Relativistic effects in chemistry: more common than you thought

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.

Free Difluorobis(pentafluoroethyl)phosphoranide Ion and Its Ligand Properties

In this contribution, we report on the "free" [P(C₂F₅)₂F₂]^- ion and its ligand properties in transition metal complex chemistry. For this purpose, Ag[P(C₂F₅)₂F₂] was treated with [{(Et₂N)₃P═N}₃P═NHC(CH₃)₃]Cl ([EtP₄H]Cl) to yield [EtP₄H][P(C₂F₅)₂F₂], featuring a weakly coordinating phosphazenium cation. Due to the weak interaction between the cation and anion, the [P(C₂F₅)₂F₂]^- ion meets the so-called pseudo-gas-phase conditions. To determine the Tolman electronic parameter, [EtP₄H][Ni(CO)₃{P(C₂F₅)₂F₂}] was prepared from [EtP₄H][P(C₂F₅)₂F₂] and [Ni(CO)₄], facilitating the classification of the P(C₂F₅)₂F₂ moiety as a moderately π-acidic ligand. By treatment of [EtP₄H][P(C₂F₅)₂F₂] with [AuCl(tht)], the neutral tetrahydrothiophene ligand was substituted by the phosphoranide ion, yielding [EtP₄H][AuCl{P(C₂F₅)₂F₂}]. When Ag[P(C₂F₅)₂F₂] was treated with [AuCl(tht)], on the other hand, the chloride was substituted. Transmetalation reactions of this type proved to be an efficient transfer method of the P(C₂F₅)₂F₂ moiety, as further demonstrated by the reactions of Ag[P(C₂F₅)₂F₂] with [FeCl(CO)₂Cp], [FeCl(CO)₂Cp*], and [PdCl₂(NCMe)₂]. Surprisingly, P(C₂F₅)₂F demonstrated fluorinating abilities toward [FeCl(CO)₂Cp], [FeCl(CO)₂Cp*], [AuCl(tht)], and [PdCl₂(NCMe)₂]. Apparently, fluorido transition metal complexes were generated in situ under the formation of P(C₂F₅)₂Cl. The fluorido iron and palladium complexes transfer their fluoride ions onto P(C₂F₅)₂F, yielding the respective phosphoranido complexes, featuring the P(C₂F₅)₂F₂ moiety.

Transient Radiation-Induced Berkelium(III) and Californium(III) Redox Chemistry in Aqueous Solution

Despite the significant impact of radiation-induced redox reactions on the accessibility and lifetimes of actinide oxidation states, fundamental knowledge of aqueous actinide metal ion radiation chemistry is limited, especially for the late actinides. A quantitative understanding of these intrinsic radiation-induced processes is essential for investigating the fundamental properties of these actinides. We present here a picosecond electron pulse reaction kinetics study into the radiation-induced redox chemistry of trivalent berkelium (Bk(III)) and californium (Cf(III)) ions in acidic aqueous solutions at ambient temperature. New and first-of-a-kind, second-order rate coefficients are reported for the transient radical-induced reduction of Bk(III) and Cf(III) by the hydrated electron (e_aq^-) and hydrogen atom (H^•), demonstrating a significant reactivity (up to 10¹¹ M^-1 s^-1) indicative of a preference of these metals to adopt divalent states. Additionally, we report the first-ever second-order rate coefficients for the transient radical-induced oxidation of these elements by a reaction with hydroxyl (^•OH) and nitrate (NO₃^•) radicals, which also exhibited fast reactivity (ca. 10⁸ M^-1 s^-1). Transient Cf(II), Cf(IV), and Bk(IV) absorption spectra are also reported. Overall, the presented data highlight the existence of rich, complex, intrinsic late actinide radiation-induced redox chemistry that has the potential to influence the findings of other areas of actinide science.

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.

Designing 3d metal oxides: selecting optimal density functionals for strongly correlated materials

Transition metal oxides (TMOs) have remarkable physicochemical properties, are non-toxic, and have low cost and high annual production, thus they are commonly studied for various technological applications. Density functional theory (DFT) can help to optimize TMO materials by providing insights into their electronic, optical and thermodynamic properties, and hence into their structure-performance relationships, over a wide range of solid-state structures and compositions. However, this is underpinned by the choice of the exchange-correlation (XC) functional, which is critical to accurately describe the highly localized and correlated 3d-electrons of the transition metals in TMOs. This tutorial review presents a benchmark study of density functionals (DFs), ranging from generalized gradient approximation (GGA) to range-separated hybrids (RSH), with the all-electron def2-TZVP basis set, comparing magneto-electro-optical properties of 3d TMOs against experimental observations. The performance of the DFs is assessed by analyzing the band structure, density of states, magnetic moment, structural static and dynamic parameters, optical properties, spin contamination and computational cost. The results disclose the strengths and weaknesses of the XC functionals, in terms of accuracy, and computational efficiency, suggesting the unprecedented PBE0-1/5 as the best candidate. The findings of this work contribute to necessary developments of XC functionals for periodic systems, and materials science modelling studies, particularly informing how to select the optimal XC functional to obtain the most trustworthy description of the ground-state electron structure of 3d TMOs.

