Paramagnetic NMR Shielding Tensors Based on Scalar Exact Two-Component and Spin-Orbit Perturbation Theory

Large Hyperfine Coupling Arising from Pseudo-2S Ground States in a Series of Lutetium(II) Metallocene Complexes

The synthesis of molecules with strong coupling between electronic and nuclear spins represents an important challenge in molecular quantum information science. Here, we report the synthesis and characterization of the divalent lutetium metallocene complexes Lu(CpMe5)(CpiPr5) (CpMe5 = pentamethylcyclopentadienyl; CpiPr5 = pentaisopropylcyclopentadienyl), Lu(CpiPr4Et)2 (CpiPr4Et = ethyltetraisopropylcyclopentadienyl), and Lu(CpiPr4)2 (CpiPr4 = tetraisopropylcyclopentadienyl). The molecular structures of these complexes, as determined through single-crystal X-ray diffraction, feature a common bent sandwich geometry, with average Cp-Lu-Cp angles ranging from 159.9° to 152.6°. Analysis of continuous-wave electron paramagnetic resonance (EPR) spectra for the complexes reveals nearly isotropic g tensors with only a slight deviation from that of a free electron. Moreover, an extremely large splitting of the eight-line spectra indicates the presence of strong hyperfine coupling, and simulations provide isotropic hyperfine coupling constants of Aiso = 4.38, 4.30, and 4.17 GHz across the series, where the value of Aiso is found to decrease as the Cp-Lu-Cp angle becomes more acute. Notably, these values are the largest yet observed for any lanthanide complex. Moreover, EPR and computational analysis show that the large values of Aiso stem from large s-orbital character─up to 41.2%─in the corresponding singly occupied molecular orbitals. To our knowledge, this degree of s-character in a molecular orbital is the largest yet reported for an open-shell isolable complex. These results outline a general strategy toward the isolation of paramagnetic molecules with strong hyperfine coupling and highly isotropic doublet electronic ground states.

Isolation of a planar π-aromatic Bi5- ring in a cobalt-based inverse-sandwich-type complex

Monocyclic π-aromatic compounds are ubiquitous throughout almost all fields of natural sciences-as synthons in industrial processes, as ligands of metal complexes for catalysis or sensing and as bioactive molecules. Planar organocycles stand out through their specific way of overcoming electron deficiency by a non-localizable set of (4n + 2)π electrons. By contrast, all-metal aromatic monocycles are still rare, as metal atoms prefer to form clusters with multiply bonded atoms instead. This limits the knowledge and potential of corresponding compounds in chemical syntheses or for innovative materials. Here we report the successful generation of Bi5-, the heaviest analogue of (C5H5)-. Its use as a ligand in [{IMesCo}2(µ,η5:η5-Bi5)] (1) was realized by reacting (TlBi3)2- with [(IMes)2CoCl] (where IMes is bis(1,3-(2,4,6-trimethylphenyl))imidazol-2-ylidene) in ortho-difluorobenzene. Compound 1 is mixed-valence Co0/CoI as verified by µ-SQUID measurements and density functional theory, and embeds the planar Bi5- cycle in an inverse-sandwich-type manner. Capturing Bi5- represents a landmark in the chemistry of all-metal aromatic molecules and defines a new era for aromatic compounds.

Application of the noncollinear Scalmani-Frisch formalism to current density functional theory

We generalize the noncollinear formalism proposed by Scalmani and Frisch [J. Chem. Theory Comput. 8, 2193 (2012)] to include the particle and spin current densities for meta-generalized gradient approximations and local hybrid functionals. This allows us to fully include the impact of spin-orbit coupling in relativistic calculations and for applications to finite magnetic fields. For the latter, we use London atomic orbitals to ensure gauge origin invariance. It is shown that this formalism is superior to the more common canonical noncollinear approach in relativistic calculations, as it naturally includes all three spin current densities in the closed-shell limit and avoids the projection onto the spin magnetization vector. This is important to easily restore rotational invariance in this limit. In addition, the Scalmani-Frisch approach can be made numerically stable and may lead to a nonvanishing local magnetic torque. However, both formalisms are rotationally invariant for open-shell systems and in finite magnetic fields.

A General and Transferable Local Hybrid Functional for Electronic Structure Theory and Many-Fermion Approaches

Density functional theory has become the workhorse of quantum physics, chemistry, and materials science. Within these fields, a broad range of applications needs to be covered. These applications range from solids to molecular systems, from organic to inorganic chemistry, or even from electrons to other Fermions, such as protons or muons. This is emphasized by the plethora of density functional approximations that have been developed for various cases. In this work, two new local hybrid exchange-correlation density functionals are constructed from first-principles, promoting generality and transferability. We show that constraint satisfaction can be achieved even for admixtures with full exact exchange, without sacrificing accuracy. The performance of the new functionals CHYF-PBE and CHYF-B95 is assessed for thermochemical properties, excitation energies, Mössbauer isomer shifts, NMR spin-spin coupling constants, NMR shieldings and shifts, magnetizabilities, and EPR hyperfine coupling constants. Here, the new density functional shows excellent performance throughout all tests and is numerically robust only requiring small grids for converged results. Additionally, both functionals can easily be generalized to arbitrary Fermions as shown for electron-proton correlation energies. Therefore, we outline that density functionals generated in this way are general purpose tools for quantum mechanical studies.

