Multiconfigurational Calculations Research Articles

Magnetic exchange interaction between unpaired π- and d-electrons in nanographene-metal coordination complexes

Abstract The combination of open-shell nanographenes (NGs) and magnetic transition metals holds great promise for generating various new quantum phases applicable in spintronics and quantum information technologies. However, a crucial aspect in accomplishing this is to comprehend the magnetic exchange interactions between unpaired π- and d-electrons, a topic that has been seldom addressed. In this study, we focus on magnetic π-d exchange interactions between open-shell NGs and a magnetic coordination center of Fe or Co by employing scanning tunneling microscopy and spectroscopy. We synthesize two sets of NG-metal coordination complexes on a Au(111) substrate, secured by coordination bonds of carboxyl acid-Fe (Co). Through analysis of the excitation spectra, we observe a characteristic exchange coupling of 9 meV (5 meV) between the unpaired π-electron and the Fe (Co) d-shell electrons. Our experimental findings are qualitatively in agreement with multiconfigurational quantum chemistry calculations. This work evidences that a substantial magnetic exchange coupling can be achieved and engineered in metal-organic coordination systems, paving the way for designing and customizing extended radical metal-organic frameworks with precisely tailored magnetic properties.

Approximate Hamiltonians from a Linear Vibronic Coupling Model for Solution-Phase Spin Dynamics.

The linear vibronic coupling (LVC) model is an approach for approximating how a molecular Hamiltonian changes in response to small changes in molecular geometry. The LVC framework thus has the ability to approximate molecular Hamiltonians at low computational expense but with quality approaching multiconfigurational ab initio calculations, when the change in geometry compared to the reference calculation used to parametrize it is small. Here, we show how the LVC approach can be used to project approximate spin Hamiltonians of a solvated lanthanide complex along a room-temperature molecular dynamics trajectory. As expected, the LVC approximation is less accurate as the geometry diverges from that at which the model was parametrized. We examine the accuracy of the predicted Hamiltonians by performing time-dependent quantum simulations of the spin dynamics of the molecule, with reference to the dynamics obtained using spin Hamiltonians projected from ab initio calculations at each step. We find that quantitatively accurate behavior is obtained when LVC parametrizations are performed at least every 10 fs during the trajectory.

Photoluminescence of a Uranium(IV) Alkoxide Complex.

In this report, we describe the photoluminescence of a homoleptic uranium(IV) alkoxide complex. Excitation of [Li(THF)]2[UIV(O t Bu)6] leads to the first example of photoluminescence from a well-defined actinide complex originating from an f-f excitation, supported by second order multiconfigurational electronic structure calculations including spin-orbit coupling. These calculations show strong spin-orbit coupling between the excited triplet and singlet states for the 5f-orbital manifold, which leads to a long-lived excited state lifetime of 0.85 s at low temperature. The photophysical properties of homoleptic uranium(V) and uranium(VI) tertbutoxide complexes are also presented; we find that oxidation of the uranium(IV) alkoxide results in quenching of luminescence in [Li(THF)][UV(O t Bu)6] and [UVI(O t Bu)6]. This is attributed to competing ligand to metal charge transfer absorption processes shifted to lower energy upon oxidation of the actinide center, which mask the relevant f-f transitions in the visible region of the electronic absorption spectrum.

Efficient Simulation of Inhomogeneously Correlated Systems Using Block Interaction Product States.

The strength of the density matrix renormalization group (DMRG) in handling strongly correlated systems lies in its unbiased and simultaneous treatment of identical sites that are both energetically degenerate and spatially similar, as typically encountered in physical models. However, this very feature becomes a drawback when DMRG is applied to quantum chemistry calculations for large, realistic correlated systems. This is because entangled orbitals often span broad ranges in both energy and space, with their interactions being notably inhomogeneous. In this study, we suggest addressing the strong intrafragment correlations and weak interfragment correlations separately, utilizing a large-scale multiconfigurational calculation framework grounded in the block interaction product state formulation. The strong intrafragment correlation can be encapsulated in several electronic states located on fragments, which are obtained by considering the entanglement between fragments and their environments. Moreover, we incorporate non-Abelian spin-SU(2) symmetry in our work to target the desired states we interested with well-defined particle number and spin, providing deeper insights into the corresponding chemical processes. The described method has been examined in various chemical systems and demonstrates high efficiency in addressing the inhomogeneous effects in strong correlation quantum chemistry.