Insights into the value of statistical models, solvent, and relativistic effects for investigating Re complexes of 2-(4'-aminophenyl)benzothiazole: a potential spectroscopic probe

Cancer affects a major part of the worldwide population, and, to minimize deaths, the diagnosis in the early stages of the disease is fundamental. Thus, to improve diagnosis and treatment new potential spectroscopic probes are crucial. Benzothiazole derivates present antitumor properties and are highly selective and interact strongly with the enzyme phosphoinositide 3-kinase (PI3K), which was associated with cell proliferation and breast cancer cells. In this paper, the rhenium shielding tensors (¹⁸⁷Re(σ)) and hydrogen and carbon chemical shifts (¹H(δ) and ¹³C(δ)) of the Re(CO)₃(NNO) complex conjugated with 2-(4'-aminophenyl)benzothiazole (ReABT) were evaluated. A statistical HCA model was used to analyze the best DFT protocol to compute σ and δ values and to evaluate the relativistic effects, both in the basis set and Hamiltonian as well as the functionals M06L or PBE0. The best protocol was applied to obtain ¹⁸⁷Re(σ) of the ReABT complex in different environments (gas phase, solution, and in the active site of the PI3K enzyme). The results point out that ¹⁸⁷Re(σ) values of the ReABT complex change significantly when the complex is docked in the PI3K enzyme.

Polarizable Embedding Complex Polarization Propagator in Four- and Two-Component Frameworks

Explicit embedding methods combined with the complex polarization propagator (CPP) enable the modeling of spectroscopy for increasingly complex systems with a high density of states. We present the first derivation and implementation of the CPP in four- and exact-two-component (X2C) polarizable embedding (PE) frameworks. We denote the developed methods PE-4c-CPP and PE-X2C-CPP, respectively. We illustrate the methods by estimating the solvent effect on ultraviolet-visible (UV-vis) and X-ray atomic absorption (XAS) spectra of [Rh(H₂O)₆]³⁺ and [Ir(H₂O)₆]³⁺ immersed in aqueous solution. We moreover estimate solvent effects on UV-vis spectra of a platinum complex that can be photochemically activated (in water) to kill cancer cells. Our results clearly show that the inclusion of the environment is required: UV-vis and (to a lesser degree) XAS spectra can become qualitatively different from vacuum calculations. Comparison of PE-4c-CPP and PE-X2C-CPP methods shows that X2C essentially reproduces the solvent effect obtained with the 4c methods.

Uranium: The Nuclear Fuel Cycle and Beyond

This review summarizes the recent developments regarding the use of uranium as nuclear fuel, including recycling and health aspects, elucidated from a chemical point of view, i.e., emphasizing the rich uranium coordination chemistry, which has also raised interest in using uranium compounds in synthesis and catalysis. A number of novel uranium coordination features are addressed, such the emerging number of U(II) complexes and uranium nitride complexes as a promising class of materials for more efficient and safer nuclear fuels. The current discussion about uranium triple bonds is addressed by quantum chemical investigations using local vibrational mode force constants as quantitative bond strength descriptors based on vibrational spectroscopy. The local mode analysis of selected uranium nitrides, N≡U≡N, U≡N, N≡U=NH and N≡U=O, could confirm and quantify, for the first time, that these molecules exhibit a UN triple bond as hypothesized in the literature. We hope that this review will inspire the community interested in uranium chemistry and will serve as an incubator for fruitful collaborations between theory and experimentation in exploring the wealth of uranium chemistry.

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.

SOiCI and iCISO: combining iterative configuration interaction with spin-orbit coupling in two ways

The near-exact iCIPT2 approach for strongly correlated systems of electrons, which stems from the combination of iterative configuration interaction (iCI, an exact solver of full CI) with configuration selection for static correlation and second-order perturbation theory (PT2) for dynamic correlation, is extended to the relativistic domain. In the spirit of spin separation, relativistic effects are treated in two steps: scalar relativity is treated by the infinite-order, spin-free part of the exact two-component (X2C) relativistic Hamiltonian, whereas spin-orbit coupling (SOC) is treated by the first-order, Douglas-Kroll-Hess-like SOC operator derived from the same X2C Hamiltonian. Two possible combinations of iCIPT2 with SOC are considered, i.e., SOiCI and iCISO. The former treats SOC and electron correlation on an equal footing, whereas the latter treats SOC in the spirit of state interaction, by constructing and diagonalizing an effective spin-orbit Hamiltonian matrix in a small number of correlated scalar states. Both double group and time reversal symmetries are incorporated to simplify the computation. Pilot applications reveal that SOiCI is very accurate for the spin-orbit splitting (SOS) of heavy atoms, whereas the computationally very cheap iCISO can safely be applied to the SOS of light atoms and even of systems containing heavy atoms when SOC is largely quenched by ligand fields.

