An efficient implementation of two-component relativistic exact-decoupling methods for large molecules

Probing the performance of DFT in the structural characterization of [FeFe] hydrogenase models

In this work, a series of DFT and DFT-D methods is combined with double-ζ basis sets to benchmark their performance in predicting the structures of five newly synthesized hexacarbonyl diiron complexes with a bridging ligand featuring a μ-S2C3 motif in a ring-containing unit functionalized with aromatic groups. Such complexes have been considered as [FeFe] hydrogenase catalytic site models with potential for eco-friendly energetic applications. According to this assessment, r2SCAN is identified as the density functional recommended for the reliable description of the molecular and crystal structures of the herein studied models. However, the butterfly (μ-S)2Fe2 core of the models demonstrates a minor deformation of its optimized geometry obtained from both molecular and periodic calculations. The FeFe bond length is slightly underestimated while the FeS bonds tend to be too long. Adding the D3(BJ) correction to r2SCAN does not lead to any improvement in the calculated structures.

Accelerating Relativistic Exact-Two-Component Density Functional Theory Calculations with Graphical Processing Units

Numerical integration of the exchange-correlation potential is an inherently parallel problem that can be significantly accelerated by graphical processing units (GPUs). In this Letter, we present the first implementation of GPU-accelerated exchange-correlation potential in the GauXC library for relativistic, 2-component density functional theory. By benchmarking against copper, silver, and gold coinage metal clusters, we demonstrate the speed and efficiency of our implementation, achieving significant speedup compared to CPU-based calculations. One GPU card provides computational power equivalent to roughly 400 CPU cores in the context of this work. The speedup further increases for larger systems, highlighting the potential of our approach for future, more computationally demanding simulations. Our implementation supports arbitrary angular momentum basis functions, enabling the simulation of systems with heavy elements and providing substantial speedup to relativistic electronic structure calculations. This advancement paves the way for more efficient and extensive computational studies in the field of density functional theory.

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.

Core-electron contributions to the magnetic response of molecules with heavy elements and their significance in aromaticity assessments

This study delves into the magnetic response of core electrons and their influence on the global magnetic response of planar and three-dimensional systems containing heavy elements, employing the removing valence electron (RVE) approximation. We also explore electronic aromaticity indices to understand the potential role of core electrons on electron delocalization in the absence of an external perturbation. The study reveals that core electrons significantly contribute to the overall magnetic response, especially to the magnetic shielding, affecting the interpretation of aromaticity. In contrast, the calculation of the electronic aromaticity indices suggests a negligible participation of the core electrons on electron delocalization. Despite their widespread use, the study emphasizes caution in labeling systems as strongly aromatic based solely on shielding function computations. It is noteworthy to emphasize the limitations associated with each aromaticity criterion; particularly in the context of magnetic shielding function calculations, the core-electron effect contamination is undeniable. Hence, the integration of various criteria becomes imperative for attaining a comprehensive understanding of magnetic responses within complex systems.

Relativistic Coupled Cluster with Completely Renormalized and Perturbative Triples Corrections

We have implemented noniterative triples corrections to the energy from coupled-cluster with single and double excitations (CCSD) within the 1-electron exact two-component (1eX2C) relativistic framework. The effectiveness of both the CCSD(T) and the completely renormalized (CR) CC(2,3) approaches are demonstrated by performing all-electron computations of the potential energy curves and spectroscopic constants of copper, silver, and gold dimers in their ground electronic states. Spin-orbit coupling effects captured via the 1eX2C framework are shown to be crucial for recovering the correct shape of the potential energy curves, and the correlation effects due to triples in these systems change the dissociation energies by about 0.1-0.2 eV or about 4-7%. We also demonstrate that relativistic effects and basis set size and contraction scheme are significantly more important in Au2 than in Ag2 or Cu2.

Circular dichroism of relativistically-moving chiral molecules

Understanding the impact of the relativistic motion of a chiral molecule on its optical response is a prime challenge for fundamental science, but it also has a direct practical relevance in our search for extraterrestrial life. To contribute to these significant developments, we describe a multi-scale computational framework that combines quantum chemistry calculations and full-wave optical simulations to predict the chiral optical response from molecules moving at relativistic speeds. Specifically, the effect of a relativistic motion on the transmission circular dichroism (TCD) of three life-essential biomolecules, namely, B-DNA, chlorophyll a, and chlorophyll b, is investigated. Inspired by previous experiments to detect interstellar chiral molecules, we assume that the molecules move between a stationary observer and a light source, and we study the rotationally averaged TCD as a function of the speed of the molecule.We find that the TCD spectrum that contains the signatures of the molecules shifts with increasing speed to shorter wavelengths, with the effects already being visible for moderate velocities.

Mechanistic Principles of Hydrogen Evolution in the Membrane-Bound Hydrogenase

The membrane-bound hydrogenase (Mbh) from Pyrococcus furiosus is an archaeal member of the Complex I superfamily. It catalyzes the reduction of protons to H2 gas powered by a [NiFe] active site and transduces the free energy into proton pumping and Na+/H+ exchange across the membrane. Despite recent structural advances, the mechanistic principles of H2 catalysis and ion transport in Mbh remain elusive. Here, we probe how the redox chemistry drives the reduction of the proton to H2 and how the catalysis couples to conformational dynamics in the membrane domain of Mbh. By combining large-scale quantum chemical density functional theory (DFT) and correlated ab initio wave function methods with atomistic molecular dynamics simulations, we show that the proton transfer reactions required for the catalysis are gated by electric field effects that direct the protons by water-mediated reactions from Glu21L toward the [NiFe] site, or alternatively along the nearby His75L pathway that also becomes energetically feasible in certain reaction steps. These local proton-coupled electron transfer (PCET) reactions induce conformational changes around the active site that provide a key coupling element via conserved loop structures to the ion transport activity. We find that H2 forms in a heterolytic proton reduction step, with spin crossovers tuning the energetics along key reaction steps. On a general level, our work showcases the role of electric fields in enzyme catalysis and how these effects are employed by the [NiFe] active site of Mbh to drive PCET reactions and ion transport.

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.

