Computational Quantum Chemistry for Multiple-Site Heisenberg Spin Couplings Made Simple: Still Only One Spin-Flip Required

The resolution of the weak-exchange limit made rigorous, simple and general in binuclear complexes

The correct interpretation of magnetic properties in the weak-exchange regime has remained a challenging task for several decades. In this regime, the effective exchange interaction between local spins is quite weak, of the same order of magnitude or smaller than the various anisotropic terms, which generates a complex set of levels characterized by spin mixing. Although the model multispin Hamiltonian in the absence of local orbital momentum, , is considered good enough to map the experimental energies at zero field and in the strong-exchange limit, theoretical works pointed out limitations of this simple model. This work revives the use of ĤMS from a new theoretical perspective, detailing point-by-point a strategy to correctly map the computational energies and wave functions onto ĤMS, thus validating it regardless of the exchange limit. We will distinguish two cases, based on experimentally characterized dicobalt(II) complexes from the literature. If centrosymmetry imposes alignment of the various rank-2 tensors constitutive of ĤMS in the first case, the absence of any symmetry element prevents such alignment in the second case. In such a context, the strategy provided herein becomes a powerful tool to rationalize the experimental magnetic data, since it is capable of fully and rigorously extracting the multispin model without any assumption on the orientation of its constitutive tensors. Furthermore, the strategy allows to question the use of the spin Hamiltonian approach by explicitly controlling the projection norms on the model space, which is showcased in the second complex where local orbital momentum could have occurred (distorted octahedra). Finally, previous theoretical data related to a known dinickel(II) complex is reinterpreted, clarifying initial wanderings regarding the weak exchange limit.

Systematic determination of coupling constants in spin clusters from broken-symmetry mean-field solutions

Quantum-chemical calculations aimed at deriving magnetic coupling constants in exchange-coupled spin clusters commonly utilize a broken-symmetry (BS) approach. This involves calculating several distinct collinear spin configurations, predominantly by density-functional theory. The energies of these configurations are interpreted in terms of the Heisenberg model, H̃=∑i Collapse

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Shadan Ghassemi Tabrizi Department of Chemistry, University of Potsdam, Karl-Liebknecht-Str. 24-25, D-14476 Potsdam-Golm, Germany

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High-Spin Blatter's Triradicals

Robust organic triradicals with high-spin quartet ground states provide promising applications in molecular magnets, spintronics, etc. In this context, a triradical based on Blatter's radical has been synthesized recently, having two low-lying non-degenerate doublet states with a quartet ground state. The traditional broken-symmetry (BS)-DFT computed doublet-quartet energy gaps are reported to be somewhat overestimated in comparison to the experimentally observed values. In this work, we have employed different ab initio methods on this prototypical system to obtain more accurate doublet-quartet energy gaps for this triradical. The spin-constraint broken-symmetry (CBS)-DFT method has been used to reduce the overestimation of energy gaps from BS-DFT. To address the issues of spin-contamination and the multireference nature of low-spin states affecting the DFT methods, we have computed the energy gaps using appropriately state-averaged CASSCF and NEVPT2 computations. Using a series of active spaces, our calculations are shown to provide quite accurate values in concordance with the experimentally observed results. Furthermore, we have proposed and modeled another two triradicals based on Blatter's radical, which are of interest for experimental synthesis and characterization. Our computations show that all these triradicals also have a quartet ground state with a similar energy difference between the excited doublet states.

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.

Evaluation of Neel Temperatures from Fully Self-Consistent Broken-Symmetry GW and High-Temperature Expansion: Application to Cubic Transition-Metal Oxides

Using fully self-consistent thermal broken-symmetry GW, we construct effective magnetic Heisenberg Hamiltonians for a series of transition metal oxides (NiO, CoO, FeO, and MnO), capturing a rigorous but condensed description of the magnetic states. Then applying high-temperature expansion, we find the decomposition coefficients for spin susceptibility and specific heat. The radius of convergence of the found series determines the Neel temperature. The NiO, CoO, and FeO contain a small ferromagnetic interaction between the nearest neighbors (NNs) and the dominant antiferromagnetic interaction between the next-nearest neighbors (NNNs). For them, the derived Neel temperatures are in good agreement with experiment. The case of MnO is different because both NN and NNN couplings are antiferromagnetic and comparable in magnitude, for which the error in the estimated Neel temperature is larger, which is a signature of additional effects not captured by electronic structure calculations.