Unified one-electron Hamiltonian formalism of spin-orbit Jahn-Teller and pseudo-Jahn-Teller problems in tetrahedral and octahedral symmetries

Heavy element compounds with high symmetries often feature both spin-orbit coupling and vibronic coupling. This is especially true for systems with tetrahedral and octahedral symmetries, whose electronic states may be three-fold degenerate and experience complicated Jahn-Teller and pseudo-Jahn-Teller interactions. To accurately describe these interactions, high quality spin-orbit vibronic Hamiltonian operators are needed. In this study, we present a unified one-electron Hamiltonian formalism for spin-orbit vibronic interactions for systems in all tetrahedral and octahedral symmetries. The formalism covers all spin-orbit Jahn-Teller and pseudo-Jahn-Teller problems in the symmetries with arbitrary types and arbitrary numbers of vibrational modes, and generates Hamiltonian expansion formulas of arbitrarily high order.

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.

Hydrogen Bonding with Polonium

Hydrogen bonding (H-bonding) with heavier chalcogens such as polonium and tellurium is almost unexplored owing to their lower electronegativities, providing us an opportunity to delve into the uncharted territory of...

Hyperfine Coupling Constants in Local Exact Two-Component Theory

We present a highly efficient implementation of the electron-nucleus hyperfine coupling matrix within the one-electron exact two-component (X2C) theory. The complete derivative of the X2C Hamiltonian is formed, that is, the derivatives of the unitary decoupling transformation are considered. This requires the solution of the response and Sylvester equations, consequently increasing the computational costs. Therefore, we apply the diagonal local approximation to the unitary decoupling transformation (DLU). The finite nucleus model is employed for both the scalar potential and the vector potential. Two-electron picture-change effects are modeled with the (modified) screened nuclear spin-orbit approach. Our implementation is fully integral direct and OpenMP-parallelized. An extensive benchmark study regarding the Hamiltonian, the basis set, and the density functional approximation is carried out for a set of 12-17 transition-metal compounds. The error introduced by DLU is negligible, and the DLU-X2C Hamiltonian accurately reproduces its four-component "fully" relativistic parent results. Functionals with a large amount of Hartree-Fock exchange such as CAM-QTP-02 and ωB97X-D are generally favorable. The pure density functional r²SCAN performs remarkably and even outperforms the common hybrid functionals TPSSh and CAM-B3LYP. Fully uncontracted basis sets or contracted quadruple-ζ bases are required for accurate results. The capability of our implementation is demonstrated for [Pt(C₆Cl₅)₄]^- with more than 4700 primitive basis functions and four rare-earth single-molecule magnets: [La(OAr*)₃]^-, [Lu(NR₂)₃]^-, [Lu(OAr*)₃]^-, and [TbPc₂]^-. Here, the results with the spin-orbit DLU-X2C Hamiltonian are in an excellent agreement with the experimental findings of all Pt, La, Lu, and Tb molecules.

Unified one-electron Hamiltonian formalism of spin-orbit Jahn-Teller and pseudo-Jahn-Teller problems in axial symmetries

Spin-orbit coupling and vibronic coupling are both closely related to orbital degeneracy of electronic states. Both types of coupling play significant roles in determining properties of heavy element compounds and shall be treated on the same footing. In this work, we derive a unified one-electron Hamiltonian formalism for spin-orbit and vibronic interactions for systems in all axial symmetries. The one-electron formalism is usually adequate as the spin-orbit interaction can often be approximated as a one-electron interaction. For the first time, the formalism covers spin-orbit and vibronic couplings in all axial symmetries from C₁ to D_∞h, arbitrary types of vibrational modes in those symmetries, and an arbitrary number of those modes and gives Hamiltonian expansions up to an arbitrary order.

Iridium(VII)-Corrole Terminal Carbides Should Exist as Stable Compounds

Scalar-relativistic DFT calculations with multiple exchange-correlation functionals and large basis sets foreshadow the existence of stable iridium(VII)-corrole terminal carbide derivatives. For the parent compound Ir[Cor](C), OLYP/STO-TZ2P calculations predict a short Ir-C bond distance of 1.69 Å, a moderately domed macrocycle with no indications of ligand noninnocence, a surprisingly low electron affinity of ∼1.1 eV, and a substantial singlet-triplet gap of ∼1.8 eV. These results, and their essential invariance with respect to the choice of the exchange-correlation functional, lead us to posit that Ir(VII)-corrole terminal carbide complexes should be isolable and indefinitely stable under ambient conditions.