Investigation of Isolated IrF5 -, IrF6 - Anions and M[IrF6] (M=Na, K, Rb, Cs) Ion Pairs by Matrix-Isolation Spectroscopy and Relativistic Quantum-Chemical Calculations

The molecular IrF5 -, IrF6 - anions and M[IrF6] (M=Na, K, Rb, Cs) ion pairs were prepared by co-deposition of laser-ablated alkali metal fluorides MF with IrF6 and isolated in solid neon or argon matrices under cryogenic conditions. The free anions were obtained as well by co-deposition of IrF6 with laser-ablated metals (Ir or Pt) as electron sources. The products were characterized in a combined analysis of matrix IR spectroscopy and electronic structure calculations using two-component quasi-relativistic DFT methods accounting for spin-orbit coupling (SOC) effects as well as multi-reference configuration-interaction (MRCI) approaches with SOC. Inclusion of SOC is crucial in the prediction of spectra and properties of IrF6 - and its alkali-metal ion pairs. The observed IR bands and the computations show that the IrF6 - anion adopts an Oh structure in a nondegenerate ground state stabilized by SOC effects, and not a distorted D4h structure in a triplet ground state as suggested by scalar-relativistic calculations. The corresponding "closed-shell" M[IrF6] ion pairs with C3v symmetry are stabilized by coordination of an alkali metal ion to three F atoms, and their structural change in the series from M=Na to Cs was proven spectroscopically. There is no evidence for the formation of IrF7, IrF7 - or M[IrF7] (M=Na, K, Rb, Cs) ion pairs in our experiments.

A novel approach for estimating the strength of argentophilic and aurophilic interactions using QTAIM parameters

Metallophilic interactions, specifically argentophilic (Ag⋯Ag) and aurophilic (Au⋯Au) interactions, play a crucial role in stabilizing various molecular and solid-state structures. In this manuscript, we present a convenient method to estimate the strength of argentophilic and aurophilic interactions based on quantum theory of atoms in molecules (QTAIM) parameters evaluated at the bond critical points connecting the metal centres. We employ density functional theory (DFT) calculations and the QTAIM parameters to develop this energy predictor. To validate the reliability and applicability of our method, we test it using a selection of X-ray crystal structures extracted from the cambridge structural database (CSD), where argentophilic and aurophilic interactions are known to be significant in their solid-state arrangements. This method offers a distinct advantage in systems where multiple interactions, beyond metallophilic interactions, contribute to the overall stability of the structure. By employing our approach, researchers can distinctly quantify the strength of argentophilic and aurophilic interactions, facilitating a deeper understanding of their impact on molecular and solid-state properties. This method fills a critical gap in the existing literature, offering a valuable tool to researchers seeking to unravel the intricate interactions in metal-containing compounds.

Chelate Complexes of 3d Transition Metal Ions─A Challenge for Electronic-Structure Methods?

Different electronic-structure methods were assessed for their ability to predict two important properties of the industrially relevant chelating agent nitrilotriacetic acid (NTA): its selectivity with respect to six different first-row transition metal ions and the spin-state energetics of its complex with Fe(III). The investigated methods encompassed density functional theory (DFT), the random phase approximation (RPA), coupled cluster (CC) theory, and the auxiliary-field quantum Monte Carlo (AFQMC) method, as well as the complete active space self-consistent field (CASSCF) method and the respective on-top methods: second-order N-electron valence state perturbation theory (NEVPT2) and multiconfiguration pair-density functional theory (MC-PDFT). Different strategies for selecting active spaces were explored, and the density matrix renormalization group (DMRG) approach was used to solve the largest active spaces. Despite somewhat ambiguous multi-reference diagnostics, most methods gave relatively good agreement with experimental data for the chemical reactions connected to the selectivity, which only involved transition-metal complexes in their high-spin state. CC methods yielded the highest accuracy followed by range-separated DFT and AFQMC. We discussed in detail that even higher accuracies can be obtained with NEVPT2, under the prerequisite that consistent active spaces along the entire chemical reaction can be selected, which was not the case for reactions involving Fe(III). A bigger challenge for electronic-structure methods was the prediction of the spin-state energetics, which additionally involved lower spin states that exhibited larger multi-reference diagnostics. Conceptually different, typically accurate methods ranging from CC theory via DMRG-NEVPT2 in combination with large active spaces to AFQMC agreed well that the high-spin state is energetically significantly favored over the other spin states. This was in contrast to most DFT functionals and RPA which yielded a smaller stabilization and some common DFT functionals and MC-PDFT even predicting the low-spin state to be energetically most favorable.

The p-block challenge: assessing quantum chemistry methods for inorganic heterocycle dimerizations

The elements of the p-block of the periodic table are of high interest in various chemical and technical applications like frustrated Lewis-pairs (FLP) or opto-electronics. However, high-quality benchmark data to assess approximate density functional theory (DFT) for their theoretical description are sparse. In this work, we present a benchmark set of 604 dimerization energies of 302 "inorganic benzenes" composed of all non-carbon p-block elements of main groups III to VI up to polonium. This so-called IHD302 test set comprises two classes of structures formed by covalent bonding and by weaker donor-acceptor (WDA) interactions, respectively. Generating reliable reference data with ab initio methods is challenging due to large electron correlation contributions, core-valence correlation effects, and especially the slow basis set convergence. To compute reference values for these dimerization reactions, after thorough testing, we applied a computational protocol using state-of-the-art explicitly correlated local coupled cluster theory termed PNO-LCCSD(T)-F12/cc-VTZ-PP-F12(corr.). It includes a basis set correction at the PNO-LMP2-F12/aug-cc-pwCVTZ level. Based on these reference data, we assess 26 DFT methods in combination with three different dispersion corrections and the def2-QZVPP basis set, five composite DFT approaches, and five semi-empirical quantum mechanical methods. For the covalent dimerizations, the r2SCAN-D4 meta-GGA, the r2SCAN0-D4 and ωB97M-V hybrids, and the revDSD-PBEP86-D4 double-hybrid functional are found to be the best-performing methods among the evaluated functionals of the respective class. However, since def2 basis sets for the 4th period are not associated to relativistic pseudo-potentials, we obtained significant errors in the covalent dimerization energies (up to 6 kcal mol-1) for molecules containing p-block elements of the 4th period. Significant improvements were achieved for systems containing 4th row elements by using ECP10MDF pseudopotentials along with re-contracted aug-cc-pVQZ-PP-KS basis sets introduced in this work with the contraction coefficients taken from atomic DFT (PBE0) calculations. Overall, the IHD302 set represents a challenge to contemporary quantum chemical methods. This is due to a large number of spatially close p-element bonds which are underrepresented in other benchmark sets, and the partial covalent bonding character for the WDA interactions. The IHD302 set may be helpful to develop more robust and transferable approximate quantum chemical methods in the future.