Magnetic exchange interactions in binuclear and tetranuclear iron(III) complexes described by spin-flip DFT and Heisenberg effective Hamiltonians

Low-energy spectra of single-molecule magnets (SMMs) are often described by Heisenberg Hamiltonians. Within this formalism, exchange interactions between magnetic centers determine the ground-state multiplicity and energy separation between the ground and excited states. In this contribution, we extract exchange coupling constants (J) for a set of iron (III) binuclear and tetranuclear complexes from all-electron calculations using non-collinear spin-flip time-dependent density functional theory (NC-SF-TDDFT). For 12 binuclear complexes with J-values ranging from -6 to -132 cm^-1 , our benchmark calculations using the short-range hybrid ωPBEh functional and 6-31G(d,p) basis set agree well with the experimentally derived values (mean absolute error of 4.7 cm^-1 ). For the tetranuclear SMMs, the computed J constants are within 6 cm^-1 from the experimentally derived values. We explore the range of applicability of the Heisenberg model by analyzing bonding patterns in these Fe(III) complexes using natural orbitals (NO), their occupations, and the number of effectively unpaired electrons. The results illustrate the efficiency of the spin-flip protocol for computing the exchange couplings and the utility of the NO analysis in assessing the validity of effective spin Hamiltonians.

Symmetry Breaking Slows Convergence of the ADAPT Variational Quantum Eigensolver

Because quantum simulation of molecular systems is expected to provide the strongest advantage over classical computing methods for systems exhibiting strong electron correlation, it is critical that the performance of VQEs be assessed for strongly correlated systems. For classical simulation, strong correlation often results in symmetry breaking of the Hartree-Fock reference, leading to Löwdin's well-known "symmetry dilemma", whereby accuracy in the energy can be increased by breaking spin or spatial symmetries. Here, we explore the impact of symmetry breaking on the performance of ADAPT-VQE using two strongly correlated systems: (i) the "fermionized" anisotropic Heisenberg model, where the anisotropy parameter controls the correlation in the system, and (ii) symmetrically stretched linear H₄, where correlation increases with increasing H-H separation. In both of these cases, increasing the level of correlation of the system leads to spontaneous symmetry breaking (parity and S^2, respectively) of the mean-field solutions. We analyze the role that symmetry breaking in the reference states and orbital mappings of the fermionic Hamiltonians have in the compactness and performance of ADAPT-VQE. We observe that improving the energy of the reference states by breaking symmetry has a deleterious effect on ADAPT-VQE by increasing the length of the ansatz necessary for energy convergence and exacerbating the problem of "gradient troughs".

Validation of the Green's Function Approximation for the Calculation of Magnetic Exchange Couplings

In this work, we assess the potential of the Green's function approximation to predict isotropic magnetic exchange couplings and to reproduce the standard broken-symmetry energy difference approach for transition metal complexes. To this end, we have selected a variety of heterodinuclear, homodinuclear, and polynuclear systems containing 3d transition metal centers and computed the couplings using both the Green's function and energy difference methods. The Green's function approach is shown to have mixed results for the cases tested. For dinuclear complexes with large strength couplings (≳50 cm^-1), the Green's function method is unable to reliably reproduce the energy difference values. However, for weaker dinuclear couplings, the Green's function approach acceptably reproduces broken-symmetry energy difference couplings. In polynuclear cases, the Green's function approximation worked remarkably well, especially for Fe^III complexes. On the other hand, for a Ni^II polynuclear complex, qualitatively wrong couplings are predicted. Overall, the evaluation of exchange couplings from local rigid magnetization rotations offers a powerful alternative to time-consuming energy differences methods for large polynuclear transition metal complexes, but to achieve a quantitative agreement, some improvements to the method are needed.

The many-body electronic interactions of Fe(II)–porphyrin

Fe(II)–porphyrin complexes exhibit a diverse range of electronic interactions between the metal and macrocycle. Herein, the incremental full configuration interaction method is applied to the entire space of valence orbitals of a Fe(II)–porphyrin model using a modest basis set. A novel visualization framework is proposed to analyze individual many-body contributions to the correlation energy, providing detailed maps of this complex’s highly correlated electronic structure. This technique is used to parse the numerous interactions of two low-lying triplet states (3A2g and 3Eg) and to show that strong metal d–d and macrocycle π–π orbital interactions preferentially stabilize the 3A2g state. d–π interactions, on the other hand, preferentially stabilize the 3Eg state and primarily appear when correlating six electrons at a time. Ultimately, the Fe(II)–porphyrin model’s full set of 88 valence electrons are correlated in 275 orbitals, showing the interactions up to the 4-body level, which covers the great majority of correlations in this system.