Creation of an unexpected plane of enhanced covalency in cerium(III) and berkelium(III) terpyridyl complexes

Controlling the properties of heavy element complexes, such as those containing berkelium, is challenging because relativistic effects, spin-orbit and ligand-field splitting, and complex metal-ligand bonding, all dictate the final electronic states of the molecules. While the first two of these are currently beyond experimental control, covalent M‒L interactions could theoretically be boosted through the employment of chelators with large polarizabilities that substantially shift the electron density in the molecules. This theory is tested by ligating BkIII with 4'-(4-nitrophenyl)-2,2':6',2"-terpyridine (terpy*), a ligand with a large dipole. The resultant complex, Bk(terpy*)(NO3)3(H2O)·THF, is benchmarked with its closest electrochemical analog, Ce(terpy*)(NO3)3(H2O)·THF. Here, we show that enhanced Bk‒N interactions with terpy* are observed as predicted. Unexpectedly, induced polarization by terpy* also creates a plane in the molecules wherein the M‒L bonds trans to terpy* are shorter than anticipated. Moreover, these molecules are highly anisotropic and rhombic EPR spectra for the CeIII complex are reported.

Paramagnetic NMR Shielding Tensors and Ring Currents: Efficient Implementation and Application to Heavy Element Compounds

We present an efficient implementation of paramagnetic NMR shielding tensors and shifts in a nonrelativistic and scalar-relativistic density functional theory framework. For the latter, we make use of the scalar exact two-component Hamiltonian in its local approximation, and generally we apply the well established (multipole-accelerated) resolution of the identity approximation and the seminumerical exchange approximation. The perturbed density matrix of a paramagnetic NMR shielding calculation is further used to study the magnetically induced current density and ring currents of open-shell systems as illustrated for [U@Bi₁₂]^3-. [U@Bi₁₂]^3- features delocalized highest occupied molecular orbitals and sustains a net diatropic ring current of ca. 18 nA/T through the Bi₁₂ torus similar to the all-metal aromatic heavy-element cluster [Th@Bi₁₂]^4-.

DFT Functionals for Modeling of Polyethylene Chains Cross-Linked by Metal Atoms. DLPNO-CCSD(T) Benchmark Calculations

Density functional theory (DFT) functionals for calculations of binding energies (BEs) of the polyethylene (PE) chains cross-linked by selected metal atoms (M) are benchmarked against DLPNO-CCSD(T) and DLPNO-CCSD(T1) data. PEX-M-PEX complexes as compared with plain parallel PEX···PEX chains with X = 3-9 carbon atoms are model species characterized by a cooperative effect of covalent C-M-C bonds and interchain dispersion interactions. The accuracy of DLPNO-CC methods was assessed by a comparison of BEs with the canonical CCSD(T) results for small PE3-M-PE3 complexes. Functionals for PEX···PEX and closed-shell PEX-M-PEX complexes (M = Be, Mg, Zn) were benchmarked against DLPNO-CCSD(T) BEs; open-shell complexes (M = Li, Ag, Au) were benchmarked against the DLPNO-CCSD(T1) method with iterative triples. Three dispersion corrections were combined with 25 DFT functionals for calculations of BEs with respect to PEX-M and PEX fragments employing def2-TZVPP and def2-QZVPP basis sets. Accuracy to within 5% for the closed-shell PEX-M-PEX complexes was achieved with five functionals. Less accurate are functionals for the open-shell PEX-M-PEX complexes; only two functionals deviate by less than 15% from DLPNO-CCSD(T1). Particularly problematic were PEX-Li-PEX complexes. A reasonable overall performance across all complexes in terms of the mean absolute percentage error is found for the range-separated hybrid functionals ωB97X-D3 and CAM-B3LYP/D3(BJ)-ABC.

Combining Machine Learning and Computational Chemistry for Predictive Insights Into Chemical Systems

Machine learning models are poised to make a transformative impact on chemical sciences by dramatically accelerating computational algorithms and amplifying insights available from computational chemistry methods. However, achieving this requires a confluence and coaction of expertise in computer science and physical sciences. This Review is written for new and experienced researchers working at the intersection of both fields. We first provide concise tutorials of computational chemistry and machine learning methods, showing how insights involving both can be achieved. We follow with a critical review of noteworthy applications that demonstrate how computational chemistry and machine learning can be used together to provide insightful (and useful) predictions in molecular and materials modeling, retrosyntheses, catalysis, and drug design.

