Relativistic effects in chemistry: more common than you thought

Toward Accurate Spin-Orbit Splittings from Relativistic Multireference Electronic Structure Theory

Most nonrelativistic electron correlation methods can be adapted to account for relativistic effects, as long as the relativistic molecular spinor integrals are available, from either a four-, two-, or one-component mean-field calculation. However, relativistic multireference correlation methods remain a relatively unexplored area, with mixed evidence regarding the improvements brought by perturbative treatments. We report, for the first time, the implementation of state-averaged four-component relativistic multireference perturbation theories to second and third order based on the driven similarity renormalization group (DSRG). With our methods, named 4c-SA-DSRG-MRPT2 and 3, we find that the dynamical correlation included on top of 4c-CASSCF references can significantly improve the spin-orbit splittings in p-block elements and potential energy surfaces when compared to 4c-CASSCF and 4c-CASPT2 results. We further show that 4c-DSRG-MRPT2 and 3 are applicable to these systems over a wide range of the flow parameter, with systematic improvement from second to third order in terms of both improved error statistics and reduced sensitivity with respect to the flow parameter.

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.

Relativistic Fully Self-Consistent GW for Molecules: Total Energies and Ionization Potentials

The fully self-consistent GW (scGW) method with an iterative solution of the Dyson equation provides a consistent approach for describing the ground and excited states without any dependence on the mean-field reference. In this work, we present a relativistic version of scGW for molecules containing heavy elements using the exact two-component (X2C) Coulomb approximation. We benchmark SOC-81 data set containing closed shell heavy elements for the first ionization potential using the fully self-consistent GW as well as one-shot GW. The self-consistent GW provides superior results compared to G0W0 with PBE reference and comparable results to G0W0 with PBE0 while also removing the starting point dependence. The photoelectron spectra obtained at the X2C level demonstrate very good agreement with the experimental spectra. We also observe that scGW provides very good estimation of ionization potential for the inner d-shell orbitals. Additionally, using the well-conserved total energy, we investigate the equilibrium bond length and harmonic frequencies of a few halogen dimers using scGW. Overall, our findings demonstrate the applicability of the fully self-consistent GW method for accurate ionization potential, photoelectron spectra, and total energies in finite systems with heavy elements with a reasonable computational scaling.

Consistent Second-Order Treatment of Spin-Orbit Coupling and Dynamic Correlation in Quasidegenerate N-Electron Valence Perturbation Theory

We present a formulation and implementation of second-order quasidegenerate N-electron valence perturbation theory (QDNEVPT2) that provides a balanced and accurate description of spin-orbit coupling and dynamic correlation effects in multiconfigurational electronic states. In our approach, the energies and wave functions of electronic states are computed by treating electron repulsion and spin-orbit coupling operators as equal perturbations to the nonrelativistic complete active-space wave functions, and their contributions are incorporated fully up to the second order. The spin-orbit effects are described using the Breit-Pauli (BP) or exact two-component Douglas-Kroll-Hess (DKH) Hamiltonians within spin-orbit mean-field approximation. The resulting second-order methods (BP2- and DKH2-QDNEVPT2) are capable of treating spin-orbit coupling effects in nearly degenerate electronic states by diagonalizing an effective Hamiltonian expanded in a compact non-relativistic basis. For a variety of atoms and small molecules across the entire periodic table, we demonstrate that DKH2-QDNEVPT2 is competitive in accuracy with variational two-component relativistic theories. BP2-QDNEVPT2 shows high accuracy for the second- and third-period elements, but its performance deteriorates for heavier atoms and molecules. We also consider the first-order spin-orbit QDNEVPT2 approximations (BP1- and DKH1-QDNEVPT2), among which DKH1-QDNEVPT2 is reliable but less accurate than DKH2-QDNEVPT2. Both DKH1- and DKH2-QDNEVPT2 hold promise as efficient and accurate electronic structure methods for treating electron correlation and spin-orbit coupling in a variety of applications.

Simulating transient X-ray photoelectron spectra of Fe(CO)5 and its photodissociation products with multireference algebraic diagrammatic construction theory

Accurate simulations of transient X-ray photoelectron spectra (XPS) provide unique opportunities to bridge the gap between theory and experiment in understanding the photoactivated dynamics in molecules and materials. However, simulating X-ray photoelectron spectra along a photochemical reaction pathway is challenging as it requires accurate description of electronic structure incorporating core-hole screening, orbital relaxation, electron correlation, and spin-orbit coupling in excited states or at nonequilibrium ground-state geometries. In this work, we employ the recently developed multireference algebraic diagrammatic construction theory (MR-ADC) to investigate the core-ionized states and X-ray photoelectron spectra of Fe(CO)5 and its photodissociation products (Fe(CO)4, Fe(CO)3) following excitation with 266 nm light. The simulated transient Fe 3p and CO 3σ XPS spectra incorporating spin-orbit coupling and high-order electron correlation effects are shown to be in a good agreement with the experimental measurements by Leitner et al. [J. Chem. Phys., 2018, 149, 044307]. Our calculations suggest that core-hole screening, spin-orbit coupling, and ligand-field splitting effects are similarly important in reproducing the experimentally observed chemical shifts in transient Fe 3p XPS spectra of iron carbonyl complexes. Our results also demonstrate that the MR-ADC methods can be very useful in interpreting the transient XPS spectra of transition metal compounds.