Small tensor product distributed active space (STP-DAS) framework for relativistic and non-relativistic multiconfiguration calculations: Scaling from 109 on a laptop to 1012 determinants on a supercomputer

Despite the power and flexibility of configuration interaction (CI) based methods in computational chemistry, their broader application is limited by an exponential increase in both computational and storage requirements, particularly due to the substantial memory needed for excitation lists that are crucial for scalable parallel computing. The objective of this work is to develop a new CI framework, namely, the small tensor product distributed active space (STP-DAS) framework, aimed at drastically reducing memory demands for extensive CI calculations on individual workstations or laptops, while simultaneously enhancing scalability for extensive parallel computing. Moreover, the STP-DAS framework can support various CI-based techniques, such as complete active space (CAS), restricted active space, generalized active space, multireference CI, and multireference perturbation theory, applicable to both relativistic (two- and four-component) and non-relativistic theories, thus extending the utility of CI methods in computational research. We conducted benchmark studies on a supercomputer to evaluate the storage needs, parallel scalability, and communication downtime using a realistic exact-two-component CASCI (X2C-CASCI) approach, covering a range of determinants from 109 to 1012. Additionally, we performed large X2C-CASCI calculations on a single laptop and examined how the STP-DAS partitioning affects performance.

Beyond Spin Models in Orbitally Degenerate Open-Shell Nanographenes.

The study of open-shell nanographenes has relied on a paradigm where spins are the only low-energy degrees of freedom. Here we show that some nanographenes can host low-energy excitations that include strongly coupled spin and orbital degrees of freedom. The key ingredient is the existence of orbital degeneracy, as a consequence of leaving the benzenoid/half-filling scenario. We analyze the case of nitrogen-doped triangulenes, using both density-functional theory and Hubbard model multiconfigurational and random-phase approximation calculations. We find a rich interplay between orbital and spin degrees of freedom that confirms the need to go beyond the spin-only paradigm, opening a new avenue in this field of research.

A Four-Coordinate Pr4+ Imidophosphorane Complex.

The imidophosphorane ligand, [NPtBu3]- (tBu=tert-butyl), enables isolation of a pseudo-tetrahedral, tetravalent praseodymium complex, [Pr4+(NPtBu3)4] (1-Pr), which is characterized by a suite of physical characterization methods including single-crystal X-ray diffraction, electron paramagnetic resonance, and L3-edge X-ray near-edge spectroscopies. Variable-temperature direct-current magnetic susceptibility data, supported by multiconfigurational quantum chemical calculations, demonstrate that the electronic structure diverges from the isoelectronic Ce3+ analogue, driven by increased crystal field. The four-coordinate environment around Pr4+ in 1-Pr, which is unparalleled in reported extended solid systems, provides a unique opportunity to study the interplay between crystal field splitting and spin-orbit coupling in a molecular tetravalent lanthanide within a pseudo-tetrahedral coordination geometry.

A Four‐Coordinate Pr4+ Imidophosphorane Complex

AbstractThe imidophosphorane ligand, [NPtBu3]− (tBu=tert‐butyl), enables isolation of a pseudo‐tetrahedral, tetravalent praseodymium complex, [Pr4+(NPtBu3)4] (1‐Pr), which is characterized by a suite of physical characterization methods including single‐crystal X‐ray diffraction, electron paramagnetic resonance, and L3‐edge X‐ray near‐edge spectroscopies. Variable‐temperature direct‐current magnetic susceptibility data, supported by multiconfigurational quantum chemical calculations, demonstrate that the electronic structure diverges from the isoelectronic Ce3+ analogue, driven by increased crystal field. The four‐coordinate environment around Pr4+ in 1‐Pr, which is unparalleled in reported extended solid systems, provides a unique opportunity to study the interplay between crystal field splitting and spin‐orbit coupling in a molecular tetravalent lanthanide within a pseudo‐tetrahedral coordination geometry.