The Quest to Simulate Excited-State Dynamics of Transition Metal Complexes

This Perspective describes current computational efforts in the field of simulating photodynamics of transition metal complexes. We present the typical workflows and feature the strengths and limitations of the different contemporary approaches. From electronic structure methods suitable to describe transition metal complexes to approaches able to simulate their nuclear dynamics under the effect of light, we give particular attention to build a bridge between theory and experiment by critically discussing the different models commonly adopted in the interpretation of spectroscopic experiments and the simulation of particular observables. Thereby, we review all the studies of excited-state dynamics on transition metal complexes, both in gas phase and in solution from reduced to full dimensionality.

Tutorial on Chemical Pressure Analysis: How Atomic Packing Drives Laves/Zintl Intergrowth in K3Au5Tl

The tight atomic packing generally exhibited by alloys and intermetallics can create the impression of their being composed of hard spheres arranged to maximize their density. As such, the atomic size factor has historically been central to explanations of the structural chemistry of these systems. However, the role atomic size plays structurally has traditionally been inferred from empirical considerations. The recently developed DFT-Chemical Pressure (CP) analysis has opened a path to investigating these effects with theory. In this article, we provide a step-by-step tutorial on the DFT-CP method for non-specialists, along with advances in the approach that broaden its applicability. A new version of the CP software package is introduced, which features an interactive system that guides the user in preparing the necessary electronic structure data and generating the CP scheme, with the results being readily visualized with a web browser (and easily incorporated into websites). For demonstration purposes, we investigate the origins of the crystal structure of K3Au5Tl, which represents an intergrowth of Laves and Zintl phase domains. Here, CP analysis reveals that the intergrowth is supported by complementary CP features of NaTl-type KTl and MgCu2-type KAu2 phases. In this way, K3Au5Tl exemplifies how CP effects can drive the merging for geometrical motifs derived from different families of intermetallics through a mechanism referred to as epitaxial stabilization.

Relativity as a Synthesis Design Principle: A Comparative Study of [3 + 2] Cycloaddition of Technetium(VII) and Rhenium(VII) Trioxo Complexes with Olefins

The difference in [3 + 2] cycloaddition reactivity between fac-[MO₃(tacn)]⁺ (M = Re, ⁹⁹Tc; tacn = 1,4,7-triazacyclononane) complexes has been reexamined with a selection of unsaturated substrates including sodium 4-vinylbenzenesulfonate, norbornene, 2-butyne, and 2-methyl-3-butyn-2-ol (2MByOH). None of the substrates was found to react with the Re cation in water at room temperature, whereas the ⁹⁹Tc reagent cleanly yielded the [3 + 2] cycloadducts. Interestingly, a bis-adduct was obtained as the sole product for 2MByOH, reflecting the high reactivity of a ⁹⁹TcO-enediolato monoadduct. On the basis of scalar relativistic and nonrelativistic density functional theory calculations of the reaction pathways, the dramatic difference in reactivity between the two metals has now been substantially attributed to differences in relativistic effects, which are much larger for the 5d metal. Furthermore, scalar-relativistic ΔG values were found to decrease along the series propene > norbornene > 2-butyne > dimethylketene, indicating major variations in the thermodynamic driving force as a function of the unsaturated substrate. The suggestion is made that scalar-relativistic effects, consisting of greater destabilization of the valence electrons of the 5d elements compared with those of the 4d elements, be viewed as a new design principle for novel ^99mTc/Re radiopharmaceuticals, as well as more generally in heavy-element coordination chemistry.

Room temperature cycloaddition reactivity of fac-[⁹⁹TcO₃(tacn)]⁺ (tacn = 1,4,7-triazacyclononane) with a variety of unsaturated substrates and the lack of such reactivity for fac-[ReO₃(tacn)]⁺ appears largely attributable to much stronger relativistic effects for Re relative to Tc, based on relativistic density functional theory calculations.

Unprecedented [d9]Cu[d10]Au coinage bonding interactions in {Cu(NH3)4[Au(CN)2]}+[Au(CN)2]- salt

The X-ray structure of the {Cu(NH₃)₄[Au(CN)₂]}⁺[Au(CN)₂]^- salt is reported showing an unprecedented [d⁹]Cu[d¹⁰]Au coinage bond. The physical nature of the interaction has been studied using DFT calculations, including the quantum theory of atoms-in-molecules, the noncovalent interaction plot and the natural bond orbital analysis, revealing the nucleophilic role of the [d¹⁰]Au metal and the electrophilic role of [d⁹]Cu metal.