Application of the Adiabatic Connection Random Phase Approximation to Electron-Nucleus Hyperfine Coupling Constants

The electron-nucleus hyperfine coupling constant is a challenging property for density functional methods. For accurate results, hybrid functionals with a large amount of exact exchange are often needed and there is no clear "one-for-all" functional which describes the hyperfine coupling interaction for a large set of nuclei. To alleviate this unfavorable situation, we apply the adiabatic connection random phase approximation (RPA) in its post-Kohn-Sham fashion to this property as a first test. For simplicity, only the Fermi-contact and spin-dipole terms are calculated within the nonrelativistic and the scalar-relativistic exact two-component framework. This requires to solve a single coupled-perturbed Kohn-Sham equation to evaluate the relaxed density matrix, which comes with a modest increase in computational demands. RPA performs remarkably well and substantially improves upon its Kohn-Sham (KS) starting point while also reducing the dependence on the KS reference. For main-group systems, RPA outperforms global, range-separated, and local hybrid functionals─at similar computational costs. For transition-metal compounds and lanthanide complexes, a similar performance as for hybrid functionals is observed. In contrast, related post-Hartree-Fock methods such as Møller-Plesset perturbation theory or CC2 perform worse than semilocal density functionals.

Beyond Electrons: Correlation and Self-Energy in Multicomponent Density Functional Theory

Post-Kohn-Sham methods are used to evaluate the ground-state correlation energy and the orbital self-energy of systems consisting of multiple flavors of different fermions. Starting from multicomponent density functional theory, suitable ways to arrive at the corresponding multicomponent random-phase approximation and the multicomponent Green's functionG W ${GW}$ approximation, including relativistic effects, are outlined. Given the importance of both of this methods in the development of modern Kohn-Sham density functional approximations, this work will provide a foundation to design advanced multicomponent density functional approximations. Additionally, theG W ${GW}$ quasiparticle energies are needed to study light-matter interactions with the Bethe-Salpeter equation.

Paramagnetic Nuclear Magnetic Resonance Shifts for Triplet Systems and Beyond with Modern Relativistic Density Functional Methods

An efficient framework for the calculation of paramagnetic NMR (pNMR) shifts within exact two-component (X2C) theory and (current-dependent) density functional theory (DFT) up to the class of local hybrid functionals (LHFs) is presented. Generally, pNMR shifts for systems with more than one unpaired electron depend on the orbital shielding contribution and a temperature-dependent term. The latter includes zero-field splitting (ZFS), hyperfine coupling (HFC), and the g-tensor. For consistency, we calculate these three tensors at the same level of theory, i.e., using scalar-relativistic X2C augmented with spin-orbit perturbation theory. Results for pNMR chemical shifts of transition-metal complexes reveal that this X2C-DFT framework can yield good results for both the shifts and the individual tensor contributions of metallocenes and related systems, especially if the HFC constant is large. For small HFC constants, the relative error is often large, and sometimes the sign may be off. 4d and 5d complexes with more complicated structures demonstrate the limitations of a fully DFT-based approach. Additionally, a Co-based complex with a very large ZFS and pronounced multireference character is not well described. Here, a hybrid DFT-multireference framework is necessary for accurate results. Our results show that X2C is sufficient to describe relativistic effects and computationally cheaper than a fully relativistic approach. Thus, it allows use of large basis sets for converged HFCs. Overall, current-dependent meta-generalized gradient approximations and LHFs show some potential; however, the currently available functionals leave a lot to be desired, and the predictive power is limited.

Zero-field splitting parameters within exact two-component theory and modern density functional theory using seminumerical integration

An efficient implementation of zero-field splitting parameters based on the work of Schmitt et al. [J. Chem. Phys. 134, 194113 (2011)] is presented. Seminumerical integration techniques are used for the two-electron spin-dipole contribution and the response equations of the spin-orbit perturbation. The original formulation is further generalized. First, it is extended to meta-generalized gradient approximations and local hybrid functionals. For these functional classes, the response of the paramagnetic current density is considered in the coupled-perturbed Kohn-Sham equations for the spin-orbit perturbation term. Second, the spin-orbit perturbation is formulated within relativistic exact two-component theory and the screened nuclear spin-orbit (SNSO) approximation. The accuracy of the implementation is demonstrated for transition-metal and diatomic main-group compounds. The efficiency is assessed for Mn and Mo complexes. Here, it is found that coarse integration grids for the seminumerical schemes lead to drastic speedups while introducing clearly negligible errors. In addition, the SNSO approximation substantially reduces the computational demands and leads to very similar results as the spin-orbit mean field Ansatz.