Revealing the Contest between Triplet-Triplet Exchange and Triplet-Triplet Energy Transfer Coupling in Correlated Triplet Pair States in Singlet Fission

Understanding the separation of the correlated triplet pair state ¹(TT) intermediate is critical for leveraging singlet fission to improve solar cell efficiency. This separation mechanism is dominated by two key interactions: (i) the exchange interaction (K) between the triplets which leads to the spin splitting of the biexciton state into ¹(TT),³(TT) and ⁵(TT) states, and (ii) the triplet-triplet energy transfer integral (t) which enables the formation of the spatially separated (but still spin entangled) state ¹(T···T). We develop a simple ab initio technique to compute both the biexciton exchange (K) and biexciton transfer coupling. Our key findings reveal new conditions for successful correlated triplet pair state dissociation. The biexciton exchange interaction needs to be ferromagnetic or negligible to the triplet energy transfer for favorable dissociation. We also explore the effect of chromophore packing to reveal geometries where these conditions are achieved for tetracene.

Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange-correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear-electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an "open teamware" model and an increasingly modular design.

Equation-of-Motion Coupled-Cluster Protocol for Calculating Magnetic Properties: Theory and Applications to Single-Molecule Magnets

We present a new computational protocol for computing macroscopic magnetic properties of transition-metal complexes using the equation-of-motion coupled-cluster (EOM-CC) framework. The approach follows a two-step state-interaction scheme: we first compute zero-order states using nonrelativistic EOM-CC and then use these states to evaluate matrix elements of the spin-orbit and Zeeman operators. Diagonalization of the resulting Hamiltonian yields spin-orbit- and field-perturbed eigenstates. Temperature- and field-dependent magnetization and susceptibility are computed by numerical differentiation of the partition function. To compare with powder-sample experiments, these quantities are numerically averaged over field orientations. We applied this protocol to several single-molecule magnets (SMMs) with Fe(II) and Fe(III) in trigonal pyramidal, linear, and trigonal bipyramidal coordination environments. We described the underlying electronic structure by the electron-attachment (EOM-EA) and spin-flip (EOM-SF) variants of EOM-CC. The computed energy barriers for spin inversion, and macroscopic magnetization and susceptibility agree well with experimental data. Trends in magnetic anisotropy and spin-reversal energy barriers are explained in terms of a molecular orbital picture rigorously distilled from spinless transition density matrices between many-body states. The results illustrate excellent performances of EOM-CC in describing magnetic behavior of mononuclear transition-metal SMMs.

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.

Spin-flip methods in quantum chemistry

This Perspective discusses salient features of the spin-flip approach to strong correlation and describes different methods that sprung from this idea. The spin-flip treatment exploits the different physics of low-spin and high-spin states and is based on the observation that correlation is small for same-spin electrons. By using a well-behaved high-spin state as a reference, one can access problematic low-spin states by deploying the same formal tools as in the excited-state treatments (i.e., linear response, propagator, or equation-of-motion theories). The Perspective reviews applications of this strategy within wave function and density functional theory frameworks as well as the extensions for molecular properties and spectroscopy. The utility of spin-flip methods is illustrated by examples. Limitations and proposed future directions are also discussed.

Spin splittings from first-order symmetry-adapted perturbation theory without single-exchange approximation

The recently proposed spin-flip symmetry-adapted perturbation theory (SF-SAPT) first-order exchange energy [Patkowski et al., J. Chem. Phys. 148, 164110 (2018)] enables the standard open-shell SAPT approach to treat arbitrary spin states of the weakly interacting complex. Here, we further extend first-order SF-SAPT beyond the single-exchange approximation to a complete treatment of the exchanges of electrons between monomers. This new form of the exchange correction replaces the single-exchange approximation with a more moderate single-spin-flip approximation. The newly developed expressions are applied to a number of small test systems to elucidate the quality of both approximations. They are also applied to the singlet-triplet splittings in pancake bonded dimers. The accuracy of the single-exchange approximation deteriorates at short intermolecular separations, especially for systems with few electrons and for the high-spin state of the complex. In contrast, the single-spin-flip approximation is exact for interactions involving a doublet molecule and remains highly accurate for any number of unpaired electrons. Because the single-exchange approximation affects the high-spin and low-spin states of pancake bonded complexes evenly, the resulting splitting values are of similar accuracy to those produced by the formally more accurate single-spin-flip approximation.