The relativistic effects on the methane activation by gold(I) cations

The reactivity of gold has been investigated for a long time. Here, we performed an in-depth analysis of relativistic effects over the chemical kinetic properties of elementary reactions associated with methane activation by gold(I) cations, CH₄ + Au⁺ ↔ AuCH₂ ⁺ + H₂. The global reaction is modeled as a two-step process, CH₄ + Au⁺ ↔ HAuCH₃ ⁺ ↔ AuCH₂ ⁺ + H₂. Moreover, the barrierless dissociation of the initial adduct between reactants, AuCH₄ ⁺, is discussed as well. Higher-order relativistic treatments are used to provide corrections beyond the commonly considered scalar effects of relativistic effective core potentials (RECPs). Although the scalar relativistic contributions predominate, lowering the forward barrier heights by 48.4 and 36.1 kcal mol^-1, the spin-orbit coupling effect can still provide additional reductions of these forward barrier heights by as much as 9% (1.0 and 2.2 kcal mol^-1). The global reaction proceeds rapidly at low temperatures to the intermediate attained after the first hydrogen transfer, HAuCH₃ ⁺. The relativistic corrections beyond the ones from RECPs are still able to double the rate constant of the CH₄ + Au⁺ → HAuCH₃ ⁺ process at 300 K, while the reverse reaction becomes five times slower. The formation of global products from this intermediate only becomes significant at much higher temperatures (∼1500 K upward). The scalar relativistic contributions decrease the dissociation energy of the initial adduct, AuCH₄ ⁺, into the global products by 105.8 kcal mol^-1, while the spin-orbit effect provides an extra lowering of 1.8 kcal mol^-1.

A Spectroscopic Validation of the Improved Lennard-Jones Model

The Lennard-Jones (LJ) and Improved Lennard-Jones (ILJ) potential models have been deeply tested on the most accurate CCSD(T)/CBS electronic energies calculated for some weakly bound prototype systems. These results are important to plan the correct application of such models to systems at increasing complexity. CCSD(T)/CBS ground state electronic energies were determined for 21 diatomic systems composed by the combination of the noble gas atoms. These potentials were employed to calculate the rovibrational spectroscopic constants, and the results show that for 20 of the 21 pairs the ILJ predictions agree more effectively with the experimental data than those of the LJ model. The CCSD(T)/CBS energies were also used to determine the β parameter of the ILJ form, related to the softness/hardness of the interacting partners and controlling the shape of the potential well. This information supports the experimental finding that suggests the adoption of β≈9 for most of the systems involving noble gas atoms. The He-Ne and He-Ar molecules have a lifetime of less than 1ps in the 200-500 K temperature range, indicating that they are not considered stable under thermal conditions of gaseous bulks. Furthermore, the controversy concerning the presence of a "virtual" or a "real" vibrational state in the He2 molecule is discussed.

NMR Indirect Spin-Spin Coupling Constants in a Modern Quasi-Relativistic Density Functional Framework

A quasi-relativistic implementation of NMR indirect spin-spin coupling constants is presented. The exact two-component (X2C) Hamiltonian and its diagonal local approximation to the unitary decoupling transformation (DLU) are utilized together with the (modified) screened nuclear spin-orbit approach. In a restricted kinetic balance, the finite nucleus model is available for both the scalar and vector potentials. The implementation supports density functionals up to the fourth rung of Jacob's ladder, i.e., (range-separated) hybrid and local hybrid functionals based on a seminumerical ansatz. We assess the quality of our quasi-relativistic X2C approach by comparison with "fully" relativistic four-component results for small main-group molecules and alkynyl compounds. The mean absolute error introduced by the DLU scheme is less than 0.05 × 10¹⁹ T J^-2 of the reduced coupling constant for the small main-group molecules and 0.5 Hz for the alkynyl compounds. Thus, the error is significantly smaller than finite nucleus size effects for heavy elements. The basis set convergence and the impact of different density functional approximations are further studied. We propose a simple scheme to develop segmented-contracted relativistic all-electron basis sets for NMR spin-spin couplings. Our implementation allows us to perform calculations of extended molecules with reasonable computational effort, which is illustrated for the ¹J(¹¹⁹Sn, ³¹P) coupling constant of a low-valent tin phosphinidenide complex. The corresponding results are in good agreement with the experimental findings.

