Chem. Lett. 2019, 10, 6762-6770]. Here, we show that this analysis may lead to an incorrect interpretation of the J-coupling mechanism. Instead, we propose an orbital-entanglement analysis that is based on the two-electron density and that provides a coherent picture of the contributing exchange pathways, which seems fully consistent with the available J values. For this purpose, we used a prototypical bis-μ-oxo binuclear manganese complex ([Mn₂O₂(NH₃)₈]⁴⁺) and demonstrated that its antiferromagnetism (J < 0), calculated by using the active space composed of all valence p_O and d_Mn orbitals, correlates well with the largest elements in the differential low-spin vs high-spin entanglement map. These elements correspond to interactions between the pairs of d_Mn orbitals mediated by the oxo-bridging out-of-plane p orbitals, representing the π superexchange pathway. We also show that the reduction of active space to manifold of the singly occupied magnetic orbitals does not lead to discrepancy between the calculated J values and entanglement maps. This contrasts to analysis of mutual information, which suggests the "direct" d_Mn-d_Mn interactions to play a dominant role for the J coupling, irrespective of the size of active space as well as of the antiferromagnetism expected. The failure is attributed to the large contribution of spin entanglement contained in the mutual information of the low-spin state, which may be regarded as the origin of the different complexity of its wave function and electron density.

Successes, challenges, and opportunities for quantum chemistry in understanding metalloenzymes for solar fuels research

Overview of the rich and diverse contributions of quantum chemistry to understanding the structure and function of the biological archetypes for solar fuel research, photosystem II and hydrogenases.