Time-Resolved Serial Femtosecond Crystallography of Cytochrome c Oxidase by use of a photolabile oxygen cage and novel insights of the catalytic site from high level multiconfigurational ab initio calculations

Time-Resolved Serial Femtosecond Crystallography of Cytochrome c Oxidase by use of a photolabile oxygen cage and novel insights of the catalytic site from high level multiconfigurational ab initio calculations

Coercive Fields Exceeding 30 T in the Mixed-Valence Single-Molecule Magnet (CpiPr5)2Ho2I3.

Mixed-valence dilanthanide complexes of the type (CpiPr5)2Ln2I3 (CpiPr5 = pentaisopropylcyclopentadienyl; Ln = Gd, Tb, Dy) featuring a direct Ln-Ln σ-bonding interaction have been shown to exhibit well-isolated high-spin ground states and, in the case of the Tb and Dy variants, a strong axial magnetic anisotropy that gives rise to a large magnetic coercivity. Here, we report the synthesis and characterization of two new mixed-valence dilanthanide compounds in this series, (CpiPr5)2Ln2I3 (1-Ln; Ln = Ho, Er). Both compounds feature a Ln-Ln bonding interaction, the first such interaction in any molecular compounds of Ho or Er. Like the Tb and Dy congeners, both complexes exhibit high-spin ground states arising from strong spin-spin coupling between the lanthanide 4f electrons and a single σ-type lanthanide-lanthanide bonding electron. Beyond these similarities, however, the magnetic properties of the two compounds diverge. In particular, 1-Er does not exhibit observable magnetic blocking or slow magnetic relaxation, while 1-Ho exhibits magnetic blocking below 28 K, which is the highest temperature among Ho-based single-molecule magnets, and a spin reversal barrier of 556(4) cm-1. Additionally, variable-field magnetization data collected for 1-Ho reveal a coercive field of greater than 32 T below 8 K, more than 6-fold higher than observed for the bulk magnets SmCo5 and Nd2Fe14B, and the highest coercive field reported to date for any single-molecule magnet or molecule-based magnetic material. Multiconfigurational calculations, supported by far-infrared magnetospectroscopy data, reveal that the stark differences in magnetic properties of 1-Ho and 1-Er arise from differences in the local magnetic anisotropy of the lanthanide centers.

Natural Orbitals and Targeted Non-Orthogonal Orbital Sets for Atomic Hyperfine Structure Multiconfiguration Calculations

Hyperfine structure constants have many applications, but are often hard to calculate accurately due to large and canceling contributions from different terms of the hyperfine interaction operator, and also from different closed and spherically symmetric core subshells that break up due to electron correlation effects. In multiconfiguration calculations, the wave functions are expanded in terms of configuration state functions (CSFs) built from sets of one-electron orbitals. The orbital sets are typically enlarged within the layer-by-layer approach. The calculations are energy-driven, and orbitals in each new layer of correlation orbitals are spatially localized in regions where the weighted total energy decreases the most, overlapping and breaking up different closed core subshells in an irregular pattern. As a result, hyperfine structure constants, computed as expectation values of the hyperfine operators, often show irregular or oscillating convergence patterns. Large orbital sets, and associated large CSF expansions, are needed to obtain converged values of the hyperfine structure constants. We analyze the situation for the states of the {2s22p3,2s22p23p,2s22p24p} odd and {2s22p23s,2s2p4,2s22p24s,2s22p23d} even configurations in N I, and show that the convergence with respect to the increasing sets of orbitals is radically improved by introducing separately optimized orbital sets targeted for describing the spin- and orbital-polarization effects of the 1s and 2s core subshells that are merged with, and orthogonalized against, the ordinary energy-optimized orbitals. In the layer-by-layer approach, the spectroscopic orbitals are kept frozen from the initial calculation and are not allowed to relax in response to the introduced layers of correlation orbitals. To compensate for this lack of variational freedom, the orbitals are transformed to natural orbitals prior to the final calculation based on single and double substitutions from an increased multireference set. The use of natural orbitals has an important impact on the states of the 2s22p23s configuration, bringing the corresponding hyperfine interaction constants in closer agreement with experiment. Relying on recent progress in methodology, the multiconfiguration calculations are based on configuration state function generators, cutting down the time for spin-angular integration by factors of up to 50, compared to ordinary calculations.