Theoretical Photoelectron Spectroscopy of Metal-Metal Quintuple Bonds: Relativity-Driven Reordering of Frontier Orbitals

A recent reinvestigation of the gas-phase photoelectron spectra of Group 6 metal-metal quadruple-bonded complexes with scalar-relativistic DFT calculations showed that common exchange-correlation functionals reproduce the lowest ionization potentials in a semiquantitative manner. The finding encouraged us to undertake a DFT study of metal-metal quintuple bonds in a set of bisamidinato complexes with the formula MI 2[HC(NR)2]2 (M = Cr, Mo, W; R = H, Ph, 2,6-iPr2C6H3) and idealized D 2h symmetry. Scalar-relativistic OLYP/STO-TZ2P calculations indicated significant shifts in valence orbital energies among the three metals, which translate to lower first ionization potentials, higher electron affinities, and lower HOMO-LUMO gaps for the W complexes relative to their Cr and Mo counterparts. These differences are largely attributable to substantially larger relativistic effects in the case of tungsten relative to those of its lighter congeners.

Simulating Vibronic Spectra by Direct Application of Doktorov Formulas on a Superconducting Quantum Simulator

In this work, a direct quantum implementation of the Doktorov formulas for calculating the vibronic spectrum of molecules under the harmonic approximation is presented. It is applied to the three-atom molecules H2O, SO2, ClO2, HS2, and ZnOH. The method solves the classically hard problem of estimating the Franck-Condon (FC) factors by using the Duschinsky matrices as the only input via the Doktorov quantum circuit. This has the advantage of avoiding basis changes, artificial squeezing parameters, and symmetry dependencies. In other words, it is a general method for three-atom molecules that can easily be generalized to bigger molecules. The results are compared with other quantum algorithms and classical anharmonic algorithms. Furthermore, the circuit requirements are studied in order to estimate its applicability on real superconducting quantum hardware.

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.

Interoperable workflows by exchanging grid-based data between quantum-chemical program packages

Quantum-chemical subsystem and embedding methods require complex workflows that may involve multiple quantum-chemical program packages. Moreover, such workflows require the exchange of voluminous data that go beyond simple quantities, such as molecular structures and energies. Here, we describe our approach for addressing this interoperability challenge by exchanging electron densities and embedding potentials as grid-based data. We describe the approach that we have implemented to this end in a dedicated code, PyEmbed, currently part of a Python scripting framework. We discuss how it has facilitated the development of quantum-chemical subsystem and embedding methods and highlight several applications that have been enabled by PyEmbed, including wave-function theory (WFT) in density-functional theory (DFT) embedding schemes mixing non-relativistic and relativistic electronic structure methods, real-time time-dependent DFT-in-DFT approaches, the density-based many-body expansion, and workflows including real-space data analysis and visualization. Our approach demonstrates, in particular, the merits of exchanging (complex) grid-based data and, in general, the potential of modular software development in quantum chemistry, which hinges upon libraries that facilitate interoperability.

Theoretical Photoelectron Spectroscopy of Quadruple-Bonded Dimolybdenum(II,II) and Ditungsten(II,II) Paddlewheel Complexes: Performance of Common Density Functional Theory Methods

We have revisited the gas-phase photoelectron spectra of quadruple-bonded dimolybdenum(II,II) and ditungsten(II,II) paddlewheel complexes with modern density functional theory methods and obtained valuable calibration of four well-known exchange-correlation functionals, namely, BP86, OLYP, B3LYP*, and B3LYP. All four functionals were found to perform comparably, with discrepancies between calculated and experimental ionization potentials ranging from <0.1 to ∼0.5 eV, with the lowest errors observed for the classic pure functional BP86. All four functionals were found to reproduce differences in ionization potentials (IPs) between analogous Mo2 and W2 complexes, as well as large, experimentally observed ligand field effects on the IPs, with near-quantitative accuracy. The calculations help us interpret a number of differences between analogous Mo2 and W2 complexes through the lens of relativistic effects. Thus, relativity results in not only significantly lower IPs for the W2 complexes but also smaller HOMO-LUMO gaps and different triplet states relative to their Mo2 counterparts.

Stabilizing Monoatomic Two-Coordinate Bismuth(I) and Bismuth(II) Using a Redox Noninnocent Bis(germylene) Ligand

The formation of isolable monatomic BiI complexes and BiII radical species is challenging due to the pronounced reducing nature of metallic bismuth. Here, we report a convenient strategy to tame BiI and BiII atoms by taking advantage of the redox noninnocent character of a new chelating bis(germylene) ligand. The remarkably stable novel BiI cation complex 4, supported by the new bis(iminophosphonamido-germylene)xanthene ligand [(P)GeII(Xant)GeII(P)] 1, [(P)GeII(Xant)GeII(P) = Ph2P(NtBu)2GeII(Xant)GeII(NtBu)2PPh2, Xant = 9,9-dimethyl-xanthene-4,5-diyl], was synthesized by a two-electron reduction of the cationic BiIIII2 precursor complex 3 with cobaltocene (Cp2Co) in a molar ratio of 1:2. Notably, owing to the redox noninnocent character of the germylene moieties, the positive charge of BiI cation 4 migrates to one of the Ge atoms in the bis(germylene) ligand, giving rise to a germylium(germylene) BiI complex as suggested by DFT calculations and X-ray photoelectron spectroscopy (XPS). Likewise, migration of the positive charge of the BiIIII2 cation of 3 results in a bis(germylium)BiIIII2 complex. The delocalization of the positive charge in the ligand engenders a much higher stability of the BiI cation 4 in comparison to an isoelectronic two-coordinate Pb0 analogue (plumbylone; decomposition below -30 °C). Interestingly, 4[BArF] undergoes a reversible single-electron transfer (SET) reaction (oxidation) to afford the isolable BiII radical complex 5 in 5[BArF]2. According to electron paramagnetic resonance (EPR) spectroscopy, the unpaired electron predominantly resides at the BiII atom. Extending the redox reactivity of 4[OTf] employing AgOTf and MeOTf affords BiIII(OTf)2 complex 7 and BiIIIMe complex 8, respectively, demonstrating the high nucleophilic character of BiI cation 4.