Optical gain and entanglement through dielectric confinement and electric field in InP quantum dots

Quantum dots are widely recognized for their advantageous light-emitting properties. Their excitonic fine structure along with the high quantum yields offers a wide range of possibilities for technological applications. However, especially for the case of colloidal QDs, there are still characteristics and properties which are not adequately controlled and downgrade their performance for applications which go far beyond the simple light emission. Such a challenging task is the ability to manipulate the energetic ordering of exciton and biexciton emission and subsequently control phenomena such as Auger recombination, optical gain and photon entanglement. In the present work we attempt to engineer this ordering for the case of InP QDs embedded in polymer matrix, by means of their size, the dielectric confinement and external electric fields. We employ well tested, state of the art theoretical methods, in order to explore the conditions under which the exciton-biexciton configuration creates the desired conditions either for optical gain or photon entanglement. Indeed, this appears to be feasible for QDs with small diameters (1 nm, 1.5 nm) embedded in a host material with high dielectric constant and additional external electric fields. These findings offer a new design principle which might be complementary to the well-established type II core-shell QDs approach for achieving electron-hole separation.

Dirac-Coulomb-Breit Molecular Mean-Field Exact-Two-Component Relativistic Equation-of-Motion Coupled-Cluster Theory

We present a relativistic equation-of-motion coupled-cluster with single and double excitation formalism within the exact two-component framework (X2C-EOM-CCSD), where both scalar relativistic effects and spin-orbit coupling are variationally included at the reference level. Three different molecular mean-field treatments of relativistic corrections, including the one-electron, Dirac-Coulomb, and Dirac-Coulomb-Breit Hamiltonian, are considered in this work. Benchmark calculations include atomic excitations and fine-structure splittings arising from spin-orbit coupling. Comparison with experimental values and relativistic time-dependent density functional theory is also carried out. The computation of the oscillator strength using the relativistic X2C-EOM-CCSD approach allows for studies of spin-orbit-driven processes, such as the spontaneous phosphorescence lifetime.

Overdestabilization vs Overstabilization in the Theoretical Analysis of f-Orbital Covalency

The complex nature of the f-orbital electronic structures and their interaction with the chemical environment pose significant computational challenges. Advanced computational techniques that variationally include scalar relativities and spin-orbit coupling directly at the molecular orbital level have been developed to address this complexity. Among these, variational relativistic multiconfigurational multireference methods stand out for their high accuracy and systematic improvement in studies of f-block complexes. Additionally, these advanced methods offer the potential for calibrating low-scaling electronic structure methods such as density functional theory. However, studies on the Cl K-edge X-ray absorption spectra of the [Ce(III)Cl6]3- and [Ce(IV)Cl6]2- complexes show that time-dependent density functional theory with approximate exchange-correlation kernels can lead to inaccuracies, resulting in an overstabilization of 4f orbitals and incorrect assessments of covalency. In contrast, approaches utilizing small active space wave function methods may understate the stability of these orbitals. The results herein demonstrate the need for large active space, multireference, and variational relativistic methods in studying f-block complexes.

Unveiling Hidden Shake-Up Features in the Uranyl M4-Edge Spectrum

The M4,5-edge high energy resolution X-ray absorption near-edge structure (HR-XANES) spectra of actinyls offer valuable insights into the electronic structure and bonding properties of heavy-element complexes. To conduct a comprehensive spectral analysis, it is essential to employ computational methods that accurately account for relativistic effects and electron correlation. In this work, we utilize variational relativistic multireference configurational interaction methods to compute and analyze the X-ray M4-edge absorption spectrum of uranyl. By employing these advanced computational techniques, we achieve excellent agreement between the calculated spectral features and experimental observations. Moreover, the calculations unveil significant shake-up features, which arise from the intricate interplay between strongly correlated 3d core-electron and ligand excitations. This research provides important theoretical insights into the spectral characteristics of heavy-element complexes. Furthermore, it establishes the foundation for utilizing M4,5-edge spectroscopy as a means to investigate the chemical activities of such complexes. By leveraging this technique, we can gain a deeper understanding of the bonding behavior and reactivity of heavy-element compounds.

Benchmark Study on the Calculation of 207Pb NMR Chemical Shifts

A benchmark set for the computation of 207Pb nuclear magnetic resonance (NMR) chemical shifts is presented. The PbS50 set includes conformer ensembles of 50 lead-containing molecular compounds and their experimentally measured 207Pb NMR chemical shifts. Various bonding motifs at the Pb center with up to seven bonding partners are included. Six different solvents were used in the measurements. The respective shifts lie in the range between +10745 and -5030 ppm. Several calculation settings are assessed by evaluating computed 207Pb NMR shifts for the use with different density functional approximations (DFAs), relativistic approaches, treatment of the conformational space, and levels for geometry optimization. Relativistic effects were included explicitly with the zeroth order regular approximation (ZORA), for which only the spin-orbit variant was able to yield reliable results. In total, seven GGAs and three hybrid DFAs were tested. Hybrid DFAs significantly outperform GGAs. The most accurate DFAs are mPW1PW with a mean absolute deviation (MAD) of 429 ppm and PBE0 with an MAD of 446 ppm. Conformational influences are small as most compounds are rigid, but more flexible structures still benefit from Boltzmann averaging. Including explicit relativistic treatments such as SO-ZORA in the geometry optimization does not show any significant improvement over the use of effective core potentials (ECPs).

Comparison of Variational and Perturbative Spin-Orbit Coupling within Two-Component CASSCF

The modeling of spin-orbit coupling (SOC) remains a challenge in computational chemistry due to the high computational cost. With the rising popularity of spin-driven processes and f-block metals in chemistry and materials science, it is incumbent on the community to develop accurate multiconfigurational SOC methods that scale to large systems and understand the limits of different treatments of SOC. Herein, we introduce an implementation of perturbative SOC in scalar-relativistic two-component CASSCF (srX2C-CASSCF-SO). Perspectives on the limitations and accuracy of srX2C-CASSCF-SO are presented via benchmark calculations.