ExROPPP: Fast, accurate, and spin-pure calculation of the electronically excited states of organic hydrocarbon radicals.

Recent years have seen an explosion of interest in organic radicals due to their promise for highly efficient organic light-emitting diodes and molecular qubits. However, accurately and inexpensively computing their electronic structure has been challenging, especially for excited states, due to the spin-contamination problem. Furthermore, while alternacy or "pseudoparity" rules have guided the interpretation and prediction of the excited states of closed-shell hydrocarbons since the 1950s, similar general rules for hydrocarbon radicals have not to our knowledge been found yet. In this article, we present solutions to both of these challenges. First, we combine the extended configuration interaction singles method with Pariser-Parr-Pople (PPP) theory to obtain a method that we call ExROPPP (Extended Restricted Open-shell PPP) theory. We find that ExROPPP computes spin-pure excited states of hydrocarbon radicals with comparable accuracy to experiment as high-level general multi-configurational quasi-degenerate perturbation theory calculations but at a computational cost that is at least two orders of magnitude lower. We then use ExROPPP to derive widely applicable rules for the spectra of alternant hydrocarbon radicals, which are completely consistent with our computed results. These findings pave the way for highly accurate and efficient computation and prediction of the excited states of organic radicals.

A comprehensive exploration of structural and electronic properties of molybdenum clusters

Molybdenum clusters, characterized by their unique structure and intriguing catalytic properties, have gained significant attention in recent years. In several existing studies, density functional theory (DFT) methods have been used to find the lowest energy Mo clusters and explore their electronic and magnetic structure. In all cases, with the exception of a single recent study, where a genetic algorithm was employed, initial geometries of the clusters, prior to geometry optimization, were chosen using heuristic approaches based on symmetry considerations and known structures. DFT calculations were performed using different types of pseudopotentials, from soft to hard, and different types of basis sets. However, no comprehensive study has yet been carried out in which a DFT method with the best control on its precision would be complemented by a reliable global minimum search method to find the lowest energy Mo clusters. In this work, we employ a combination of a plane wave-based DFT method and ab initio random structure searching technique to find the lowest energy clusters of up to 10 Mo atoms. In each case, the search has been performed for clusters with different spin multiplicities, which enabled us to explore their magnetic structure. The results are compared for both hard and soft pseudopotentials stressing the importance of treating more electrons explicitly, in agreement with some of the previous studies. For most of the low-energy magnetic structures found, we investigate the distribution of their spin densities, and for all low energy clusters, we confirm their stability by calculating their vibrational structure. For a few smallest clusters, the results of multiconfigurational quantum chemistry calculations are also discussed. Finally, free energies of the Mo clusters, within the quasi-harmonic approximation, are also calculated and discussed.

Time dependent vibrational electronic coupled cluster (VECC) theory for non-adiabatic nuclear dynamics.

A time-dependent vibrational electronic coupled-cluster (VECC) approach is proposed to simulate photo-electron/UV-VIS absorption spectra as well as time-dependent properties for non-adiabatic vibronic models, going beyond the Born-Oppenheimer approximation. A detailed derivation of the equations of motion and a motivation for the ansatz are presented. The VECC method employs second-quantized bosonic construction operators and a mixed linear and exponential ansatz to form a compact representation of the time-dependent wave-function. Importantly, the method does not require a basis set, has only a few user-defined inputs, and has a classical (polynomial) scaling with respect to the number of degrees of freedom (of the vibronic model), resulting in a favorable computational cost. In benchmark applications to small models and molecules, the VECC method provides accurate results compared to multi-configurational time-dependent Hartree calculations when predicting short-time dynamical properties (i.e., photo-electron/UV-VIS absorption spectra) for non-adiabatic vibronic models. To illustrate the capabilities, the VECC method is also successfully applied to a large vibronic model for hexahelicene with 14 electronic states and 63 normal modes, developed in the group by Aranda and Santoro [J. Chem. Theory Comput. 17, 1691, (2021)].