Relativistic effects and pressure-induced phase transition in CsAu

Cesium auride (CsAu) is an intriguing compound formed by two metals that, upon reacting, exhibits properties of an ionic salt. In this study, we employ computer simulations to explore the influence of relativistic effects on the structure and some physical properties of CsAu, as well as on a potential pressure-induced structural phase transition, the effect of high pressures on its electronic gap, and the possible transition to a conducting state. We have found that including relativistic effects reduces the lattice parameter of CsAu and brings its volumetric properties closer to the trend observed in alkali halides. It also enhances the charge transfer from cesium to gold, resulting in a difference of up to 0.15e, at ambient pressure, between non-relativistic and fully relativistic calculations. Additionally, upon increasing pressure, in the absence of intervening structural phase transitions, the closing of CsAu's band gap is expected at approximately 31.5 GPa. The inclusion of relativistic effects stabilizes the CsAu Pm3̄m structure and shifts the transition pressure to a possible high-pressure P4/mmm phase from 2 GPa (non-relativistic calculation) to 14 GPa (fully-relativistic calculation). Both the Pm3̄m and P4/mmm structures become dynamically unstable around 15 GPa, thus suggesting that the tetragonal structure may be an intermediate state towards a truly stable high-pressure CsAu phase.

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.

The Lack of Triplet Fusion for an Intramolecular Singlet Fission Chromophore: The Expected, the Unexpected, and a Reconciliation

Singlet fission (SF) has the potential to play a key role in photovoltaics since it generates a larger number of longer-lived triplet excitons after photoabsorption. Intramolecular SF (iSF) is of special interest since it enables tuning of SF efficiency by adjusting interchromophore configuration through covalent interaction. However, as elaborated in the present work, iSF chromophores are doomed to dissatisfy one general thermodynamic criterion for all SF chromophores, intramolecular or not: E(T2) ≥ 2E(T1), and therefore, the fusion of two triplet excitons to one triplet exciton is thermodynamically favorable. In our nonadiabatic quantum dynamics simulation for a model iSF chromophore, this expected fusion does not occur, because of the inefficient intersystem crossing hidden under the cover of internal conversion of the triplet fusion. A reconciliation is achieved between the dissatisfaction of E(T2) ≥ 2E(T1) and the large tetraradical character for general iSF chromophores.

Predicting the FCI Energy of Large Systems to Chemical Accuracy from Restricted Active Space Density Matrix Renormalization Group Calculations

We theoretically derive and validate with large scale simulations a remarkably accurate power law scaling of errors for the restricted active space density matrix renormalization group (DMRG-RAS) method [J. Phys. Chem. A 126, 9709] in electronic structure calculations. This yields a new extrapolation method, DMRG-RAS-X, which reaches chemical accuracy for strongly correlated systems such as the chromium dimer, dicarbon up to a large cc-pVQZ basis and even a large chemical complex such as the FeMoco with significantly lower computational demands than those of previous methods. The method is free of empirical parameters, performed robustly and reliably in all examples we tested, and has the potential to become a vital alternative method for electronic structure calculations in quantum chemistry and more generally for the computation of strong correlations in nuclear and condensed matter physics.

Exploring Locality in Molecular Dirac-Coulomb-Breit Calculations: A Perspective

The Dirac-Coulomb-Breit (DCB) operator is widely recognized for its ability to accurately capture relativistic effects and spin-physics in molecular calculations. However, due to its high computational cost, there is a need to develop low-scaling approximations without compromising accuracy. To tackle this challenge, it becomes essential to gain a deeper understanding of the DCB operator's behavior. This work aims to explore local integral approximations, shedding light on the locality of the parts of the charge-current distribution due to the small component. In particular, we propose an atomic Breit approximation that leverages an analysis of the behavior observed in a series of gold chains. Through benchmark studies of metal complexes, we evaluated the accuracy and performance of the proposed atomic Breit approximation. This work provides a comprehensive understanding of the behavior of the charge-current distribution in terms of its contributions from its AO basis constituents, facilitating the development of low-scaling methods that strike a balance between computational efficiency and accuracy.

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.

The Unified Hamiltonian Formalism of Spin-Orbit Jahn-Teller and Pseudo-Jahn-Teller Problems in All Axial Symmetries

Spatial degeneracy of electronic states closely connects spin-orbit coupling and vibronic coupling, which together determine properties of materials, especially heavy element compounds. Accurate description of those materials entails accurate mathematical formulas for spin-orbit vibronic Hamiltonians. For the first time ever, we in this work derive the Hamiltonian formalism to describe all spin-orbit Jahn-Teller and pseudo-Jahn-Teller vibronic problems in all axial symmetries. The conventional one-electron approximation of spin-orbit coupling, which was the foundation of all previous studies in this field, is not involved in the present work. Actually, the present formalism is applicable to all time-reversal symmetric hermitian Hamiltonian that has a Rank-1 dependence on the spin operator, without any restriction on the type and the number of term symbols and vibrational modes.

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.

Towards structural optimization of gold nanoclusters with quantum Monte Carlo

We study the prospects of using quantum Monte Carlo techniques (QMC) to optimize the electronic wavefunctions and atomic geometries of gold compounds. Complex gold nanoclusters are widely studied for diverse biochemical applications, but the dynamic correlation and relativistic effects in gold set the bar high for reliable, predictive simulation methods. Here we study selected ground state properties of few-atom gold clusters by using density functional theory (DFT) and various implementations of the variational Monte Carlo (VMC) and diffusion Monte Carlo. We show that the QMC methods mitigate the exchange-correlation (XC) approximation made in the DFT approach: the average QMC results are more accurate and significantly more consistent than corresponding DFT results based on different XC functionals. Furthermore, we use demonstrate structural optimization of selected thiolated gold clusters with between 1 and 3 gold atoms using VMC forces. The optimization workflow is demonstrably consistent, robust, and its computational cost scales with nb, where b < 3 and n is the system size. We discuss the implications of these results while laying out steps for further developments.