Economical quasi-Newton unitary optimization of electronic orbitals

We present an efficient quasi-Newton orbital solver optimized to reduce the number of gradient evaluations and other computational steps of comparable cost. The solver optimizes orthogonal orbitals by sequences of unitary rotations generated by the (preconditioned) limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm equipped with trust-region step restriction. The low-rank structure of the L-BFGS inverse Hessian is exploited when solving the trust-region problem. The efficiency of the proposed "Quasi-Newton Unitary Optimization with Trust-Region" (QUOTR) solver is compared to that of the standard Roothaan-Hall approach accelerated by the Direct Inversion of Iterative Subspace (DIIS), and other exact and approximate Newton solvers for mean-field (Hartree-Fock and Kohn-Sham) problems.

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.

Scalar Relativistic Effects with Multiwavelets: Implementation and Benchmark

The importance of relativistic effects in quantum chemistry is widely recognized, not only for heavier elements but throughout the periodic table. At the same time, relativistic effects are strongest in the nuclear region, where the description of electrons through a linear combination of atomic orbitals becomes more challenging. Furthermore, the choice of basis sets for heavier elements is limited compared with lighter elements where precise basis sets are available. Thanks to the framework of multiresolution analysis, multiwavelets provide an appealing alternative to overcoming this challenge: they lead to robust error control and adaptive algorithms that automatically refine the basis set description until the desired precision is reached. This allows one to achieve a proper description of the nuclear region. In this work, we extended the multiwavelet-based code MRChem to the scalar zero-order regular approximation framework. We validated our implementation by comparing the total energies for a small set of elements and molecules. To confirm the validity of our implementation, we compared both against a radial numerical code for atoms and the plane-wave-based code EXCITING.

Range Separation Method for Density Functional Theory Based on Two-Electron Infinite-Order Two-Component Hamiltonian

The range separation method for density functional theory (DFT) was extended to a two-component relativistic theory based on the unitary transformation of one- and two-electron operators and a density operator. In the framework of the spin-free infinite-order two-component Hamiltonian, we implemented several types of two-electron integrals of range-separated two-electron interactions arising from the unitary transformation. Numerical assessments were performed using long-range-corrected (LC)-DFT, which utilizes the range separation of an exchange functional. The present method successfully reproduced the reference values obtained by the four-component LC-DFT calculations when the whole unitary transformations of one-electron, full-range, and range-separated two-electron operators and a density operator were considered. An efficient scheme for the unitary transformation, which is termed the local unitary transformation (LUT), was also applied to the range-separated two-electron term and other operators. The LUT method reduced the computational costs of the LC-DFT calculations significantly without any loss of accuracy.

A DFT perspective on organometallic lanthanide chemistry

Computational studies of the coordination chemistry and bonding of lanthanides have grown in recent decades as the need for understanding the distinct physical, optical, and magnetic properties of these compounds increased. Density functional theory (DFT) methods offer a favorable balance of computational cost and accuracy in lanthanide chemistry and have helped to advance the discovery of novel oxidation states and electronic configurations. This Frontier article examines the scope and limitations of DFT in interpreting structural and spectroscopic data of low-valent lanthanide complexes, elucidating periodic trends, and predicting their properties and reactivity, presented through selected examples.

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.

Ligand-Induced Symmetry Breaking as the Origin of Multiexponential Photoluminescence Decay in CdSe Quantum Dots

The bright photoluminescence (PL) of colloidal CdSe quantum dots (QDs) makes them interesting for optical applications. For most of them, well-defined PL properties, dominated by a single excitonic state, are required. However, in many PL experiments with QD ensembles, multiexponential decay was observed. On the basis of spin-orbit density functional theory and screened configuration interaction calculations, we show that highly symmetric and defect-free CdSe QDs with diameters of 1.7 and 2.0 nm possess a multiexponential low-temperature PL at the single-dot level. This is a consequence of ligand-induced symmetry breaking with a subsequent rearrangement of the lowest eight excitonic states in two sets of four singly degenerate excitonic states. For each set, the lowest state is dark and the other three are bright. We find that the splitting between the sets can be modified by the coverage and choice of the ligand, which facilitates the engineering of the PL properties of CdSe QDs.

Improving One-Electron Exact-Two-Component Relativistic Methods with the Dirac-Coulomb-Breit-Parameterized Effective Spin-Orbit Coupling

In photochemical processes, spin-orbit coupling plays a crucial role in determining the outcome of the reaction. However, the exact treatment of the Dirac-Coulomb-Breit two-electron operator required for rigorous inclusion of spin-orbit coupling is computationally prohibitive. To address this challenge, we present a Dirac-Coulomb-Breit-parameterized screened-nuclear spin-orbit factor to approximate two-electron spin-orbit couplings in the effective one-electron spin-orbit Hamiltonian. We propose two schemes, the universal and row-dependent parameterizations, to further improve the accuracy of the method. Benchmark calculations on both atomic and molecular systems are performed and compared to results from the computationally expensive four-component Dirac-Coulomb-Breit method. The Dirac-Coulomb-Breit-parameterized approach offers a more computationally feasible method for accurate spin-orbit coupling calculations.

NMR chemical shift of confined 129Xe: coordination number, paramagnetic channels and molecular dynamics in a cryptophane-A biosensor

Advances in hyperpolarisation and indirect detection have enabled the development of xenon nuclear magnetic resonance (NMR) biosensors (XBSs) for molecule-selective sensing in down to picomolar concentration. Cryptophanes (Crs) are popular cages for hosting the Xe "spy". Understanding the microscopic host-guest chemistry has remained a challenge in the XBS field. While early NMR computations of XBSs did not consider the important effects of host dynamics and explicit solvent, here we model the motionally averaged, relativistic NMR chemical shift (CS) of free Xe, Xe in a prototypic CrA cage and Xe in a water-soluble CrA derivative, each in an explicit H2O solvent, over system configurations generated at three different levels of molecular dynamics (MD) simulations. We confirm the "contact-type" character of the Xe CS, arising from the increased availability of paramagnetic channels, magnetic couplings between occupied and virtual orbitals through the short-ranged orbital hyperfine operator, when neighbouring atoms are in contact with Xe. Remarkably, the Xe CS in the present, highly dynamic and conformationally flexible situations is found to depend linearly on the coordination number of the Xe atom. We interpret the high- and low-CS situations in terms of the magnetic absorption spectrum and choose our preference among the used MD methods based on comparison with the experimental CS. We check the role of spin-orbit coupling by comparing with fully relativistic CS calculations. The study outlines the computational workflow required to realistically model the CS of Xe confined in dynamic cavity structures under experimental conditions, and contributes to microscopic understanding of XBSs.