Iron Kβ X-ray Emission Spectroscopy: The Origin of Spectral Features from Atomic to Molecular Systems Using Multi-configurational Calculations.

Kβ X-ray emission spectroscopy (XES) is widely used to fingerprint the local spin of transition-metal ions, including in pump-probe experiments, to identify excited states or in chemical and biological reactions to characterize short-lived intermediates. In this study, the spectra of ferrous and ferric complexes for various spin states were measured experimentally and described theoretically through restricted active space (RAS) calculations including dynamic correlations. Through the RAS calculations from simple atomic models to complex molecular systems, spectral effects such as the exchange interactions, crystal-field strength, and covalent orbital mixing were evaluated and discussed. The calculations find that only the spectral features of low-spin cases show a dependence on the crystal-field strength, particularly for ferrous low spin. The effect of the covalent orbital mixing strength on the first moment of the Kβ1,3 main line and the Kβ1,3-Kβ' energy splitting is quantitatively described. Clear relationships are found within a given nominal spin but less between different spin states, which calls for careful selection of reference spectra in future experiments. This study further advances our understanding of the correlation between changes in experimental spectral features and their corresponding electronic structure information.

Carbene Complexes of Plutonium: Structure, Bonding, and Divergent Reactivity to Lanthanide Analogs.

Organoplutonium chemistry was established in 1965, yet structurally authenticated plutonium-carbon bonds remain rare being limited to π-bonded carbocycle and σ-bonded isonitrile and hydrocarbyl derivatives. Thus, plutonium-carbenes, including alkylidenes and N-heterocyclic carbenes (NHCs), are unknown. Here, we report the preparation and characterization of the diphosphoniomethanide-plutonium complex [Pu(BIPMTMSH)(I)(μ-I)]2 (1Pu, BIPMTMSH = (Me3SiNPPh2)2CH) and the diphosphonioalkylidene-plutonium complexes [Pu(BIPMTMS)(I)(DME)] (2Pu, BIPMTMS = (Me3SiNPPh2)2C) and [Pu(BIPMTMS)(I)(IMe4)2] (3Pu, IMe4 = C(NMeCMe)2), thus disclosing non-actinyl transneptunium multiple bonds and transneptunium NHC complexes. These Pu-C double and dative bonds, along with cerium, praseodymium, samarium, uranium, and neptunium congeners, enable lanthanide-actinide and actinide-actinide comparisons between metals with similar ionic radii and isoelectronic 4f5 vs 5f5 electron-counts within conserved ligand fields over 12 complexes. Quantum chemical calculations reveal that the orbital-energy and spatial-overlap terms increase from uranium to neptunium; however, on moving to plutonium the orbital-energy matching improves but the spatial overlap decreases. The bonding picture that emerges is more complex than the traditional picture of the bonding of lanthanides being ionic and early actinides being more covalent but becoming more ionic left to right. Multiconfigurational calculations on 2M and 3M (M = Pu, Sm) account for the considerably more complex UV/vis/NIR spectra for 5f5 2Pu and 3Pu compared to 4f5 2Sm and 3Sm. Supporting the presence of Pu═C double bonds in 2Pu and 3Pu, 2Pu exhibits metallo-Wittig bond metathesis involving the highest atomic number element to date, reacting with benzaldehyde to produce the alkene PhC(H)═C(PPh2NSiMe3)2 (4) and "PuOI". In contrast, 2Ce and 2Pr do not react with benzaldehyde to produce 4.

Divergent Bimetallic Mechanisms in Copper(II)-Mediated C-C, N-N, and O-O Oxidative Coupling Reactions.