Quantum computations in heavy noble-gas hydride cations: Reference energies and new spectroscopic data

Computational quantum chemistry has become a powerful tool with a wide range of possibilities to solve chemical-physical problems. As a result of this, the interest in the applications of computational quantum chemistry has expanded considerably, and has opened up novel research opportunities. In particular, those related to the characterization of heavy-atoms complexes, as most electronic structure calculations for such systems struggle with the problem posed by the large number of electrons present in them, and consequently, the introduction of relativistic effects. The present study performed an exhaustive assess to characterized the uncommon NgH+ (Ng = Kr, Xe, and Rn) hydride cations in order to provide accurate rovibrational data of their isotopes to assist in the laboratory characterization or even their astronomical detection. Scalar relativistic effects were included, and the ground and first electronically exited states potential curves were obtained from benchmark ab initio CCSD(T)/CBS and MRCI+Q electronic structure calculations. Next, such interaction potentials, correctly extended to long-range asymptotic regions, were employed in quantum bound state calculations and molecular spectroscopic constants were determined for the most abundance 84Kr, 132Xe, and 222Rn isotopes. Our results were discussed in comparison with available experimental and previous theoretical estimates, aiming to treat accuracy issues. The new sets provide reference data that could serve for spectroscopic characterization of such low abundance and high radioactive species.

Polyvalent s-block elements: A missing link challenges the periodic law of chemistry for the heavy elements

The Periodic Law of Chemistry is one of the great discoveries in cultural history. Elements behaving chemically similar are empirically merged in groups G of a Periodic Table, each element with G valence electrons per neutral atom, and with upper limit G for the oxidation and valence numbers. Here, we report that among the usually mono- or di-valent s-block elements (G = 1 or 2), the heaviest members (87Fr, 88Ra, 119E, and 120E) with atomic numbers Z = 87, 88, 119, 120 form unusual 5- or 6-valent compounds at ambient conditions. Together with well-reported basic changes of valence at the end of the 6d-series, in the whole 7p-series, and for 5g6f-elements, it indicates that at the bottom of common Periodic Tables, the classic Periodic Law is not as straightforward as commonly expected. Specifically, we predict the feasible experimental synthesis of polyvalent [RaL-n] (n = 4, 6) compounds.

Understanding Polaritonic Chemistry from Ab Initio Quantum Electrodynamics

In this review, we present the theoretical foundations and first-principles frameworks to describe quantum matter within quantum electrodynamics (QED) in the low-energy regime, with a focus on polaritonic chemistry. By starting from fundamental physical and mathematical principles, we first review in great detail ab initio nonrelativistic QED. The resulting Pauli-Fierz quantum field theory serves as a cornerstone for the development of (in principle exact but in practice) approximate computational methods such as quantum-electrodynamical density functional theory, QED coupled cluster, or cavity Born-Oppenheimer molecular dynamics. These methods treat light and matter on equal footing and, at the same time, have the same level of accuracy and reliability as established methods of computational chemistry and electronic structure theory. After an overview of the key ideas behind those ab initio QED methods, we highlight their benefits for understanding photon-induced changes of chemical properties and reactions. Based on results obtained by ab initio QED methods, we identify open theoretical questions and how a so far missing detailed understanding of polaritonic chemistry can be established. We finally give an outlook on future directions within polaritonic chemistry and first-principles QED.

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.

Direct Conversion of Methane to Propylene

Nonoxidative coupling of methane exhibits promising prospect in that it affords value-added hydrocarbons and hydrogen with high atom economy. However, challenge remains in direct, selective conversion of methane to more valuable hydrocarbons like olefins. The current work presents a catalyst with well-dispersed Ta atoms anchored by graphitic C3N4-supported phthalocyanine. Such a catalyst is able to convert methane selectively to ethylene and propylene at a relatively low temperature (350 °C). The conception of the active center and construction of the catalyst have been described, and the origins of the catalytic performance are discussed.

Coordination Behavior of the Tellurium Incorporated Mercuraazametallamacrocycle and Investigation of d10 ⋅⋅⋅d10 Interactions between Closed Shell (Ag+ Hg2+ ) Metal Ions

Herein, a new tellurium and mercury containing mercuraazametallamacrocycle has been prepared via (2+2) condensation of bis(o-aminophenyl)telluride and bis(o-formylphenyl)mercury(II). The isolated bright yellow solid of mercuraazametallamacrocycle has adopted unsymmetrical figure-of-eight conformation in the crystal structure. To study the metallophilic interactions between closed shell metal ions, the macrocyclic ligand has been treated with two equiv. of AgOTf (OTf=trifluoromethansulfonate) and AgBF4 , which afforded greenish-yellow bimetallic silver complexes. The isolated silver complexes displayed intramolecular Hg⋅⋅⋅Ag, Te⋅⋅⋅Ag interactions as well as intermolecular Hg⋅⋅⋅Hg interactions and formed an extended 1D molecular chain by directing six atoms to interact as TeII ⋅⋅⋅AgI ⋅⋅⋅HgII ⋅⋅⋅HgII ⋅⋅⋅AgI ⋅⋅⋅TeII in a non linear fashion. The Hg⋅⋅⋅Ag, Te⋅⋅⋅Ag interactions have also been studied in solution by 199 Hg, 125 Te NMR spectroscopy, absorption, and emission spectroscopy. In DFT calculations, the Atom in Molecule (AIM) analysis, non-covalent interactions (NCI), natural bonding orbital (NBO) analysis strongly supported for experimental evidences and revealed that the intermolecular Hg⋅⋅⋅Hg interaction is stronger than the intramolecular Hg⋅⋅⋅Ag interactions.