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.

φ-Aromaticity in prismatic {Bi6}-based clusters

The occurrence of aromaticity in organic molecules is widely accepted, but its occurrence in purely metallic systems is less widespread. Molecules comprising only metal atoms (M) are known to be able to exhibit aromatic behaviour, sustaining ring currents inside an external magnetic field along M-M connection axes (σ-aromaticity) or above and below the plane (π-aromaticity) for cyclic or cage-type compounds. However, all-metal compounds provide an extension of the electrons' mobility also in other directions. Here, we show that regular {Bi₆} prisms exhibit a non-localizable molecular orbital of f-type symmetry and generate a strong ring current that leads to a behaviour referred to as φ-aromaticity. The experimentally observed heterometallic cluster [{CpRu}₃Bi₆]^-, based on a regular prismatic {Bi₆} unit, displays aromatic behaviour; according to quantum chemical calculations, the corresponding hypothetical Bi₆^2- prism shows a similar behaviour. By contrast, [{(cod)Ir}₃Bi₆] features a distorted Bi₆ moiety that inhibits φ-aromaticity.

State Interaction Linear Response Time-Dependent Density Functional Theory with Perturbative Spin-Orbit Coupling: Benchmark and Perspectives

Spin-orbit coupling (SOC) is an important driving force in photochemistry. In this work, we develop a perturbative spin-orbit coupling method within the linear response time-dependent density function theory framework (TDDFT-SO). A full state interaction scheme, including singlet-triplet and triplet-triplet coupling, is introduced to describe not only the coupling between the ground and excited states, but also between excited states with all couplings between spin microstates. In addition, expressions to compute spectral oscillator strengths are presented. Scalar relativity is included variationally using the second-order Douglas-Kroll-Hess Hamiltonian, and the TDDFT-SO method is validated against variational SOC relativistic methods for atomic, diatomic, and transition metal complexes to determine the range of applicability and potential limitations. To demonstrate the robustness of TDDFT-SO for large-scale chemical systems, the UV-Vis spectrum of Au₂₅(SR)₁₈ ^- is computed and compared to experiment. Perspectives on the limitation, accuracy, and capability of perturbative TDDFT-SO are presented via analyses of benchmark calculations. Additionally, an open-source Python software package (PyTDDFT-SO) is developed and released to interface with the Gaussian 16 quantum chemistry software package to perform this calculation.

Ion-Selective Assembly of Supertetrahedral Selenido Germanate Clusters for Alkali Metal Ion Capture and Separation

Supertetrahedral chalcogenido (semi)metalate cluster-based frameworks possess high selectivity for alkali metal cations, matching the specific charge density of their inner surfaces, which enables their use as ion-exchange materials. Aggregates of the supertetrahedral chalcogenido metalate cluster offer even new perspectives for metal ion capture and separation. Herein, we report on ionothermal preparation of two corresponding model compounds, (C₂C₁Im)₇[Cs@Ge^II₄(Ge^IV₄Se₁₀)₄] (1) and (C₂C₁Im)₁₀[Na₅(CN)₆@Cu₆(Ge₄Se₁₀)₄(Cu)] (2). Their formation is reliant on one specific cation type each, Cs⁺ for 1 and Na⁺ for 2, thus providing promising separation potential during crystallization. Compound 1 is based on the largest discrete binary selenido germanate cluster reported to date and the first mixed-valent chalcogenido germanate(II/IV) supertetrahedron. Moreover, it adds to the few examples of chalcogenides capable of capturing Cs⁺ ions. Its high selectivity for Cs⁺ compared to that of Li⁺, Na⁺, K⁺, and Rb⁺ was confirmed by single-crystal X-ray diffraction, energy-dispersive X-ray spectroscopy, and electrospray ionization mass spectrometry. Quantum chemical studies indicate that smaller ions, K⁺ and Rb⁺, could also be embedded in an isolated cluster assembly, but as the cluster aggregate slightly distorts for crystallization, the selectivity for Cs⁺ becomes exclusive in the salt. The anionic substructure of compound 2 is based on a two-dimensional network of supramolecular assemblies and exhibits an exclusive preference for Na⁺. This work thus provides the first comprehensive insight into the selective incorporation of specific alkali metal ions into supramolecular aggregates of supertetrahedral chalcogenide clusters, as a promising basis for new ion trapping techniques─especially for heavy alkali metal ions that pose environmental challenges.

Simulating Spin-Orbit Coupling with Quasidegenerate N-Electron Valence Perturbation Theory

We present the first implementation of spin-orbit coupling effects in fully internally contracted second-order quasidegenerate N-electron valence perturbation theory (SO-QDNEVPT2). The SO-QDNEVPT2 approach enables the computations of ground- and excited-state energies and oscillator strengths combining the description of static electron correlation with an efficient treatment of dynamic correlation and spin-orbit coupling. In addition to SO-QDNEVPT2 with the full description of one- and two-body spin-orbit interactions at the level of two-component Breit-Pauli Hamiltonian, our implementation also features a simplified approach that takes advantage of spin-orbit mean-field approximation (SOMF-QDNEVPT2). The accuracy of these methods is tested for the group 14 and 16 hydrides, 3d and 4d transition metal ions, and two actinide dioxides (neptunyl and plutonyl dications). The zero-field splittings of group 14 and 16 molecules computed using SO-QDNEVPT2 and SOMF-QDNEVPT2 are in good agreement with the available experimental data. For the 3d transition metal ions, the SO-QDNEVPT2 method is significantly more accurate than SOMF-QDNEVPT2, while no substantial difference in the performance of two methods is observed for the 4d ions. Finally, we demonstrate that for the actinide dioxides the results of SO-QDNEVPT2 and SOMF-QDNEVPT2 are in good agreement with the data from previous theoretical studies of these systems. Overall, our results demonstrate that SO-QDNEVPT2 and SOMF-QDNEVPT2 are promising multireference methods for treating spin-orbit coupling with a relatively low computational cost.