Copper-catalyzed aerobic oxidative coupling of diaryl imines provides a route for conversion of ammonia to hydrazine. The present study uses experimental and density functional theory computational methods to investigate the mechanism of N-N bond formation, and the data support a mechanism involving bimolecular coupling of Cu-coordinated iminyl radicals. Computational analysis is extended to CuII-mediated C-C, N-N, and O-O coupling reactions involved in the formation of cyanogen (NC-CN) from HCN, 1,3-butadiyne from ethyne (i.e., Glaser coupling), hydrazine from ammonia, and hydrogen peroxide from water. The results reveal two different mechanistic pathways. Heteroatom ligands with an uncoordinated lone pair (iminyl, NH2, OH) undergo charge transfer to CuII, generating ligand-centered radicals that undergo facile bimolecular radical-radical coupling. Ligands lacking a lone pair (CN and CCH) form bridged binuclear diamond-core structures that undergo C-C coupling. This mechanistic bifurcation is rationalized by analysis of spin densities in key intermediates and transition states, as well as multiconfigurational calculations. Radical-radical coupling is especially favorable for N-N coupling owing to energetically favorable charge transfer in the intermediate and thermodynamically favorable product formation.

A homoleptic Fe(iv) ketimide complex with a low-lying excited state.

The reaction of 4 equiv. of Li(N[double bond, length as m-dash]C( t Bu)Ph) with FeIICl2 results in isolation of [Li(Et2O)]2[FeII(N[double bond, length as m-dash]C( t Bu)Ph)4] (1), in good yields. The reaction of 1 with 1 equiv. of I2 leads to formation of [FeIV(N[double bond, length as m-dash]C( t Bu)Ph)4] (2), in moderate yields. 57Fe Mössbauer spectroscopy confirms the Fe(iv) oxidation state of 2, and X-ray crystallography reveals that 2 has a square planar coordination geometry along with several intramolecular H⋯C interactions. Furthermore, SQUID magnetometry indicates a small magnetic moment at room temperature, suggestive of an accessible S = 1 state. Both density functional theory and multiconfigurational calculations were done to elucidate the nature of the ground state. Consistent with the experimental results, the ground state was found to be an S = 0 state with an S = 1 excited state close in energy.

Revisiting the role of octahedral symmetry in the interpretation of spectroscopic properties of [OsF6]2- and PtF6 complexes.

The electronic structure of [OsF6]2- and PtF6 complexes was studied by means of CASSCF/NEVPT2 multiconfigurational calculations, including spin-orbital coupling, which is very relevant in the case of these metals. From these calculations, it is possible to establish that in the octahedral symmetry (Oh), the ground state is non-magnetic (Jeff = 0) because of the strong ligand field, and the interaction with paramagnetic excited states is almost negligible, resulting in a non-magnetic behavior, which is in agreement with the experimental evidence.

Photoisomerization mechanism of iminoguanidinium receptors from spectroscopic methods and quantum chemical calculations.

The hydrazone functional group, when coupled with a pyridyl substituent, offers a unique class of widely tunable photoswitches, whose E-to-Z photoisomerization equilibria can be controlled through intramolecular hydrogen bonding between the N-H hydrazone donor and the pyridyl acceptor. However, little is known about the photoisomerization mechanism in this class of compounds. To address this issue, we report a pyridine-appended iminoguanidinium photoswitch that is functionally related to acylhydrazones and provides insight into the photoisomerization processes between the E and Z configurations. The E-to-Z photoisomerization of the E-2-pyridyl-iminoguanidinium cation (2PyMIG) in DMSO, prepared as the bromide salt, was quantified by 1H NMR, and probed in real time with ultrafast laser spectroscopy. The photoisomerization process occurs on a picosecond timescale, resulting in low fluorescence yields. The multiconfigurational reaction path found with the growing string method features a small barrier (4.3 or 6.5 kcal mol-1) that the E isomer in the π-π* state must overcome to reach the minimum energy conical intersection (MECI) connecting the E and Z isomers of 2PyMIG. While two possible pathways exist depending on the orientation of the pyridine ring, both exhibit the same qualitative features along the path and at their MECIs, involving simultaneous changes in the CCNN and CNNC dihedral angles. Furthermore, the ground state barrier for pyridine ring rotation is readily accessible, thus a low barrier pathway to the experimentally observed Z isomer exists for both MECIs leading to a transition from the E isomer to photoproduct. Combining multiconfigurational reaction path calculations using growing string method with time-resolved fluorescence spectroscopy provided crucial insights into the photoisomerization process of 2PyMIG, resulting in both the computational and experimental results pointing to rapid photoisomerization via a surface crossing between the singlet π-π* and the ground states.