Electronic Structure and Anion Photoelectron Spectroscopy of Uranium-Gold Clusters UAun-, n = 3-7

A collaborative effort between experiment and theory toward elucidating the electronic and molecular structures of uranium-gold clusters is presented. Anion photoelectron spectra of UAun-(n = 3-7) were taken at the third (355 nm) and fourth (266 nm) harmonics of a Nd:YAG laser, as well as excimer (ArF 193 nm) photon energies, where the experimental adiabatic electron affinities and vertical detachment energies values were measured. Complementary first-principles calculations were subsequently carried out to corroborate experimentally determined electron detachment energies and to determine the geometry and electronic structure for each cluster. Except for the ring-like neutral isomer of UAu6 where one unpaired electron is spread over the Au atoms, all other neutral and anionic UAun clusters (n = 3-7) were calculated to possess open-shell electrons with the unpaired electrons localized on the central U atom. The smaller clusters closely resemble the analogous UFn species, but significant deviations are seen starting with UAu5 where a competition between U-Au and Au-Au bonding begins to become apparent. The UAu6 system appears to mark a transition where Au-Au interactions begin to dominate, where both a ring-like and two heavily distorted octahedral structures around the central U atom are calculated to be nearly isoenergetic. With UAu7, only ring-like structures are calculated. Overall, the calculated electron detachment energies are in good agreement with the experimental values.

Corrigendum: Coupled cluster theory on modern heterogeneous supercomputers

[This corrects the article DOI: 10.3389/fchem.2023.1154526.].

Effective quantum electrodynamics: One-dimensional model of the relativistic hydrogen-like atom

We consider a one-dimensional effective quantum electrodynamics (QED) model of the relativistic hydrogen-like atom using delta-potential interactions. We discuss the general exact theory and the Hartree-Fock approximation. The present one-dimensional effective QED model shares the essential physical feature of the three-dimensional theory: the nuclear charge polarizes the vacuum state (creation of electron-positron pairs), which results in a QED Lamb-type shift of the bound-state energy. Yet, this 1D effective QED model eliminates some of the most serious technical difficulties of the three-dimensional theory coming from renormalization. We show how to calculate the vacuum-polarization density at zeroth order in the two-particle interaction and the QED Lamb-type shift of the bound-state energy at first order in the two-particle interaction. The present work may be considered a step toward the development of a quantum-chemistry effective QED theory of atoms and molecules.

Coupled cluster theory on modern heterogeneous supercomputers

This study examines the computational challenges in elucidating intricate chemical systems, particularly through ab-initio methodologies. This work highlights the Divide-Expand-Consolidate (DEC) approach for coupled cluster (CC) theory-a linear-scaling, massively parallel framework-as a viable solution. Detailed scrutiny of the DEC framework reveals its extensive applicability for large chemical systems, yet it also acknowledges inherent limitations. To mitigate these constraints, the cluster perturbation theory is presented as an effective remedy. Attention is then directed towards the CPS (D-3) model, explicitly derived from a CC singles parent and a doubles auxiliary excitation space, for computing excitation energies. The reviewed new algorithms for the CPS (D-3) method efficiently capitalize on multiple nodes and graphical processing units, expediting heavy tensor contractions. As a result, CPS (D-3) emerges as a scalable, rapid, and precise solution for computing molecular properties in large molecular systems, marking it an efficient contender to conventional CC models.

Spectroscopic and Magnetic Studies of Co(II) Scorpionate Complexes: Is There a Halide Effect on Magnetic Anisotropy?

The observation of single-molecule magnetism in transition-metal complexes relies on the phenomenon of zero-field splitting (ZFS), which arises from the interplay of spin-orbit coupling (SOC) with ligand-field-induced symmetry lowering. Previous studies have demonstrated that the magnitude of ZFS in complexes with 3d metal ions is sometimes enhanced through coordination with heavy halide ligands (Br and I) that possess large free-atom SOC constants. In this study, we systematically probe this "heavy-atom effect" in high-spin cobalt(II)-halide complexes supported by substituted hydrotris(pyrazol-1-yl)borate ligands (Tp^tBu,Me and Tp^Ph,Me). Two series of complexes were prepared: [Co^IIX(Tp^tBu,Me)] (1-X; X = F, Cl, Br, and I) and [Co^IIX(Tp^Ph,Me)(Hpz^Ph,Me)] (2-X; X = Cl, Br, and I), where Hpz^Ph,Me is a monodentate pyrazole ligand. Examination with dc magnetometry, high-frequency and -field electron paramagnetic resonance, and far-infrared magnetic spectroscopy yielded axial (D) and rhombic (E) ZFS parameters for each complex. With the exception of 1-F, complexes in the four-coordinate 1-X series exhibit positive D-values between 10 and 13 cm^-1, with no dependence on halide size. The five-coordinate 2-X series exhibit large and negative D-values between -60 and -90 cm^-1. Interpretation of the magnetic parameters with the aid of ligand-field theory and ab initio calculations elucidated the roles of molecular geometry, ligand-field effects, and metal-ligand covalency in controlling the magnitude of ZFS in cobalt-halide complexes.

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.