Fully Tin-Coated Coinage Metal Ions: A Pincer-Type Bis-stannylene Ligand for Exclusive Tetrahedral Complexation

The synthesis of a novel bis-stannylene pincer ligand and its complexation with coinage metals (Cu^I , Ag^I and Au^I ) are described. All coinage metal centres are in tetrahedral coordination environments in the solid state and are exclusively coordinated by four neutral Sn^II donors. ¹¹⁹ Sn NMR provided information about the behaviour in solution. All of the isolated compounds have photoluminescent properties, and these were investigated at low and elevated temperatures. Compared to the free bis-stannylene ligand, coordination to coinage metals led to an increase in the luminescence intensity. The new compounds were investigated in detail through all-electron relativistic density functional theory (DFT) calculations.

Modeling Nonresonant X-ray Emission of Second- and Third-Period Elements without Core-Hole Reference States and Empirical Parameters

Nonresonant X-ray emission (XE) energies and oscillator strengths are obtained using the effective potential of the generalized Kohn-Sham semi-canonical projected random phase approximation (GKS-spRPA) method. XE energies are estimated as a difference between the valence and core ionization eigenvalues, while the oscillator strengths are obtained within a frozen orbital approximation. This straightforward approach provides accurate XE energies without any need for core-hole reference states, empirical shifting parameters, or tuning of density functionals. To account for relativistic corrections to the core orbitals, we have formulated a scalar relativistic (sr) GKS-spRPA approach based on the spin-free X2C one-electron Hamiltonian. The sr-GKS-spRPA method provides highly reliable XE energies using uncontracted basis-sets on atoms where the core-hole is created prior to emission. For the largest basis-sets used in our study, using completely uncontracted polarized core-valence Dunning basis-sets, the mean absolute errors (MAEs) are within 0.7 eV compared to experimental reference values for a test-set consisting of 27 valence-to-core XE energies of molecules with second- and third-period elements. Considering a balance of accuracy and computational effort, we recommend the use of s-uncontracted def2-TZVP for second-period and all-uncontracted def2-TZVP for third-period elements. For this recommended basis-set, the MAE is 0.2 eV. The analytically continued sr-GKS-spRPA approach, with an O(N4) computational cost, enables efficient computation of XE spectra of molecules such as S₈ and C₆₀ with several core-hole states.

Relativistic Kramers-Unrestricted Exact-Two-Component Density Matrix Renormalization Group

In this work we develop a variational relativistic density matrix renormalization group (DMRG) approach within the exact-two-component (X2C) framework (X2C-DMRG), using spinor orbitals optimized with the two-component relativistic complete active space self-consistent field. We investigate fine-structure splittings of p- (Ga, In, Tl) and d-block (Sc, Y, La) atoms and excitation energies of monohydride molecules (GeH, SnH, and TlH) with X2C-DMRG calculations using an all-electron relativistic Hamiltonian in a Kramers-unrestricted basis. We find that X2C-DMRG yields accurate ²P and ²D splittings compared to multireference configuration interaction with singles and doubles (MRCISD). We also investigated the degree of symmetry breaking in the atomic multiplets and convergence of electron correlation in the total energies. Symmetry breaking can be large in some cases (∼30 meV); however, increasing the number of renormalized block states m for the DMRG optimization recovers the symmetry breaking by several orders of magnitude. Encouragingly, we find the convergence of electron correlation to be close to MRCISDTQ5 quality. Relativistic X2C-DMRG approaches are important for cases where spin-orbit coupling is significant and the underlying reference wave function requires a large determinantal space. We are able to obtain quantitatively correct fine-structure splittings for systems up to 10¹⁹ number of determinants with traditional CI approaches, which are currently unfeasible to converge for the field.

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

The temperature-dependent Fermi-contact and pseudocontact terms are important contributions to the paramagnetic NMR shielding tensor. Herein, we augment the scalar-relativistic (local) exact two-component (X2C) framework with spin-orbit perturbation theory including the screened nuclear spin-orbit correction for the EPR hyperfine coupling and g tensor to compute these temperature-dependent terms. The accuracy of this perturbative ansatz is assessed with the self-consistent spin-orbit two-component and four-component treatments serving as reference. This shows that the Fermi-contact and pseudocontact interaction is sufficiently described for paramagnetic NMR shifts; however, larger deviations are found for the EPR spectra and the principle components of the EPR properties of heavy elements. The impact of the perturbative treatment is further compared to that of the density functional approximation and the basis set. Large-scale calculations are routinely possible with the multipole-accelerated resolution of the identity approximation and the seminumerical exchange approximation, as shown for [CeTi₆O₃(OiPr)₉(salicylate)₆].

A Local Hybrid Exchange Functional Approximation from First Principles

Local hybrid functionals are a more flexible class of density functional approximations allowing for a position-dependent admixture of exact exchange. This additional flexibility, however, comes with a more involved mathematical form and a more complicated design. A common denominator for previously constructed local hybrid funtionals is usage of thermochemical benchmark data to construct these functionals. Herein, we design a local hybrid functional without relying on benchmark data. Instead, we construct it in a more ab initio manner, following the principles of modern meta-generalized gradient approximations and considering theoretical constrains. To achieve this, we make use of the density matrix expansion and a local mixing function based on an approximate correlation length. The accuracy of the developed density functional approximation is assessed for thermochemistry, excitation energies, polarizabilities, magnetizabilities, NMR spin-spincoupling constants, NMR shieldings and shifts, as well as EPR g-tensors and hyperfine coupling constants. Here, the new exchange functional shows a robust performance and is especially well suited for atomization energies, barrier heights, excitation energies, NMR coupling constants, and EPR properties, whereas it looses some ground for the NMR shifts.Therefore, the designed functional is a major step forwards for functionals that have been designed from first principles.

Communication: Impact of the current density on paramagnetic NMR properties

Meta-generalized gradient approximations (meta-GGAs) and local hybrid functionals generally depend on the kinetic energy density τ. For magnetic properties, this necessitates generalizations to ensure gauge invariance. In most implementations, τ is generalized by incorporating the external magnetic field. However, this introduces artifacts in the response of the density matrix and does not satisfy the iso-orbital constraint. Here, we extend previous approaches based on the current density to paramagnetic NMR shieldings and EPR g-tensors. The impact is assessed for main-group compounds and transition-metal complexes considering 25 density functional approximations. It is shown that the current density leads to substantial improvements-especially for the popular Minnesota and SCAN functional families. Thus, we strongly recommend to use the current density generalized τ in paramagnetic NMR and EPR calculations with meta-GGAs.