First-order symmetry-adapted perturbation theory for multiplet splittings

We present a symmetry-adapted perturbation theory (SAPT) for the interaction of two high-spin open-shell molecules (described by their restricted open-shell Hartree-Fock determinants) resulting in low-spin states of the complex. The previously available SAPT formalisms, except for some system-specific studies for few-electron complexes, were restricted to the high-spin state of the interacting system. Thus, the new approach provides, for the first time, a SAPT-based estimate of the splittings between different spin states of the complex. We have derived and implemented the lowest-order SAPT term responsible for these splittings, that is, the first-order exchange energy. We show that within the so-called S² approximation commonly used in SAPT (neglecting effects that vanish as fourth or higher powers of intermolecular overlap integrals), the first-order exchange energies for all multiplets are linear combinations of two matrix elements: a diagonal exchange term that determines the spin-averaged effect and a spin-flip term responsible for the splittings between the states. The numerical factors in this linear combination are determined solely by the Clebsch-Gordan coefficients: accordingly, the S² approximation implies a Heisenberg Hamiltonian picture with a single coupling strength parameter determining all the splittings. The new approach is cast into both molecular-orbital and atomic-orbital expressions: the latter enable an efficient density-fitted implementation. We test the newly developed formalism on several open-shell complexes ranging from diatomic systems (Li⋯H, Mn⋯Mn, …) to the phenalenyl dimer.

Singlet-triplet energy gaps and the degree of diradical character in binuclear copper molecular magnets characterized by spin-flip density functional theory

Molecular magnets, defined here as organic polyradicals, can be used as building blocks in the fabrication of novel and structurally diverse magnetic light-weight materials. We present a theoretical investigation of the lowest spin states of several binuclear copper diradicals. In contrast to previous studies, we consider not only the energetics of the low-lying states (which are related to the exchange-coupling parameter within the Heisenberg-Dirac-van-Vleck model), but also the character of the diradical states themselves. We use natural orbitals, their occupations, and the number of effectively unpaired electrons to quantify bonding patterns in these systems. We compare the performance of spin-flip time-dependent density functional theory (SF-TDDFT) using various functionals and effective core potentials against the wave function based approach, equation-of-motion spin-flip coupled-cluster method with single and double substitutions (EOM-SF-CCSD). We find that SF-TDDFT paired with the PBE50 and B5050LYP functionals performs comparably to EOM-SF-CCSD, with respect to both singlet-triplet gaps and states' characters. Visualization of frontier natural orbitals shows that the unpaired electrons are localized on copper centers, in some cases exhibiting slight through-bond interaction via copper d-orbitals and p-orbitals of neighboring ligand atoms. The analysis reveals considerable interactions between the formally unpaired electrons in the antiferromagnetic diradicaloids, meaning that they are poorly described by the Heisenberg-Dirac-van-Vleck model. Thus, for these systems the experimentally derived exchange-coupling parameters are not directly comparable with the singlet-triplet gaps. This explains systematic discrepancies between the computed singlet-triplet energy gaps and the exchange-coupling parameters extracted from experiment.

Spin coupling interactions in C[double bond, length as m-dash]C or B-B-cored porphyrin-mimetic graphene patch nitroxide diradicals

In view of the unique structures and promising applications of porphyrins and their derivatives, exploration of their various properties has continued to a hot topic. In this work, we combine porphyrin-mimetic graphene patches which are core-modified by a C[double bond, length as m-dash]C or a B-B unit and two nitroxide radical groups to construct a series of novel diradical molecules (the CC-cored or BB-cored molecules). The spin coupling constants (J) of diradicals were calculated at the (U)B3LYP/6-311G(d,p) level by considering the different linking modes of two nitroxide groups. The results indicate that different core modification considerably affects the J values of such diradicals, and the linking modes can tune the sizes and signs of J, changing their magnetic coupling interactions with different magnitudes and the signs of J from antiferromagnetic to ferromagnetic or vice versa. More interestingly and importantly, the spin coupling interactions of the CC-cored molecules can also be tuned by stretching the core unit C-C bond, suggesting the possibility of activating specific vibrational modes of the CC-cored diradicals by energy pulses to yield variable J coupling magnitudes. On the other hand, for the BB-cored molecules, two-electron reduction can switch or tune their magnetism from ferromagnetic to antiferromagnetic. The essence of all observations is further analyzed from the structural effects and orbital and spin density distributions. The findings about magnetic regulation in these core-modified porphyrin-mimetic graphene patch nitroxide diradicals further expand the field of molecular magnets and provide a rational theoretical basis for designing novel building blocks of magnetic functional molecular materials.

