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

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

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

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

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

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

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

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.

Unraveling the enigma of Craig-type Möbius-aromatic osmium compounds

Nuclear magnetic resonance (NMR) chemical shifts and the magnetically induced current density (MICD) susceptibility of four osmium containing molecules have been calculated at the density functional theory (DFT) level using three relativistic levels of theory. The calculations were performed at the quasi-relativistic level using an effective core potential (ECP) for Os, at the all-electron scalar exact two-component (X2C) relativistic level, and at the relativistic X2C level including spin-orbit coupling (SO-X2C). In earlier studies, the osmapentalene (1) and the osmapentalynes (2 and 3) were considered Craig-type Möbius aromatic and it was suggested that the analogous osmium compound (4) is Craig-type Möbius antiaromatic. Here, the ring-current strengths were obtained with the gauge including magnetically induced currents (GIMIC) method by integrating the MICD susceptibility passing through planes that intersect chemical bonds and by line integration of the induced magnetic field using Ampère-Maxwell's law. The ring-current calculations suggest that 1, 2 and 3 are weakly aromatic and that 4 is nonaromatic. The accuracy of the MICD susceptibility was assessed by comparing calculated NMR chemical shifts to available experimental data. Visualization of the MICD susceptibility shows that the ring current does not pass from one side of the molecular plane to the other, which means that the MICD susceptibility of the studied molecules does not exhibit any Möbius topology as one would expect for Craig-type Möbius aromatic and for Craig-type Möbius antiaromatic molecules. Thus, molecules 1-3 are not Craig-type Möbius aromatic and molecule 4 is not Craig-type Möbius antiaromatic as previously suggested. Calculations of the 1H NMR and 13C NMR chemical shifts of atoms near the Os atom show the importance of including spin-orbit effects. Overall, our study revisits the understanding of the aromaticity of organometallic molecules containing transition metals.

The aromatic nature of auracycles and diauracycles based on calculated ring-current strengths

We have calculated the magnetically induced current density susceptibility for gold-containing organometallic molecular rings using the gauge-including magnetically induced currents (GIMIC) method. The aromatic nature has been determined by calculating the strength of the magnetically induced ring current susceptibility, which is often called ring current. To our knowledge, we show here for the first time that gold-containing organometallic rings may be aromatic or antiaromatic sustaining ring currents in the presence of an external magnetic field. The calculated aromatic character of the rings agrees with the aromatic nature one expects when using Hückel's aromaticity rules. The studied auracycles and diauracycles with 4n electrons in the conjugated orbitals generally sustain a weak paratropic ring current, whereas those having 4n + 2 electrons in the conjugated orbitals sustain a diatropic ring current that is almost as strong as that of benzene. The number of electrons are obtained by assuming that each C, N and Au atom of the ring contribute one electron, and a H atom connected to a N atom in the ring increases the number of electrons by one. An electron-attracting ligand at Au removes one electron from the ring. Formation of a short Au-Au bonding diauracycles reduces the number of electrons in the ring by two.

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.

Benchmark Study on the Calculation of 207Pb NMR Chemical Shifts

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

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

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

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

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