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 function G 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, the G W ${GW}$ quasiparticle energies are needed to study light-matter interactions with the Bethe-Salpeter equation.

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.

Current density functional framework for spin-orbit coupling: Extension to periodic systems.

Spin-orbit coupling induces a current density in the ground state, which consequently requires a generalization for meta-generalized gradient approximations. That is, the exchange-correlation energy has to be constructed as an explicit functional of the current density, and a generalized kinetic energy density has to be formed to satisfy theoretical constraints. Herein, we generalize our previously presented formalism of spin-orbit current density functional theory [Holzer et al., J. Chem. Phys. 157, 204102 (2022)] to non-magnetic and magnetic periodic systems of arbitrary dimension. In addition to the ground-state exchange-correlation potential, analytical derivatives such as geometry gradients and stress tensors are implemented. The importance of the current density is assessed for band gaps, lattice constants, magnetic transitions, and Rashba splittings. In the latter, the impact of the current density may be larger than the deviation between different density functional approximations.

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

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

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

Invited for the cover of this issue are the groups of P. W. Roesky (Karlsruhe) and F. Weigend (Marburg). The image depicts coinage metal cores with tetrahedrally coordinating tin atoms. Read the full text of the article at 10.1002/chem.202203583.

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.

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.

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

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

A local hybrid exchange functional approximation from first principles.

Local hybrid functionals are a more flexible class of density functional approximations, allowing for a position-dependent admixture of exact exchange. This additional flexibility, however, comes with a more involved mathematical form and a more complicated design. A common denominator for previously constructed local hybrid functionals is the 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 constraints. 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, nuclear magnetic resonance (NMR) spin-spin coupling 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 loses some ground for the NMR shifts. Therefore, the designed functional is a major step forward for functionals that have been designed from first principles.

Impact of the current density on paramagnetic NMR properties.

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

Current density functional framework for spin-orbit coupling.

Relativistic two-component density functional calculations are carried out in a non-collinear formalism to describe spin-orbit interactions, where the exchange-correlation functional is constructed as a generalization of the non-relativistic density functional approximation. Contrary to non-relativistic density functional theory (DFT), spin-orbit coupling, however, leads to a non-vanishing paramagnetic current density. Density functionals depending on the kinetic energy density, such as meta-generalized gradient approximations, should therefore be constructed in the framework of current DFT (CDFT). The latter has previously exclusively been used in the regime of strong magnetic fields. Herein, we present a consistent CDFT approach for relativistic DFT, including spin-orbit coupling. Furthermore, we assess the importance of the current density terms for ground-state energies, excitation energies, nuclear magnetic resonance shielding, and spin-spin coupling constants, as well as hyperfine coupling constants, Δg-shifts, and the nuclear quadrupole interaction tensor in electron paramagnetic resonance (EPR) spectroscopy. The most notable changes are found for EPR properties. The impact of the current-dependent terms rises with the number of unpaired electrons, and consequently, the EPR properties are more sensitive toward CDFT. Considerable changes are observed for the strongly constrained and appropriately normed functionals, as well as the B97M family and TASK. The current density terms are less important when exact exchange is incorporated. At the same time, the current-dependent kernel ensures the stability of response calculations in all cases. We, therefore, strongly recommend to use the framework of CDFT for self-consistent spin-orbit calculations.

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-.

Low-Valent Group†14 Phosphinidenide Complexes [({SIDipp}P)2 M] Exhibit P-M pπ-pπ Interaction (M=Ge, Sn, Pb).

Herein, the synthesis of new low-valent Group 14 phosphinidenide complexes [({SIDipp}P)2 M] exhibiting P-M pπ-pπ interactions (SIDipp=1,3-bis(2,6-diisopropylphenyl)-imidazolidin-2-ylidene, M=Ge, Sn, Pb), is presented. These compounds were investigated by means of structural, spectroscopic, and quantum-chemical methods. Furthermore, the monosubstituted compounds [(SIDippP)MX]2 (M=Sn, X=Cl; M=Pb, X=Br) are presented, which show dimeric structures instead of multiple bonding interaction.

TURBOMOLE: Modular program suite for ab initio quantum-chemical and condensed-matter simulations.

TURBOMOLE is a collaborative, multi-national software development project aiming to provide highly efficient and stable computational tools for quantum chemical simulations of molecules, clusters, periodic systems, and solutions. The TURBOMOLE software suite is optimized for widely available, inexpensive, and resource-efficient hardware such as multi-core workstations and small computer clusters. TURBOMOLE specializes in electronic structure methods with outstanding accuracy-cost ratio, such as density functional theory including local hybrids and the random phase approximation (RPA), GW-Bethe-Salpeter methods, second-order Møller-Plesset theory, and explicitly correlated coupled-cluster methods. TURBOMOLE is based on Gaussian basis sets and has been pivotal for the development of many fast and low-scaling algorithms in the past three decades, such as integral-direct methods, fast multipole methods, the resolution-of-the-identity approximation, imaginary frequency integration, Laplace transform, and pair natural orbital methods. This review focuses on recent additions to TURBOMOLE's functionality, including excited-state methods, RPA and Green's function methods, relativistic approaches, high-order molecular properties, solvation effects, and periodic systems. A variety of illustrative applications along with accuracy and timing data are discussed. Moreover, available interfaces to users as well as other software are summarized. TURBOMOLE's current licensing, distribution, and support model are discussed, and an overview of TURBOMOLE's development workflow is provided. Challenges such as communication and outreach, software infrastructure, and funding are highlighted.

Error-consistent segmented contracted all-electron relativistic basis sets of double- and triple-zeta quality for NMR shielding constants.

