Relativistic and correlation effects on molecular properties. I. The dihalogens F2, Cl2, Br2, I2, and At2

Synthesis Principle and Practice with Radioactive Iodines and Astatine: Advances Made So Far

Radioactive iodines and astatine, possessing distinct exploitable nuclear properties, play indispensable roles in the realms of nuclear imaging and therapy. Their analogous chemical characteristics shape the design, preparation, and substrate range for tracers labeled with these radiohalogens through interconnected radiosynthetic chemistry. This perspective systematically explores the labeling methods by types of halogenating reagents─nucleophilic and electrophilic─underpinning the rational design of such compounds. It delves into the rapidly evolving synthetic strategies and reactions in radioiodination and radioastatination over the past decade, comparing their intrinsic relationships and highlighting variations. This comparative analysis illuminates potential radiosynthetic methods for exploration. Moreover, stability concerns related to compounds labeled with radioactive iodines and astatine are addressed, offering valuable insights for radiochemists and physicians alike.

Potential energy curves for F2, Cl2, and Br2 with the i-DMFT method

The potential energy curves for dihalogens (F2, Cl2, and Br2) are calculated with the i-DMFT method proposed recently [Wang and Baerends, Phys. Rev. Lett. 128, 013001]. All electrons are correlated in a set of self-consistent-field eigenvalue equations, with the orbital occupation numbers obeying the Fermi-Dirac distribution. The only input is the dissociation energies of the molecules, which are usually available from an experimental database. The quality of the computed potential energy curve is examined by extracting spectroscopic parameters and rotation-vibration energy levels, which are compared with experiment data and other theoretical calculations.

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.

Coordination of Mercury(II) in Water Promoted over Hydrolysis in Solvated Clusters [Hg(H2O)1-6](aq)2+: Insights from Relativistic Effects and Free Energy Analysis

Understanding the nature of the interaction between mercury(II) ions, Hg2+, and water molecules is crucial to describe the stability and chemical behavior of structures formed during solvation, as well as the conditions that favor the Hg2+ coordination or inducing water hydrolysis. In our study, we explored exhaustively the potential energy surface of Hg2+ with up to six water molecules. We analyzed electronic and Gibbs free energies, binding, and nuclear magnetic resonance parameters. We used the zeroth-order regular approximation Hamiltonian, including scalar and spin-orbit relativistic corrections for free energy calculations and geometry optimizations to explore the interplay between electron correlation and relativistic effects. We analyzed intermolecular interactions with energy decomposition analysis, quantum theory of atoms in molecules, and natural bond orbital. Additionally, we used the four-component Dirac Hamiltonian to compute solvent effect on the magnetic shielding and J-coupling constants. Our results revealed that the water hydrolysis by Hg2+ requires a minimum of three water molecules. We found that the interaction between Hg2+ and water molecules is an orbital interaction due to relativistic effects and the most stable structures are opened-shape clusters, reducing the number of oxygen-mercury contacts and maximizing the formation of hydrogen bonds among water molecules. In these types of clusters, Hg2+ promotes the water hydrolysis over coordination with oxygen atoms. However, when we considered the change associated with the transfer of a cluster from the ideal gas to a solvated system, our solvation free energy analysis revealed that closed-shape clusters are more favorable, maximizing the number of oxygen-mercury contacts and reducing the formation of hydrogen bonds among water molecules. This finding suggests that, under room conditions, the coordination of Hg2+ is more favorable than hydrolysis. Our results have significant implications for understanding Hg2+ behavior in water, helping to develop targeted strategies for mercury remediation and management, and contributing to advancements in the broader field of environmental chemistry.

Investigating the Heaviest Halogen: Lessons Learned from Modeling the Electronic Structure of Astatine's Small Molecules

We present a systematic study of electron-correlation and relativistic effects in diatomic molecular species of the heaviest halogen astatine (At) within relativistic single- and multireference coupled-cluster approaches and relativistic density functional theory. We establish revised reference ab initio data for the ground states of At₂, HAt, AtAu, and AtO⁺ using a highly accurate relativistic effective core potential model and in-house basis sets developed for accurate modeling of molecules with large spin-orbit effects. Spin-dependent relativistic effects on chemical bonding in the ground state are comparable to the binding energy or even exceed it in At₂. Electron-correlation effects near the equilibrium internuclear separation are mostly dynamical and can be adequately captured using single-reference CCSD(T). However, bond elongation in At₂ and, especially, AtO⁺ results in rapid manifestation of its multireference character. While useful for evaluating the spin-orbit effects on the ground-state bonding and properties, the two-component density functional theory lacks predictive power, especially in combination with popular empirically adjusted exchange-correlation functionals. This drawback supports the necessity to develop new functionals for reliable quantum-chemical models of heavy-element compounds with strong relativistic effects.