On the instability of iodides of heavy main group atoms in their higher oxidation state

The inert pair effect-the tendency of the s orbital of heavy atoms to stay unreactive, is a consequence of the relativistic contraction of the s orbitals. While the manifestations of this on the reactivity depend on the nature of the substituents, this aspect is often overlooked. Divalent Pb prefers inorganic substituents, whereas tetravalent Pb prefers organic substituents. Among the inorganic substituents, again there are specific preferences-tetravalent Pb prefers F and Cl more than Br and I. It is as though the relativistic contraction of the s orbital of Pb is more significant with Br and I substituents than with Cl, F, and alkyl substituents. Herein, we address this problem using the molecular orbital approach and support it with quasi-relativistic density functional computations. We explain why typical hypervalent systems, like 12-X-6, and 10-X-5 (X is a heavy atom, the number preceding X is the number of valence electrons surrounding X, and the number after X is the coordination number) with less electronegative substituents carrying a lone pair (such as iodine), and Lewis octet molecules like PbI₄ are unstable, but their dianions (14-X-6, 12-X-5, PbI₄^2-) are not. For heavy atoms, the relativistic contraction of the s orbital renders the antibonding combination of s with ligand orbitals (σ₁*) very low-lying, making it a good acceptor of electrons. Thus, compounds where σ₁* is empty are kinetically unstable when an electron donor with appropriate energy (such as the lone pair on iodine or bromine) is present in the vicinity. Donor-acceptor interaction between σ₁* and the lone pair on I or Br (F and Cl lone pairs are energetically far away from σ₁*) is responsible for the instability of such compounds. The kinetic stability of tetraalkyl lead compounds is due to the absence of lone pairs on the alkyl substituents. This work illustrates the key factor responsible for the instability of heavy element iodides by taking into consideration the covalent nature of the bonds, while the existing explanations assume a purely ionic bonding, which is an oversimplification.

Relativistic quantum calculations to understand the contribution of f-type atomic orbitals and chemical bonding of actinides with organic ligands

The nuclear waste problem is one of the main interests of rare earth and actinide element chemistry. Studies of actinide-containing compounds are at the frontier of the applications of current theoretical methods due to the need to consider relativistic effects and approximations to the Dirac equation in them. Here, we employ four-component relativistic quantum calculations and scalar approximations to understand the contribution of f-type atomic orbitals in the chemical bonding of actinides (Ac) to organic ligands. We studied the relativistic quantum structure of an isostructural family made of Plutonium (Pu), Americium (Am), Californium (Cf), and Berkelium (Bk) atoms with the redox-active model ligand DOPO (2,4,6,8-tetra-tert-butyl-1-oxo-1H-phenoxazin-9-olate). Crystallographic structures were available to validate our calculations for all mentioned elements except for Cf. In short, state-of-the-art relativistic calculations were performed at different levels of theory to investigate the influence of relativistic and electron correlation effects on geometrical structures and bonding energies of Ac-DOPO₃ complexes (Ac = Pu, Am, Cf, and Bk): (1) the scalar (sc) and spin-orbit (so) relativistic zeroth order regular approximation (ZORA) within the hybrid density functional theory (DFT) and (2) the four-component Dirac equation with both the Dirac-Hartree-Fock (4c-DHF) and Lévy-Leblond (LL) Hamiltonians. We show that sr- and so-ZORA-DFT could be used as efficient theoretical models to first approximate the geometry and electronic properties of actinides which are difficult to synthesize or characterize, but knowing that the higher levels of theory, like the 4c-DHF, give closer results to experiments. We also performed spin-free 4c calculations of geometric parameters for the Americium and Berkelium compounds. To the best of our knowledge, this is the first time that these kinds of large actinide compounds (the largest contains 67 atoms and 421 electrons) have been studied with highly accurate four-component methods (all-electron calculations with 6131 basis functions for the largest compound). We show that relativistic effects play a key role in the contribution of f-type atomic orbitals to the frontier orbitals of Ac-DOPO₃ complexes. The analysis of the results obtained applying different theoretical schemes to calculate bonding energies is also given.

Relativistic Semistochastic Heat-Bath Configuration Interaction

In this work we present the extension of semistochastic heat-bath configuration interaction (SHCI) to work with any two-component and four-component Hamiltonian. The vertical detachment energy (VDE) of AuH₂^- and zero-field splitting (ZFS) of NpO₂²⁺ is calculated by correlating more than 100 spinors in both cases. This work demonstrates the capability of SHCI to treat problems where both relativistic effect and electron correlation are important.

Correlated Dirac-Coulomb-Breit multiconfigurational self-consistent-field methods

The fully correlated frequency-independent Dirac-Coulomb-Breit Hamiltonian provides the most accurate description of electron-electron interaction before going to a genuine relativistic quantum electrodynamics theory of many-electron systems. In this work, we introduce a correlated Dirac-Coulomb-Breit multiconfigurational self-consistent-field method within the frameworks of complete active space and density matrix renormalization group. In this approach, the Dirac-Coulomb-Breit Hamiltonian is included variationally in both the mean-field and correlated electron treatment. We also analyze the importance of the Breit operator in electron correlation and the rotation between the positive- and negative-orbital space in the no-virtual-pair approximation. Atomic fine-structure splittings and lanthanide contraction in diatomic fluorides are used as benchmark studies to understand the contribution from the Breit correlation.

A Computational Study on Closed-Shell Molecular Hexafluorides MF6 (M=S, Se, Te, Po, Xe, Rn, Cr, Mo, W, U) - Molecular Structure, Anharmonic Frequency Calculations, and Prediction of the NdF6 Molecule

Quantum chemical methods were used to study the molecular structure and anharmonic IR spectra of the experimentally known closed-shell molecular hexafluorides MF₆ (M=S, Se, Te, Xe, Mo, W, U). First, the molecular structures and harmonic frequencies were investigated using Density Functional Theory (DFT) with all-electron basis sets and explicitly considering the influence of spin-orbit coupling. Second, anharmonic frequencies and IR intensities were calculated with the CCSD(T) coupled cluster method and compared, where available, with IR spectra recorded by us. These comparisons showed satisfactory results. The anharmonic IR spectra provide means for identifying experimentally too little studied or unknown MF₆ molecules with M=Cr, Po, Rn. To the best of our knowledge, we predict the NdF₆ molecule for the first time and show it to be a true local minimum on the potential energy surface. We used intrinsic bond orbital (IBO) analyses to characterize the bonding situation in comparison with the UF₆ molecule.