Exact two-component theory becoming an efficient tool for NMR shieldings and shifts with spin-orbit coupling

We present a gauge-origin invariant exact two-component (X2C) approach within a modern density functional framework, supporting meta-generalized gradient approximations such as TPSS and range-separated hybrid functionals such as CAM-B3LYP. The complete exchange-correlation kernel is applied, including the direct contribution of the field-dependent basis functions and the reorthonormalization contribution from the perturbed overlap matrix. Additionally, the finite nucleus model is available for the electron-nucleus potential and the vector potential throughout. Efficiency is ensured by the diagonal local approximation to the unitary decoupling transformation in X2C as well as the (multipole-accelerated) resolution of the identity approximation for the Coulomb term (MARI-J, RI-J) and the seminumerical exchange approximation. Errors introduced by these approximations are assessed and found to be clearly negligible. The applicability of our implementation to large-scale calculations is demonstrated for a tin pincer-type system as well as low-valent tin and lead complexes. Here, the calculation of the Sn nuclear magnetic resonance shifts for the pincer-type ligand with about 2400 basis functions requires less than 1 h for hybrid density functionals. Further, the impact of spin-orbit coupling on the nucleus-independent chemical shifts and the corresponding ring currents of all-metal aromatic systems is studied.

TURBOMOLE: Today and Tomorrow

TURBOMOLE is a highly optimized software suite for large-scale quantum-chemical and materials science simulations of molecules, clusters, extended systems, and periodic solids. TURBOMOLE uses Gaussian basis sets and has been designed with robust and fast quantum-chemical applications in mind, ranging from homogeneous and heterogeneous catalysis to inorganic and organic chemistry and various types of spectroscopy, light-matter interactions, and biochemistry. This Perspective briefly surveys TURBOMOLE's functionality and highlights recent developments that have taken place between 2020 and 2023, comprising new electronic structure methods for molecules and solids, previously unavailable molecular properties, embedding, and molecular dynamics approaches. Select features under development are reviewed to illustrate the continuous growth of the program suite, including nuclear electronic orbital methods, Hartree-Fock-based adiabatic connection models, simplified time-dependent density functional theory, relativistic effects and magnetic properties, and multiscale modeling of optical properties.

Cleavage of an Aromatic C-C Bond in Ferrocene by Insertion of an Iridium Nitrido Nitrogen Atom

The intermolecular cleavage of C-C bonds is a rare event. Herein, we report on a late transition-metal terminal nitrido complex, which upon oxidation undergoes insertion of the nitrido nitrogen atom into the aromatic C-C bond of ferrocene. This reaction path was confirmed through ¹⁵N and deuterium isotope labeling experiments of the nitrido complex and ferrocenium, respectively. Cyclic voltammetry and UV/vis spectroscopy monitoring of the reaction revealed that oxidation is the initial step, yielding the tentative radical cationic nitrido complex, which is experimentally supported by extended X and Q-band electron paramagnetic resonance (EPR) and ENDOR, UV/vis, vT ¹H NMR, and vibrational spectroscopic data. Density functional theory (DFT) and multireference calculations of this highly reactive intermediate revealed an S = 1/2 ground state. The high reactivity can be traced to the increased electrophilicity in the oxidized complex. Based on high-level PNO-UCCSD(T) calculations and UV/vis kinetic measurements, it is proposed that the reaction proceeds by initial electrophilic exo attack of the nitrido nitrogen atom at the cyclopentadienyl ring and consecutive ring expansion to a pyridine ring.

Reducing Exact Two-Component Theory for NMR Couplings to a One-Component Approach: Efficiency and Accuracy

The self-consistent and complex spin-orbit exact two-component (X2C) formalism for NMR spin-spin coupling constants [ J. Chem. Theory Comput. 17, 2021, 3874-3994] is reduced to a scalar one-component ansatz. This way, the first-order response term can be partitioned into the Fermi-contact (FC) and spin-dipole (SD) interactions as well as the paramagnetic spin-orbit (PSO) contribution. The FC+SD terms are real and symmetric, while the PSO term is purely imaginary and antisymmetric. The relativistic one-component approach is combined with a modern density functional treatment up to local hybrid functionals including the response of the current density. Computational demands are reduced by factors of 8-24 as shown for a large tin compound consisting of 137 atoms. Limitations of the current ansatz are critically assessed for Sn, Pb, Pd, and Pt compounds, i.e. the one-component treatment is not sufficient for tin compounds featuring a few heavy halogen atoms.