Ab initio calculations in atoms, molecules, and solids, treating spin-orbit coupling and electron interaction on an equal footing

We incorporate explicit, non-perturbative treatment of spin-orbit coupling into ab initio auxiliary-field quantum Monte Carlo (AFQMC) calculations. The approach allows a general computational framework for molecular and bulk systems in which material specificity, electron correlation, and spin-orbit coupling effects can be captured accurately and on an equal footing, with favorable computational scaling vs system size. We adopt relativistic effective-core potentials that have been obtained by fitting to fully relativistic data and that have demonstrated a high degree of reliability and transferability in molecular systems. This results in a two-component spin-coupled Hamiltonian, which is then treated by generalizing the ab initio AFQMC approach. We demonstrate the method by computing the electron affinity in Pb, the bond dissociation energy in Br₂ and I₂, and solid Bi.

Relativistic four-component MRCISD+Q calculations of the six lowest valence states of molecular F[Formula: see text] anion including Breit interactions

In this study, the potential energy curves of the ground and the excited states of molecular fluorine anion (F[Formula: see text]) were investigated at multireference configuration interaction (MRCISD) with Davidson size-extensivity correction (denoted as +Q) within fully relativistic four-component relativistic framework including Breit interaction. Spectroscopic constants (R_e, ω_e, ω_ex_e, ω_ey_e, D_e,D₀,B_e, α_e, β_e, γ_e ), accurate extended Rydberg analytical form and rovibrational levels for ground state X:[Formula: see text] are presented, as well as spectroscopic constants for non dissociative excited states. For most states these spectroscopic constants are presented for the first time in literature and they are of interest for experimental studies, specially regarding electron attachment of F₂. Results suggest that inclusion of relativistic effects at 4-component level and correlation effects treated at MRCISD+Q level are needed to obtain reliable results, which we report for X:[Formula: see text] ground state's R_e, ω_e and D_e the values of 1.999 Å, 391 cm^- 1 and 1.22 eV, respectively.

Bond Dissociation Energies in Heavy Element Chalcogen and Halogen Small Molecules

Thermodynamic properties including bond dissociation energies (BDEs), heats of formation, and gas-phase acidities for the hydrides and dimers of chalcogens and halogens, H₂Y, HX, Y₂, and X₂ for Y = Se, Te, and At and X = Br, I, and At, have been predicted using the Feller-Peterson-Dixon composite-correlated molecular orbital theory approach. A full four-component CCSD(T) approach was used to calculate the spin-orbit effects on thermodynamic properties, except for Se₂, where the AoC-DHF value was used due to strong multireference effects in Se₂ for the SO calculations. The calculated results show that the At₂ BDE is quite small, 19.5 kcal/mol, with much of the low bond energy due to spin-orbit effects. H₂Po is not predicted to be stable to dehydrogenation to Po + H₂ in terms of the free energy at 298 K. In the gas phase, HAt is predicted to be a stronger acid than H₂SO₄. The current results provide insights into potential difficulties in the actual experimental observation of such species for heavy elements.

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⁺.

A joint data and model driven method for study diatomic vibrational spectra including dissociation behavior

The details of quantum multi-body interactions are so rich and subtle which make it difficult to accurately model for some situations such as the behavior of diatomic long-range vibrations. In recent years, data-driven machine learning has made remarkable achievements in capturing complex relationships that are subtle. Combining the characteristics of these two fields, we propose a joint machine learning method to obtain reliable diatomic vibrational spectra including dissociation energy by using accessible heterogeneous micro/macro information such as low lying vibrational energy levels and heat capacity. Applications of this method to CO and Br₂ in the ground state yield their state of the art of vibrational spectra including dissociation limit. The strategy introduced here is an exploration of combining the model-driven and data-driven method to cover subtle physical details that are difficult to study in a single way.

Quantum chemical topology at the spin-orbit configuration interaction level: Application to astatine compounds

We report a methodology that allows the investigation of the consequences of the spin-orbit coupling by means of the QTAIM and ELF topological analyses performed on top of relativistic and multiconfigurational wave functions. In practice, it relies on the "state-specific" natural orbitals (NOs; expressed in a Cartesian Gaussian-type orbital basis) and their occupation numbers (ONs) for the quantum state of interest, arising from a spin-orbit configuration interaction calculation. The ground states of astatine diatomic molecules (AtX with X = AtF) and trihalide anions (IAtI^- , BrAtBr^- , and IAtBr^- ) are studied, at exact two-component relativistic coupled cluster geometries, revealing unusual topological properties as well as a significant role of the spin-orbit coupling on these. In essence, the presented methodology can also be applied to the ground and/or excited states of any compound, with controlled validity up to including elements with active 5d, 6p, and/or 5f shells, and potential limitations starting with active 6d, 7p, and/or 6f shells bearing strong spin-orbit couplings.