Exact Mapping from Many-Spin Hamiltonians to Giant-Spin Hamiltonians

Thermodynamic and spectroscopic data of exchange-coupled molecular spin clusters (e.g. single-molecule magnets) are routinely interpreted in terms of two different models: the many-spin Hamiltonian (MSH) explicitly considers couplings between individual spin centers, while the giant-spin Hamiltonian (GSH) treats the system as a single collective spin. When isotropic exchange coupling is weak, the physical compatibility between both spin Hamiltonian models becomes a serious concern, due to mixing of spin multiplets by local zero-field splitting (ZFS) interactions ('S-mixing'). Until now, this effect, which makes the mapping MSH→GSH ('spin projection') non-trivial, had only been treated perturbationally (up to third order), with obvious limitations. Here, based on exact diagonalization of the MSH, canonical effective Hamiltonian theory is applied to construct a GSH that exactly matches the energies of the relevant (2S+1) states comprising an effective spin multiplet. For comparison, a recently developed strategy for the unique derivation of effective ('pseudospin') Hamiltonians, now routinely employed in ab initio calculations of mononuclear systems, is adapted to the problem of spin projection. Expansion of the zero-field Hamiltonian and the magnetic moment in terms of irreducible tensor operators (or Stevens operators) yields terms of all ranks k (up to k=2S) in the effective spin. Calculations employing published MSH parameters illustrate exact spin projection for the well-investigated [Ni(hmp)(dmb)Cl]₄ ('Ni₄ ') single-molecule magnet, which displays weak isotropic exchange (dmb=3,3-dimethyl-1-butanol, hmp^- is the anion of 2-hydroxymethylpyridine). The performance of the resulting GSH in finite field is assessed in terms of EPR resonances and diabolical points. The large tunnel splitting in the M=± 4 ground doublet of the S=4 multiplet, responsible for fast tunneling in Ni₄ , is attributed to a Stevens operator with eightfold rotational symmetry, marking the first quantification of a k=8 term in a spin cluster. The unique and exact mapping MSH→GSH should be of general importance for weakly-coupled systems; it represents a mandatory ultimate step for comparing theoretical predictions (e.g. from quantum-chemical calculations) to ZFS, hyperfine or g-tensors from spectral fittings.

The role of the multiconfigurational character of nitronyl-nitroxide in the singlet–triplet energy gap of its diradicals

CAS(2,2) reference may not be sufficient for the computation of singlet–triplet energy gap by DDCI.

Quantitative prediction and interpretation of spin energy gaps in polyradicals: the virtual magnetic balance

Open-shell organic molecules possessing more than two unpaired electrons and sufficient stability at room temperature are very unusual, but some of them were recently synthesized and promise a number of fascinating applications.

Black-box determination of temperature-dependent susceptibilities for crystalline organic radicals with complex magnetic topologies

In all but the simplest crystal structures, the identification of all relevant interactions between magnetic sites as well as the setup of magnetic model spaces, which are necessary for modeling macroscopic magnetism, are tedious and error-prone tasks. Here, we present a procedure to generate magnetic susceptibility versus temperature curves using only a crystal structure as input. The procedure, which is based on the first-principles bottom-up approach [Deumal et al., J. Phys. Chem. A, 2002, 106, 1299], is designed in a way to require as little user interference as possible. We employ quantum chemical calculations to parametrize a Heisenberg Hamiltonian, which is set up and diagonalized for different magnetic model spaces to ensure convergence of the model. We apply the procedure to several 6-oxo-verdazyl radical structures, including newly synthesized compounds, and compare the results to data we obtained from magnetic susceptibility measurements as well as published data to further benchmark our procedure. Furthermore, the different impact of certain dominating coupling constants is systematically analyzed.

Magnetic Coupling Constants in Three Electrons Three Centers Problems from Effective Hamiltonian Theory and Validation of Broken Symmetry-Based Approaches