Complexes with Atomic Gold Ions: Efficient Bis-Ligand Formation

Complexes of atomic gold with a variety of ligands have been formed by passing helium nanodroplets (HNDs) through two pickup cells containing gold vapor and the vapor of another dopant, namely a rare gas, a diatomic molecule (H₂, N₂, O₂, I₂, P₂), or various polyatomic molecules (H₂O, CO₂, SF₆, C₆H₆, adamantane, imidazole, dicyclopentadiene, and fullerene). The doped HNDs were irradiated by electrons; ensuing cations were identified in a high-resolution mass spectrometer. Anions were detected for benzene, dicyclopentadiene, and fullerene. For most ligands L, the abundance distribution of AuL_n⁺ versus size n displays a remarkable enhancement at n = 2. The propensity towards bis-ligand formation is attributed to the formation of covalent bonds in Au⁺L₂ which adopt a dumbbell structure, L-Au⁺-L, as previously found for L = Xe and C₆₀. Another interesting observation is the effect of gold on the degree of ionization-induced intramolecular fragmentation. For most systems gold enhances the fragmentation, i.e., intramolecular fragmentation in AuL_n⁺ is larger than in pure L_n⁺. Hydrogen, on the other hand, behaves differently, as intramolecular fragmentation in Au(H₂)_n⁺ is weaker than in pure (H₂)_n⁺ by an order of magnitude.

Bismuth species in the coordination sphere of transition metals: synthesis, bonding, coordination chemistry, and reactivity of molecular complexes

This contribution is focused on bismuth species in the coordination sphere of transition metals. In molecular transition metal complexes, three types of Bi-M bonding are considered, namely dative Bi→M interactions (with Bi acting as a donor), dative Bi←M interactions (with Bi acting as an acceptor) and covalent Bi-M interactions (M = transition metal). Synthetic routes to all three classes of compounds are outlined, the Bi-M bonding situation is discussed, trends in the geometric parameters and in the coordination chemistry of the compounds are addressed, and common spectroscopic properties are summarized. As an important part of this contribution, the reactivity of bismuth species in the coordination sphere of transition metal complexes in stoichiometric and catalytic reactions is highlighted.

Hydrogen bonding to metals as a probe for an inverted ligand field

Electron-rich, late transition metals are known to act as hydrogen-bonding (HBd) acceptors. In this regard, Pt(ii) centres in square-planar environments are particularly efficient. It is however puzzling that no convincing experimental evidence is currently available for the isoelectronic neighbour Au(iii) being involved in HBd interactions. We report now on the synthesis and characterisation of two series of isoleptic and isoelectronic (d8) compounds [(CF3)3Pt(L)]- and (CF3)3Au(L), where the L ligands are based on the quinoline frame and have been selected to favour HBd with the metal centre. Strong HBd interactions were actually found in the Pt(ii) compounds, based on structural and spectroscopic evidence, and they were further confirmed by theoretical calculations. In contrast, no evidence was obtained in the Au(iii) case. In order to find the reason underlying this general disparity, we undertook a detailed theoretical analysis of the model systems [(CF3)3Pt(py)]- and (CF3)3Au(py). This study revealed that the filled dz2 orbital is the HOMO in the case of Pt(ii), but is buried in the lower energy levels in the case of Au(iii). The sharply different electronic configurations involve ligand-field inversion on going from Pt to the next element Au. This is not a gradual but an abrupt change, which invalidates Au(iii) as a HBd-acceptor wherever ligand-field inversion occurs.

Halogen Bonding Involving I2 and d8 Transition-Metal Pincer Complexes

We systematically investigated iodine–metal and iodine–iodine bonding in van Koten’s pincer complex and 19 modifications changing substituents and/or the transition metal with a PBE0–D3(BJ)/aug–cc–pVTZ/PP(M,I) model chemistry. As a novel tool for the quantitative assessment of the iodine–metal and iodine–iodine bond strength in these complexes we used the local mode analysis, originally introduced by Konkoli and Cremer, complemented with NBO and Bader’s QTAIM analyses. Our study reveals the major electronic effects in the catalytic activity of the M–I–I non-classical three-center bond of the pincer complex, which is involved in the oxidative addition of molecular iodine I2 to the metal center. According to our investigations the charge transfer from the metal to the σ* antibonding orbital of the I–I bond changes the 3c–4e character of the M–I–I three-center bond, which leads to weakening of the iodine I–I bond and strengthening of the metal–iodine M–I bond, facilitating in this way the oxidative addition of I2 to the metal. The charge transfer can be systematically modified by substitution at different places of the pincer complex and by different transition metals, changing the strength of both the M–I and the I2 bonds. We also modeled for the original pincer complex how solvents with different polarity influence the 3c–4e character of the M–I–I bond. Our results provide new guidelines for the design of pincer complexes with specific iodine–metal bond strengths and introduce the local vibrational mode analysis as an efficient tool to assess the bond strength in complexes.