Relativistic spectral bounds for the general molecular potential: application to a diatomic molecule

We tackle with the Dirac equation in the presence of the general molecular potential (GMP). The spin symmetric solution of the relativistic wave equation is obtained by considering the Pekeris-type approximation scheme to deal with the centrifugal term, and in order to solve second-order differential equation, the asymptotic iteration method (AIM) is used. The closed form of the energy eigenvalue equation is found out for any values of the angular momentum quantum number. We calculate the relativistic vibrational bound state energies of 5¹Δ_g state of Na₂ molecule and compare them with the Rydberg-Klein-Rees (RKR) data. We show that relativistic vibrational energies of this molecule, which are found in the spin symmetric case, are more convenient with experimental RKR data than non-relativistic vibrational energies. We also obtain normalization constant by considering inductive approach and investigate the radial eigenfunctions and probability density functions corresponding to different eigenvalues graphically for the Na₂(5¹Δ_g) molecule.

Potential Energy Surface of Group 11 Trimers (Cu, Ag, Au): Bond Angle Isomerism in Au3

Potential energy surfaces for the group 11 trimers were generated at various levels of coupled-cluster theory to examine the effects of Jahn-Teller distortions. Our calculations show that the lowest-energy conformer for Cu₃, Ag₃, and Au₃ is the ²B₂ (∼65° isomer) without spin-orbit corrections. Spin-orbit corrections have negligible contributions to the relative energies for the angle dependence of the potential energy surfaces for Cu₃ and Ag₃. The inclusion of spin-orbit corrections for Au₃ makes the ²B₂ (∼65°) and ²A₁ (∼55°) states approximately degenerate. A novel ²B₂ isomer of Au₃ at an obtuse angle of ∼125° was also characterized, providing evidence for bond angle isomerism on the same ²B₂ potential energy surface. Spin-orbit corrections increase the barrier height between the ²B₂ (65°) and ²B₂ (125°) bond angle isomers of Au₃. The calculated symmetric stretch vibrational frequencies are in good agreement with the available experimental values. All frequencies calculated for the Au₃ ²B₂ (∼125°) state are real, and there is at least one bound bending vibration for this state. Jahn-Teller parameters are also derived for each trimer.

RAQET: Large-scale two-component relativistic quantum chemistry program package

The Relativistic And Quantum Electronic Theory (RAQET) program is a new software package, which is designed for large-scale two-component relativistic quantum chemical (QC) calculations. The package includes several efficient schemes and algorithms for calculations involving large molecules which contain heavy elements in accurate relativistic formalisms. These calculations can be carried out in terms of the two-component relativistic Hamiltonian, wavefunction theory, density functional theory, core potential scheme, and evaluation of electron repulsion integrals. Furthermore, several techniques, which have frequently been used in non-relativistic QC calculations, have been customized for relativistic calculations. This article introduces the brief theories and capabilities of RAQET with several calculation examples. © 2018 Wiley Periodicals, Inc.

Quantum calculations of At-mediated halogen bonds: on the influence of relativistic effects

If astatine is generally a stronger halogen-bond donor than iodine, an inversion is sometimes observed owing to the spin–orbit coupling.

Four-Component Relativistic State-Specific Multireference Perturbation Theory with a Simplified Treatment of Static Correlation

The relativistic multireference (MR) perturbative approach is one of the most successful tools for the description of computationally demanding molecular systems of heavy elements. We present here the ground state dissociation energy surfaces, equilibrium bond lengths, harmonic frequencies, and dissociation energies of Ag₂, Cu₂, Au₂, and I₂ computed using the four-component (4c) relativistic spinors based state-specific MR perturbation theory (SSMRPT) with improved virtual orbital complete active space configuration interaction (IVO-CASCI) functions. The IVO-CASCI method is a simple, robust, useful and lower cost alternative to the complete active space self-consistent field approach for treating quasidegenerate situations. The redeeming features of the resulting method, termed as 4c-IVO-SSMRPT, lies in (i) manifestly size-extensivity, (ii) exemption from intruder problems, (iii) the freedom of convenient multipartitionings of the Hamiltonian, (iv) flexibility of the relaxed and unrelaxed descriptions of the reference coefficients, and (v) manageable cost/accuracy ratio. The present method delivers accurate descriptions of dissociation processes of heavy element systems. Close agreement with reference values has been found for the calculated molecular constants indicating that our 4c-IVOSSMRPT provides a robust and economic protocol for determining the structural properties for the ground state of heavy element molecules with eloquent MR character as it treats correlation and relativity on equal footing.