Relativistic nonorthogonal configuration interaction: application to L2,3-edge X-ray spectroscopy

In this article, we develop a relativistic exact-two-component nonorthogonal configuration interaction (X2C-NOCI) for computing L-edge X-ray spectra. This article to our knowledge is the first time NOCI has been used for relativistic wave functions. A set of molecular complexes, including SF₆, SiCl₄ and [FeCl₆]^3-, are used to demonstrate the accuracy and computational scaling of the X2C-NOCI method. Our results suggest that X2C-NOCI is able to satisfactorily capture the main features of the L_2,3-edge X-ray absorption spectra. Excitations from the core require a large amount of orbital relaxation to yield reasonable energies and X2C-NOCI allows us to treat orbital optimization explicitly. However, the cost of computing the nonorthogonal coupling is higher than in conventional CI. Here, we propose an improved integral screening using overlap-scaled density combined with a continuous measure of the generalized Slater-Condon rules that allows us to estimate if an element is zero before attempting a two-electron integral contraction.

Exact-Two-Component Relativistic Multireference Second-Order Perturbation Theory

As the relativistic corrections become stronger for late-row elements, the fully perturbative treatment of spin-orbit coupling and dynamic correlation may become inadequate for accurate descriptions of chemical properties. In this work, we introduce a determinant-based Kramers-unrestricted exact-two-component multireference second-order perturbation (X2C-MRPT2) method which variationally includes relativistic corrections with a perturbative dynamic correlation. The restricted active space partitioning scheme is employed to provide an adjustable correlation space for the second-order perturbation treatment. The multistate perturbation theory is also developed to improve the descriptions of ground and excited states. Benchmark studies of atomic fine-structure splittings and spectroscopic constants of molecular monohydrides using X2C-MRPT2 are compared to the other perturbative and variational approaches. The results suggest that X2C-MRPT2 is a highly accurate alternative to the fully variational multireference configuration interaction method at only a small fraction of the computational cost.

Investigation of Molecular Iridium Fluorides IrFn (n=1-6): A Combined Matrix-Isolation and Quantum-Chemical Study

The photo-initiated defluorination of iridium hexafluoride (IrF6 ) was investigated in neon and argon matrices at 6 K, and their photoproducts are characterized by IR and UV-vis spectroscopies as well as quantum-chemical calculations. The primary photoproducts obtained after irradiation with λ=365 nm are iridium pentafluoride (IrF5 ) and iridium trifluoride (IrF3 ), while longer irradiation of the same matrix with λ=278 nm produced iridium tetrafluoride (IrF4 ) and iridium difluoride (IrF2 ) by Ir-F bond cleavage or F2 elimination. In addition, IrF5 can be reversed to IrF6 by adding a F atom when exposed to blue-light (λ=470 nm) irradiation. Laser irradiation (λ=266 nm) of IrF4 also generated IrF6 , IrF5 , IrF3 and IrF2 . Alternatively, molecular binary iridium fluorides IrFn (n=1-6) were produced by co-deposition of laser-ablated iridium atoms with elemental fluorine in excess neon and argon matrices under cryogenic conditions. Computational studies up to scalar relativistic CCSD(T)/triple-ζ level and two-component quasirelativistic DFT computations including spin-orbit coupling effects supported the formation of these products and provided detailed insights into their molecular structures by their characteristic Ir-F stretching bands. Compared to the Jahn-Teller effect, the influence of spin-orbit coupling dominates in IrF5 , leading to a triplet ground state with C4v symmetry, which was spectroscopically detected in solid argon and neon matrices.

Intersystem Crossings in Late-Row Elements: A Perspective

Intersystem crossing (ISC), a vital component of the electronic and nuclear transitions that compose photophysics, has been successfully simulated in light elements and transition metal complexes. Derived from the Z-dependent spin-orbit coupling (SOC), ISC is expected to be of greater importance in heavier elements, but few attempts have been made at the simulation of ISC in lanthanides or actinides. In this work, we explore several of the challenges that will need to be overcome in order to treat ISC in late-row elements, including the loss of spin as a good quantum number, the need to include SOC variationally via two- or four-component electronic structure, and the high density of states present in late-row complexes. Density functional theory (DFT) calculations are used to illustrate several of these effects, while a model Hamiltonian is used to illustrate the importance of momentum rescaling in surface hopping simulations of strongly coupled states.

Exact-Two-Component Multiconfiguration Pair-Density Functional Theory

Molecules containing late-row elements exhibit large relativistic effects. To account for both relativistic effects and electron correlation in a computationally inexpensive way, we derived a formulation of multiconfiguration pair-density functional theory with the relativistic exact-two-component Hamiltonian (X2C-MC-PDFT). In this new method, relativistic effects are included during variational optimization of a reference wave function by exact-two-component complete active-space self-consistent-field (X2C-CASSCF) theory, followed by an energy evaluation using pair-density functional theory. Benchmark studies of excited-state and ground-state fine-structure splitting of atomic species show that X2C-MC-PDFT can significantly improve the X2C-CASSCF results by introducing additional state-specific electron correlation.

Quasi-Relativistic Calculation of EPR g Tensors with Derivatives of the Decoupling Transformation, Gauge-Including Atomic Orbitals, and Magnetic Balance

We present an exact two-component (X2C) ansatz for the EPR g tensor using gauge-including atomic orbitals (GIAOs) and a magnetically balanced basis set expansion. In contrast to previous X2C and four-component relativistic ansätze for the g tensor, this implementation results in a gauge-origin-invariant formalism. Furthermore, the derivatives of the relativistic decoupling matrix are incorporated to form the complete analytical derivative of the X2C Hamiltonian. To reduce the associated computational costs, we apply the diagonal local approximation to the unitary decoupling transformation (DLU). The quasi-relativistic X2C and DLU-X2C Hamiltonians accurately reproduce the results of the parent four-component relativistic theory when accounting for two-electron picture-change effects with the modified screened nuclear spin-orbit approximation in the respective one-electron integrals and integral derivatives. According to our benchmark studies, the uncontracted Dyall and segmented-contracted Karlsruhe x2c-type basis sets perform well when compared to large even-tempered basis sets. Moreover, (range-separated) hybrid density functional approximations such as LC-ωPBE and ωB97X-D are needed to match the experimental findings. The impact of the GIAOs depends on the distribution of the spin density, and their use may change the Δg shifts by 10-50% as shown for [(C₅Me₅)₂Y(μ-S)₂Mo(μ-S)₂Y(C₅Me₅)₂]^-. Routine calculations of large molecules are possible with widely available and comparably low-cost hardware as demonstrated for [Pt(C₆Cl₅)₄]^- with 3003 basis functions and three spin-(1/2) La(II) and Lu(II) compounds, for which we observe good agreement with the experimental findings.