In the most general case of three electrons in three symmetry unrelated centers with Ŝ1 = Ŝ2 = Ŝ3 = 1/2 localized magnetic moments, the low energy spectrum consists of one quartet (Q) and two doublet (D1, D2) pure spin states. The energy splitting between these spin states can be described with the well-known Heisenberg-Dirac-Van Vleck (HDVV) model spin Hamiltonian, and their corresponding energy expressions are expressed in terms of the three different two-body magnetic coupling constants J12, J23, and J13. However, the values of all three magnetic coupling constants cannot be extracted using the calculated energy of the three spin-adapted states since only two linearly independent energy differences between pure spin states exist. This problem has been recently investigated by Reta et al. (J. Chem. Theory Comput. 2015, 11, 3650), resulting in an alternative proposal to the original Noodleman's broken symmetry mapping approach. In the present work, this proposal is validated by means of ab initio effective Hamiltonian theory, which allows a direct extraction of all three J values from the one-to-one correspondence between the matrix elements of both effective and HDVV Hamiltonian. The effective Hamiltonian matrix representation has been constructed from configuration interaction wave functions for the three spin states obtained for two model systems showing a different degree of delocalization of the unpaired electrons. These encompass a trinuclear Cu(II) complex and a π-conjugated purely organic triradical.

Helical Folding-Induced Stabilization of Ferromagnetic Polyradicals Based on Triarylmethyl Radical Derivatives

Magnetic ordering in purely organic π-conjugated materials is a challenging, rare, and desirable event. The interest lies on the unique magnetic properties derived from high-spin carbon-based polymers/macromolecules tailored through appropriate synthetic routes. Ground-breaking achievements have been reported regarding magnetic ordering in an organic polymer using spin clusters as building blocks. This strategy leads to two-dimensional extended polyradicals with a concomitant loss of appealing macroscopic properties such as expected magnetic anisotropy in elongated shaped macromolecules containing carbon-bearing radicals. Here we provide compelling evidence of a secondary structure-induced stabilization of ferromagnetic polyradicals with robust magnetic properties and strongly suggest revisiting a discarded attempt to obtain polymeric linear-like radicals. An alternative synthetic approach is also proposed, based on polyradicals obtained from discrete molecular precursors (oligomers) long enough to ensure a secondary structure, rather than from polymerization processes.

Electronic Structures of Anti-Ferromagnetic Tetraradicals: Ab Initio and Semi-Empirical Studies

The energy relationships and electronic structures of the lowest-lying spin states in several anti-ferromagnetic tetraradical model systems are studied with high-level ab initio and semi-empirical methods. The Full-CI method (FCI), the complete active space second-order perturbation theory (CASPT2), and the n-electron valence state perturbation theory (NEVPT2) are employed to obtain reference results. By comparing the energy relationships predicted from the Heisenberg and Hubbard models with ab initio benchmarks, the accuracy of the widely used Heisenberg model for anti-ferromagnetic spin-coupling in low-spin polyradicals is cautiously tested in this work. It is found that the strength of electron correlation (|U/t|) concerning anti-ferromagnetically coupled radical centers could range widely from strong to moderate correlation regimes and could become another degree of freedom besides the spin multiplicity. Accordingly, the Heisenberg-type model works well in the regime of strong correlation, which reproduces well the energy relationships along with the wave functions of all the spin states. In moderately spin-correlated tetraradicals, the results of the prototype Heisenberg model deviate severely from those of multi-reference electron correlation ab initio methods, while the extended Heisenberg model, containing four-body terms, can introduce reasonable corrections and maintains its accuracy in this condition. In the weak correlation regime, both the prototype Heisenberg model and its extended forms containing higher-order correction terms will encounter difficulties. Meanwhile, the Hubbard model shows balanced accuracy from strong to weak correlation cases and can reproduce qualitatively correct electronic structures, which makes it more suitable for the study of anti-ferromagnetic coupling in polyradical systems.

Handling Magnetic Coupling in Trinuclear Cu(II) Complexes

The problem of deriving three different two-body magnetic couplings in three electrons/three centers in a general geometric arrangement is investigated using the trinuclear Cu(II) HAKKEJ complex as a real case example. In these systems, one quartet and two doublet low lying electronic states exist, which define the magnetic spectra. However, the two possible linearly independent energy differences do not provide enough information to extract the three magnetic coupling constants. Here, we show how to obtain these parameters without making any assumption on the symmetry of the system from a combination of density functional- and wave function-based calculations. The density functional calculations explore various broken symmetry solutions and relate the corresponding energy to the expectation value of the Heisenberg Hamiltonian. This allows one to obtain all magnetic couplings, although their magnitude strongly depends on the exchange-correlation functional. Interestingly, a constant ratio between the magnetic coupling constants along a series of investigated functionals is found. This provides an additional equation to be used when relying on energy differences between spin states, which in turn allow solving the Heisenberg spectrum. The magnetic couplings thus obtained are compared to the experiment. Implications for the appropriate interpretation of the experiment and for the study of more complex systems are discussed.