The generalized maximum hardness principle revisited and applied to atoms and molecules

Part 1 of this duology is devoted to isolated atoms and molecules, and to chemical reactions between them; we introduce here basic concepts beyond the Generalized Maximum Hardness Principle, and the corresponding Minimum Polarizability Principle, and we illustrate applicability of both principles to a broad range of chemical phenomena and distinct systems in the gas phase.

Implementation of Analytical Energy Gradient of Spin-Dependent General Hartree-Fock Method Based on the Infinite-Order Douglas-Kroll-Hess Relativistic Hamiltonian with Local Unitary Transformation

An analytical energy gradient for the spin-dependent general Hartree-Fock method based on the infinite-order Douglas-Kroll-Hess (IODKH) method was developed. To treat realistic systems, the local unitary transformation (LUT) scheme was employed both in energy and energy gradient calculations. The present energy gradient method was numerically assessed to investigate the accuracy in several diatomic molecules containing fifth- and sixth-period elements and to examine the efficiency in one-, two-, and three-dimensional silver clusters. To arrive at a practical calculation, we also determined the geometrical parameters of fac-tris(2-phenylpyridine)iridium and investigated the efficiency. The numerical results confirmed that the present method describes a highly accurate relativistic effect with high efficiency. The present method can be a powerful scheme for determining geometries of large molecules, including heavy-element atoms.

Scrutinizing "Invisible" astatine: A challenge for modern density functionals

The main-group 6p elements did not receive much attention in the development of recent density functionals. In many cases it is still difficult to choose among the modern ones a relevant functional for various applications. Here, we illustrate the case of astatine species (At, Z = 85) and we report the first, and quite complete, benchmark study on several properties concerning such species. Insights on geometries, transition energies and thermodynamic properties of a set of 19 astatine species, for which reference experimental or theoretical data has been reported, are obtained with relativistic (two-component) density functional theory calculations. An extensive set of widely used functionals is employed. The hybrid meta-generalized gradient approximation (meta-GGA) PW6B95 functional is overall the best choice. It is worth noting that the range-separated HSE06 functional as well as the old and very popular B3LYP and PBE0 hybrid-GGAs appear to perform quite well too. Moreover, we found that astatine chemistry in solution can accurately be predicted using implicit solvent models, provided that specific parameters are used to build At cavities. © 2016 Wiley Periodicals, Inc.

Relativistic state-specific multireference perturbation theory incorporating improved virtual orbitals: Application to the ground state single-bond dissociation

Using four-component (4c) relativistic spinors, we present a computationally economical relativistic ab initio method for molecular systems employing our recently proposed second-order state-specific multireference perturbation theory (SSMRPT) incorporating the improved virtual orbital-complete active space configuration interaction (IVO-CASCI) reference wavefunction. The resulting method, 4c-IVO-SSMRPT [calculate one state at a time] is tested in pilot calculations on the homonuclear dimers including Li(2), Na(2), K(2), Rb(2), F(2), Cl(2), and Br(2) through the computations of the ground state potential energy curves (PECs). As SSMRPT curbs intruder effects, 4c-IVO-SSMRPT is numerically stable. To our knowledge, the SSMRPT in the 4c relativistic framework has not been explored in the past. Selective spectroscopic constants that are closely related to the correct shape and accuracy of the energy surfaces have been extracted from the computed PECs. For the halogen molecules, a relativistic destabilization of the bond has been found. Relativistic and electron correlation effects need to be incorporated to get reliable estimates. Our results are in good accordance with reference theoretical and experimental data which manifests the computational accuracy and efficiency of the new 4c-IVO-SSMRPT method. The method opens for an improved description of MR systems containing heavy elements. The inexpensiveness of IVO-CASCI makes 4c-IVO-SSMRPT method promising for studies on large systems of heavy elements.

QTAIM Analysis in the Context of Quasirelativistic Quantum Calculations

Computational chemistry currently lacks ad hoc tools for probing the nature of chemical bonds in heavy and superheavy-atom systems where the consideration of spin-orbit coupling (SOC) effects is mandatory. We report an implementation of the Quantum Theory of Atoms-In-Molecules in the framework of two-component relativistic calculations. Used in conjunction with the topological analysis of the Electron Localization Function, we show for astatine (At) species that SOC significantly lowers At electronegativity and boosts its propensity to make charge-shift bonds. Relativistic spin-dependent effects are furthermore able to change some bonds from mainly covalent to charge-shift type. The implication of the disclosed features regarding the rationalization of the labeling protocols used in nuclear medicine for (211)At radioisotope nicely illustrates the potential of the introduced methodology for investigating the chemistry of (super)heavy elements.