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.

Revisited relativistic Dirac-Hartree-Fock X-ray scattering factors. I. Neutral atoms with Z = 2-118

In this first of a series of publications, the X-ray scattering factors for neutral atoms are revisited. Using the recently developed DBSR_HF program [Zatsarinny & Froese Fischer (2016). Comput. Phys. Comm. 202, 287-303] the fully relativistic Dirac-Hartree-Fock ground-state wavefunctions for all atoms with Z = 2-118 (He-Og) have been calculated using the extended average level scheme and including both the Breit interaction correction to the electronic motion due to magnetic and retardation effects, and the Fermi distribution function for the description of the nuclear charge density. The comparison of our wavefunctions with those obtained in several previous studies in terms of the total and orbital (spinor) electronic energies, and a number of local and integrated total and orbital properties, confirmed the quality of the generated wavefunctions. The employed dense radial grid combined with the DBSR_HF's B-spline representation of the relativistic one-electron orbitals allowed for a precise integration of the X-ray scattering factors using a newly developed Fortran program SF. Following the established procedure [Maslen et al. (2006). International Tables for Crystallography, Vol. C, Section 6.1.1, pp. 554-589], the resulting X-ray scattering factors have been interpolated in the 0 ≤ sin θ/λ ≤ 2 Å^-1 and 2 ≤ sin θ/λ ≤ 6 Å^-1 ranges using the recommended analytical functions with both the four- (which is a current convention) and five-term expansions. An exhaustive comparison of the newly generated X-ray scattering factors with the International Union of Crystallography recommended values and those from a number of previous studies showed an overall good agreement and allowed identification of a number of typos and inconsistencies in the recommended quantities. A detailed analysis of the results suggests that the newly derived values may represent an excellent compromise among all the previous studies. The determined conventional interpolating functions for the two sin θ/λ intervals show, on average, the same accuracy as the recommended parametrizations. However, an extension of each expansion by only a single term provides a significant improvement in the accuracy of the interpolated values for an overwhelming majority of the atoms. As such, an updated set of the fully relativistic X-ray scattering factors and the interpolating functions for neutral atoms with Z = 2-118 can be easily incorporated into the existing X-ray diffraction software with only minor modifications. The outcomes of the undertaken research should be of interest to members of the crystallographic community who push the boundaries of the accuracy and precision of X-ray diffraction studies.

The LRESC-Loc Model to Analyze Magnetic Shieldings with Localized Molecular Orbitals

The leading electronic mechanisms of relativistic effects in the NMR magnetic shieldings of heavy-atom (HA) containing molecules are well described by the linear response with elimination of small components model (LRESC). We show here first results from a new version of the LRESC model written in terms of localized molecular orbitals (LMOs) which is coined as LRESC-Loc. Those LMOs resemble "chemist's orbitals", representing lone-pairs, atomic cores, and bonds. The whole set of relativistic effects are expressed in terms of non-ligand-dependent and ligand-dependent contributions. We show the electronic origin of trends and behavior of different mechanisms in molecular systems which contain heavy elements that belong to any of the IB to VIIA groups of the periodic table. The SO mechanism has a well-defined dependence with the LPs (LPσ and LPπ) when the HAs have them, but the non-SO mechanisms mostly depend on other LMOs. In addition we propose here that the SO mechanism can be used to characterize interactions involving LPs and the non-SO mechanisms to characterize covalent and close-shell interactions. All our main results are in accord with previous findings, though we are now able to analyze them in a different manner.

Exploring EPR Parameters of 187Re Complexes for Designing New MRI Probes: From the Gas Phase to Solution and a Model Protein Environment

Breast cancer is one of the major types of cancer around the world, and early diagnosis is essential for successful treatment. New contrast agents (CAs), with reduced toxicology, are needed to improve diagnosis. One of the most promising Magnetic Resonance Imaging (MRI) CA is based on rhenium conjugated with a benzothiazole derivate (ReABT). In this sense, DFT has been used to evaluate the best methodology for calculating the hyperfine coupling constant (Aiso) of ReABT. Then, a thermodynamic analysis was performed to confirm the stability of the complex. Furthermore, a docking study of ReABT at the enzyme PI3K active site and Aiso calculations of ReABT in the enzyme environment were carried out. The best methodology for the Aiso calculation of ReABT was using the M06L functional, SARC-ZORA-TZVP (for Re) and TZVP (for all other atoms) basis set, relativistic Hamiltonian, and the CPCM solvation model with water as the solvent which confirm that the relativistic effects are important for calculating the Aiso values. In addition, thermodynamic analysis indicates that ReABT presents a higher stability and a lower toxicity than Gd-based CAs. The docking studies point out that ReABT interacts with amino acids residues of alanine, aspartate, and lysine from the PI3K active site. Considering the enzyme environment, Aiso values decrease significantly. These findings indicate that the CA candidate ReABT could be a good candidate for a new contrast agent.