A 9.2-GHz clock transition in a Lu(II) molecular spin qubit arising from a 3,467-MHz hyperfine interaction

Spins in molecules are particularly attractive targets for next-generation quantum technologies, enabling chemically programmable qubits and potential for scale-up via self-assembly. Here we report the observation of one of the largest hyperfine interactions for a molecular system, A_iso = 3,467 ± 50 MHz, as well as a very large associated clock transition. This is achieved through chemical control of the degree of s-orbital mixing into the spin-bearing d orbital associated with a series of spin-½ La(II) and Lu(II) complexes. Increased s-orbital character reduces spin-orbit coupling and enhances the electron-nuclear Fermi contact interaction. Both outcomes are advantageous for quantum applications. The former reduces spin-lattice relaxation, and the latter maximizes the hyperfine interaction, which, in turn, generates a 9-GHz clock transition, leading to an increase in phase memory time from 1.0 ± 0.4 to 12 ± 1 μs for one of the Lu(II) complexes. These findings suggest strategies for the development of molecular quantum technologies, akin to trapped ion systems.

Relativistic Effects in Modeling the Ligand K-Edge X-ray Absorption Near-Edge Structure of Uranium Complexes

Accurate modeling of the complex electronic structure of actinide complexes requires full inclusion of relativistic effects. In this study, we examine the effect of explicit inclusion of spin-orbit coupling (SOC) versus scalar relativistic effects on the predicted spectra for heavy-element complexes. In this study, we employ a relativistic two-component Hamiltonian in the X2C form with all of the electrons in the system being considered explicitly to compare and contrast with previous studies that included the relativistic effects by means of relativistic effective core potentials (RECPs). A few uranium complexes are chosen as model systems. Comparison of the computed Cl K-edge X-ray absorption spectra with experimental data shows significantly improved agreement when a variational relativistic treatment of SOC is performed. In particular, we note the importance of SOC terms to obtain not only correct transition energies but also correct intensities for these heavy-element complexes because of the redistribution of ligand bonding character among the valence MOs. While RECPs generally agree well with all-electron scalar relativistic calculations, there are some differences in the predicted spectra of open-shell systems. These methods are still suitable for broad application to analyze the qualitative nature of transitions in X-ray absorption spectra, but caution is recommended for quantitative analysis, as SOC can be non-negligible for both open- and closed-shell heavy-element systems.

Benchmark Study on the Calculation of 119Sn NMR Chemical Shifts

A new benchmark set termed SnS51 for assessing quantum chemical methods for the computation of ¹¹⁹Sn NMR chemical shifts is presented. It covers 51 unique ¹¹⁹Sn NMR chemical shifts for a selection of 50 tin compounds with diverse bonding motifs and ligands. The experimental reference data are in the spectral range of ±2500 ppm measured in seven different solvents. Fifteen common density functional approximations, two scalar- and one spin-orbit relativistic approach are assessed based on conformer ensembles generated using the CREST/CENSO scheme and state-of-the-art semiempirical (GFN2-xTB), force field (GFN-FF), and composite DFT methods (r²SCAN-3c). Based on the results of this study, the spin-orbit relativistic method combinations of SO-ZORA with PBE0 or revPBE functionals are generally recommended. Both yield mean absolute deviations from experimental data below 100 ppm and excellent linear regression determination coefficients of ≤0.99. If spin-orbit calculations are not affordable, the use of SR-ZORA with B3LYP or X2C with ωB97X or M06 may be considered to obtain qualitative predictions if no severe spin-orbit effects, for example, due to heavy nuclei containing ligands, are expected. An empirical linear scaling correction is demonstrated to be applicable for further improvement, and respective empirical parameters are given. Conformational effects on chemical shifts are studied in detail but are mostly found to be small. However, in specific cases when the ligand sphere differs substantially between conformers, chemical shifts can change by up to several hundred ppm.

Studies of the First Electronically Excited State of 3-Fluoropyridine and Its Ionic Structure by Means of REMPI, Two-Photon MATI, One-Photon VUV-MATI Spectroscopy and Franck-Condon Analysis

3-Fluoropyridine (3-FP) has been investigated by means of two-photon resonance-enhanced multi photon ionization (REMPI), mass-analyzed threshold ionization (MATI) and one-photon vacuum-ultraviolet (VUV) MATI spectroscopy. The aim was the determination of the effect of m-fluorine substitution on the vibronic structure of the first electronically excited and ionic ground state. The S₁ excitation energy has been determined to be 35 064 ± 2 cm^-1 (4.3474 ± 0.0002 eV). Strong evidence of a distinct vibronic coupling via ν_16b and ν_{[Wag.out.,16a]} to one or both of the lowest ¹ππ* states has been found, which results in a warped S₁ minimum structure with C₁ symmetry. The adiabatic ionization energy of the ionic ground state (14a', n_N-LP orbital) has been determined to be 76 579 ± 6 cm^-1 (9.4946 ± 0.0007 eV), which is the first value reported for this state. The origin of the D₁ state (4a'', π-orbital) is located close by at 77 129 cm^-1 (9.5628 eV). As a result of the D₀-D₁ vicinity, the ionic ground state is coupled to the D₁ state via ν_{[Wag.out.,16a]} and ν_10a, which induces a twisted D₀ geometry with C₁ symmetry. Furthermore, for the first time two-photon and one-photon MATI spectra are presented together, which yield a much better understanding of the ionic vibronic structure in comparison to either of these experiments alone.