We present property-tailored all-electron relativistic Karlsruhe basis sets for the elements hydrogen to radon. The modifications described herein use at most four additional primitive basis functions and re-optimized contraction coefficients of the inner-most segment. Thus, the shielding constants are improved while maintaining the compactness of the basis set. A large set of 255 closed-shell molecules was used to assess the quality of the developed bases throughout the periodic table of elements.

The Identity of "Ternary" A/Tl/Pb or K/Tl/Bi Solid Mixtures and Binary Zintl Anions Isolated From Their Solutions.

Investigations of solid mixtures of the elemental combinations A/Tl/Pb (A=Na, K) and K/Tl/Bi indicate the presence of multiple binary and ternary Zintl phases, among them new ones containing Tl and Pb or Bi, respectively. Extractions with en/crypt-222 afford single crystals of several novel binary anions, including [Tl@Tl4 Pb8 ]4- and (Tl4 Bi3 )3- . [Tl@Tl4 Pb8 ]4- adopts a closo-type cage structure despite possessing one additional electron; it is therefore isostructural, yet not isoelectronic, with homoatomic [Tl@Tl12 ]11- obtained by solid state reactions. (Tl4 Bi3 )3- is a rare case of a pentagonal bipyramidal Zintl anion, yet the first binary one, and (unlike Tl77- ) the first one with a proper closo-type electron count. Assignment of the numbers and positions of the Tl/Pb or Tl/Bi atoms within the anionic clusters, indistinguishable in classical X-ray diffraction experiments, was achieved by means of quantum chemistry. The studies shed light on the complex situation in solid heavy element mixtures and their substantial differences from the composition of the Zintl anions obtained from them by extraction.

Efficient implementation of one- and two-component analytical energy gradients in exact two-component theory.

We present an efficient algorithm for one- and two-component analytical energy gradients with respect to nuclear displacements in the exact two-component decoupling approach to the one-electron Dirac equation (X2C). Our approach is a generalization of the spin-free ansatz by Cheng and Gauss [J. Chem. Phys. 135, 084114 (2011)], where the perturbed one-electron Hamiltonian is calculated by solving a first-order response equation. Computational costs are drastically reduced by applying the diagonal local approximation to the unitary decoupling transformation (DLU) [D. Peng and M. Reiher, J. Chem. Phys. 136, 244108 (2012)] to the X2C Hamiltonian. The introduced error is found to be almost negligible as the mean absolute error of the optimized structures amounts to only 0.01 pm. Our implementation in TURBOMOLE is also available within the finite nucleus model based on a Gaussian charge distribution. For a X2C/DLU gradient calculation, computational effort scales cubically with the molecular size, while storage increases quadratically. The efficiency is demonstrated in calculations of large silver clusters and organometallic iridium complexes.

Calculations of current densities and aromatic pathways in cyclic porphyrin and isoporphyrin arrays.

Magnetically induced current density susceptibilities have been studied for a number of cyclic ethyne- and butadiyne-bridged porphyrin and isoporphyrin arrays. The current density susceptibilities have been calculated using the gauge-including magnetically induced current (GIMIC) method, which is interfaced to the TURBOMOLE quantum chemistry code. Aromatic properties and current pathways have been analyzed and discussed by numerical integration of the current density susceptibilities passing selected chemical bonds yielding current strength susceptibilities. Despite the interrupted π-framework, zinc(ii) isoporphyrin sustains a ring current of ca. 10 nA T-1. Porphyrin and isoporphyrin dimers sustain a significant current strength at the linker, whereas the larger porphyrinoid arrays sustain mainly local ring currents. Isoporphyrin dimers with saturated meso carbons have strong net diatropic ring-current strengths of 20 nA T-1 fulfilling Hückels aromaticity rule. Porphyrin trimers and tetramers exhibit almost no current strength at the linker. The porphyrin moieties maintain their strong net diatropic ring current.

Reducing Exact Two-Component Theory for NMR Couplings to a One-Component Approach: Efficiency and Accuracy.

The self-consistent and complex spin-orbit exact two-component (X2C) formalism for NMR spin-spin coupling constants [ J. Chem. Theory Comput. 17, 2021, 3874-3994] is reduced to a scalar one-component ansatz. This way, the first-order response term can be partitioned into the Fermi-contact (FC) and spin-dipole (SD) interactions as well as the paramagnetic spin-orbit (PSO) contribution. The FC+SD terms are real and symmetric, while the PSO term is purely imaginary and antisymmetric. The relativistic one-component approach is combined with a modern density functional treatment up to local hybrid functionals including the response of the current density. Computational demands are reduced by factors of 8-24 as shown for a large tin compound consisting of 137 atoms. Limitations of the current ansatz are critically assessed for Sn, Pb, Pd, and Pt compounds, i.e. the one-component treatment is not sufficient for tin compounds featuring a few heavy halogen atoms.

φ-Aromaticity in prismatic {Bi6}-based clusters.

The occurrence of aromaticity in organic molecules is widely accepted, but its occurrence in purely metallic systems is less widespread. Molecules comprising only metal atoms (M) are known to be able to exhibit aromatic behaviour, sustaining ring currents inside an external magnetic field along M-M connection axes (σ-aromaticity) or above and below the plane (π-aromaticity) for cyclic or cage-type compounds. However, all-metal compounds provide an extension of the electrons' mobility also in other directions. Here, we show that regular {Bi6} prisms exhibit a non-localizable molecular orbital of f-type symmetry and generate a strong ring current that leads to a behaviour referred to as φ-aromaticity. The experimentally observed heterometallic cluster [{CpRu}3Bi6]-, based on a regular prismatic {Bi6} unit, displays aromatic behaviour; according to quantum chemical calculations, the corresponding hypothetical Bi62- prism shows a similar behaviour. By contrast, [{(cod)Ir}3Bi6] features a distorted Bi6 moiety that inhibits φ-aromaticity.