Frozen-Density Embedding for Including Environmental Effects in the Dirac-Kohn-Sham Theory: An Implementation Based on Density Fitting and Prototyping Techniques

Frozen density embedding (FDE) represents an embedding scheme in which environmental effects are included from first-principles calculations by considering the surrounding system explicitly by means of its electron density. In the present paper, we extend the full four-component relativistic Dirac-Kohn-Sham (DKS) method, as implemented in the BERTHA code, to include environmental and confinement effects with the FDE scheme (DKS-in-DFT FDE). The implementation, based on the auxiliary density fitting techniques, has been enormously facilitated by BERTHA's python API (PyBERTHA), which facilitates the interoperability with other FDE implementations available through the PyADF framework. The accuracy and numerical stability of this new implementation, also using different auxiliary fitting basis sets, has been demonstrated on the simple NH₃-H₂O system, in comparison with a reference nonrelativistic implementation. The computational performance has been evaluated on a series of gold clusters (Au_n, with n = 2, 4, 8) embedded into an increasing number of water molecules (5, 10, 20, 40, and 80 water molecules). We found that the procedure scales approximately linearly both with the size of the frozen surrounding environment (consistent with the underpinnings of the FDE approach) and with the size of the active system (in line with the use of density fitting). Finally, we applied the code to a series of heavy (Rn) and super-heavy elements (Cn, Fl, Og) embedded in a C₆₀ cage to explore the confinement effect induced by C₆₀ on their electronic structure. We compare the results from our simulations, with respect to more-approximate models employed in the atomic physics literature. Our results indicate that the specific interactions described by FDE are able to improve upon the cruder approximations currently employed, and, thus, they provide a basis from which to generate more-realistic radial potentials for confined atoms.

Bonding and 13 C-NMR properties of coinage metal tris(ethylene) and tris(norbornene) complexes: Evaluation of the role of relativistic effects from DFT calculations

The π-complexes of cationic coinage metal ions (Cu(I), Ag(I), Au(I)) provide useful experimental support for understanding fundamental characteristics of bonding and ¹³ C-NMR patterns of the group 11 triad. Here, we account for the role of relativistic effects on olefin-coinage metal ion interaction for cationic, homoleptic tris-ethylene, and tris-norbornene complexes, [M(η² -C₂ H₄ )₃ ]⁺ and [M(η² -C₇ H₁₀ )₃ ]⁺ (M = Cu, Ag, Au), as representative case of studies. The M-(CC) bond strength in the cationic, tris-ethylene complexes is affected sizably for Au and to a lesser extent for Ag and Cu (48.6%, 16.7%, and 4.3%, respectively), owing to the influence on the different stabilizing terms accounting for the interaction energy in the formation of coinage metal cation-π complexes. The bonding elements provided by olefin → M σ-donation and olefin ← M π-backbonding are consequently affected, leading to a lesser covalent interaction going down in the triad if the relativistic effects are ignored. Analysis of the ¹³ C-NMR tensors provides further understanding of the observed experimental values, where the degree of backbonding charge donation to π₂ *-olefin orbital is the main influence on the observed high-field shifts in comparison to the free olefin. This donation is larger for ethylene complexes and lower for norbornene counterparts. However, the bonding energy in the later complexes is slightly stabilized given by the enhancement in the electrostatic character of the interaction. Thus, the theoretical evaluation of metal-alkene bonds, and other metal-bonding situations, benefits from the incorporation of relativistic effects even in lighter counterparts, which have an increasing role going down in the group.

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.

Relativistic Effects in the Ligation of Atomic Coinage Metal Cations with O2 and C6H6: Anomalous Formation of Relativistic Mono- and Bis-adducts with Au. JOURNAL OF THE AMERICAN SOCIETY FOR MASS SPECTROMETRY 2022;33:1419-1426. [PMID: 35533366 DOI: 10.1021/jasms.2c00075] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

The interaction of the atomic coinage metal cations Cu⁺, Ag⁺, and Au⁺ with O₂, a weak ligand, and C₆H₆, a strong ligand, was investigated with measurements of rate coefficients of ligation and quantum-chemical computations of ligation energies with an eye on relativistic effects going down the periodic table. Strong "third row enhancements" were observed for both the rate coefficients of ligation and ligation energies with the O₂ ligand and for the formation of both the mono- and bis-adducts of M⁺ and the monoadduct of M⁺(C₆H₆). The computations revealed that the third-row enhancement in the ligation energy is attributable to a relativistic increase in the ligation energy. This means that rate coefficient measurements down the periodic table for the ligation of coinage metal cations with O₂ provide a probe of the relativistic effect in ligation reactions, as expected from the known dependence of the rate coefficient of ligation on the ligation energy. The much stronger benzene ligand was observed to ligate the atomic coinage metal cations with nearly 100% efficiency so that there is no, or only slightly, visible third-row enhancement despite the strong relativistic effect in the binding energy that is revealed by the calculations. Relativistic effects contribute substantially to the extraordinary stability against deligation of all the observed mono- and bis-adducts of Au⁺ relative to Ag⁺, truly a "third-row enhancement".

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].

Efficient Evaluation of the Breit Operator in the Pauli Spinor Basis

The frequency-independent Coulomb-Breit operator gives rise to the most accurate treatment of two-electron interaction in the non-quantum-electrodynamics regime. The Breit interaction in the Coulomb gauge consists of magnetic and gauge contributions. The high computational cost of the gauge term limits the application of the Breit interaction in relativistic molecular calculations. In this work, we apply the Pauli component integral-density matrix contraction scheme for gauge interaction with a maximum spin- and component separation scheme. We also present two different computational algorithms for evaluating gauge integrals. One is the generalized Obara-Saika algorithm, where the Laplace transformation is used to transform the gauge operator into Gaussian functions and the Obara-Saika recursion is used for reducing the angular momentum. The other algorithm is the second derivative of inverse Coulomb interaction evaluated with Rys-quadrature. This work improves the efficiency of performing Dirac-Hartree-Fock with variational treatment of Breit interaction for molecular systems. We use this formalism to examine relativistic trends in the periodic table, and analyze the relativistic two-electron interaction contributions in heavy-element complexes.