Fluorides of Silver Under Large Compression*

The silver-fluorine phase diagram has been scrutinized as a function of external pressure using theoretical methods. Our results indicate that two novel stoichiometries containing Ag⁺ and Ag²⁺ cations (Ag₃ F₄ and Ag₂ F₃ ) are thermodynamically stable at ambient and low pressure. Both are computed to be magnetic semiconductors under ambient pressure conditions. For Ag₂ F₅ , containing both Ag²⁺ and Ag³⁺ , we find that strong 1D antiferromagnetic coupling is retained throughout the pressure-induced phase transition sequence up to 65 GPa. Our calculations show that throughout the entire pressure range of their stability the mixed-valence fluorides preserve a finite band gap at the Fermi level. We also confirm the possibility of synthesizing AgF₄ as a paramagnetic compound at high pressure. Our results indicate that this compound is metallic in its thermodynamic stability region. Finally, we present general considerations on the thermodynamic stability of mixed-valence compounds of silver at high pressure.

Demethylation of Methylmercury in Bird, Fish, and Earthworm

Toxicity of methylmercury (MeHg) to wildlife and humans results from its binding to cysteine residues of proteins, forming MeHg-cysteinate (MeHgCys) complexes that hinder biological functions. MeHgCys complexes can be detoxified in vivo, yet how this occurs is unknown. We report that MeHgCys complexes are transformed into selenocysteinate [Hg(Sec)₄] complexes in multiple animals from two phyla (a waterbird, freshwater fish, and earthworms) sampled in different geographical areas and contaminated by different Hg sources. In addition, high energy-resolution X-ray absorption spectroscopy (HR-XANES) and chromatography-inductively coupled plasma mass spectrometry of the waterbird liver support the binding of Hg(Sec)₄ to selenoprotein P and biomineralization of Hg(Sec)₄ to chemically inert nanoparticulate mercury selenide (HgSe). The results provide a foundation for understanding mercury detoxification in higher organisms and suggest that the identified MeHgCys to Hg(Sec)₄ demethylation pathway is common in nature.

Relativistic four-component potential energy curves for the lowest 23 covalent states of molecular astatine (At2)

The potential energy curves (PECs) of all covalent states of Molecular Astatine (At₂) have been investigated in this work within a four-component relativistic framework using the MOLFDIR program package. The ground state was determined using multireference configuration interaction with all single and double excitations including Davidson size-extensivity correction (MRCISD+Q) whereas the 22 excited states were treated by complete open shell configuration interaction (COSCI). Spectroscopic constants (R_e,ω_e,ω_ex_e,ω_ey_e, D_e,B_e,α_e,β_e,T_e) are presented for all states as well as vertical excitations obtained at COSCI, MRCISD and MRCISD+Q levels. In addition, it is also presented accurate extended Rydberg analytical form for the ground state X: (1)0_g⁺.

Electron-Nucleus Hyperfine Coupling Calculated from Restricted Active Space Wavefunctions and an Exact Two-Component Hamiltonian

Exact two-component (X2C) relativistic nuclear hyperfine magnetic field operators were incorporated in X2C ab initio wavefunction calculations at the multireference restricted active space (RAS) level for calculations of nuclear hyperfine magnetic properties. Spin-orbit coupling was treated via RAS state interaction (SO-RASSI). The method was tested by calculations of electron-nucleus hyperfine coupling constants. The approach, implemented in the OpenMolcas program, overcomes a major limitation of a previous SO-RASSI implementation for hyperfine coupling that relied on nonrelativistic hyperfine operators [J. Chem. Theor. Comput. 2015, 11, 538-549] and therefore had limited applicability. Results from calculations on systems with light and heavy main group elements, transition metals, lanthanides, and one actinide complex demonstrate reasonably good agreement with experimental data, where available, as long as the active space can generate sufficient spin polarization.

Understanding Periodic and Non-periodic Chemistry in Periodic Tables

The chemical elements are the "conserved principles" or "kernels" of chemistry that are retained when substances are altered. Comprehensive overviews of the chemistry of the elements and their compounds are needed in chemical science. To this end, a graphical display of the chemical properties of the elements, in the form of a Periodic Table, is the helpful tool. Such tables have been designed with the aim of either classifying real chemical substances or emphasizing formal and aesthetic concepts. Simplified, artistic, or economic tables are relevant to educational and cultural fields, while practicing chemists profit more from "chemical tables of chemical elements." Such tables should incorporate four aspects: (i) typical valence electron configurations of bonded atoms in chemical compounds (instead of the common but chemically atypical ground states of free atoms in physical vacuum); (ii) at least three basic chemical properties (valence number, size, and energy of the valence shells), their joint variation across the elements showing principal and secondary periodicity; (iii) elements in which the (sp)8, (d)10, and (f)14 valence shells become closed and inert under ambient chemical conditions, thereby determining the "fix-points" of chemical periodicity; (iv) peculiar elements at the top and at the bottom of the Periodic Table. While it is essential that Periodic Tables display important trends in element chemistry we need to keep our eyes open for unexpected chemical behavior in ambient, near ambient, or unusual conditions. The combination of experimental data and theoretical insight supports a more nuanced understanding of complex periodic trends and non-periodic phenomena.