Cost-Efficient High-Resolution Linear Absorption Spectra through Extrapolating the Dipole Moment from Real-Time Time-Dependent Electronic-Structure Theory.

We present a novel function fitting method for approximating the propagation of the time-dependent electric dipole moment from real-time electronic structure calculations. Real-time calculations of the electronic absorption spectrum require discrete Fourier transforms of the electric dipole moment. The spectral resolution is determined by the total propagation time, i.e., the trajectory length of the dipole moment, causing a high computational cost. Our developed method uses function fitting on shorter trajectories of the dipole moment, achieving arbitrary spectral resolution through extrapolation. Numerical testing shows that the fitting method can reproduce high-resolution spectra by using short dipole trajectories. The method converges with as little as 100 a.u. dipole trajectories for some systems, though the difficulty converging increases with the spectral density. We also introduce an error estimate of the fit, reliably assessing its convergence and hence the quality of the approximated spectrum.

Accurate Relativistic Real-Time Time-Dependent Density Functional Theory for Valence and Core Attosecond Transient Absorption Spectroscopy.

First principles theoretical modeling of out-of-equilibrium processes observed in attosecond pump-probe transient absorption spectroscopy (TAS) triggering pure electron dynamics remains a challenging task, especially for heavy elements and/or core excitations containing fingerprints of scalar and spin-orbit relativistic effects. To address this, we formulate a methodology for simulating TAS within the relativistic real-time, time-dependent density functional theory (RT-TDDFT) framework, for both the valence and core energy regimes. Especially for TAS, full four-component (4c) RT simulations are feasible but computationally demanding. Therefore, in addition to the 4c approach, we also introduce the atomic mean-field exact two-component (amfX2C) Hamiltonian accounting for one- and two-electron picture-change corrections within RT-TDDFT. amfX2C preserves the accuracy of the parent 4c method at a fraction of its computational cost. Finally, we apply the methodology to study valence and near-L2,3-edge TAS processes of experimentally relevant systems and provide additional physical insights using relativistic nonequilibrium response theory.

Exact Two-Component TDDFT with Simple Two-Electron Picture-Change Corrections: X-ray Absorption Spectra Near L- and M-Edges of Four-Component Quality at Two-Component Cost.

X-ray absorption spectroscopy (XAS) has gained popularity in recent years as it probes matter with high spatial and elemental sensitivities. However, the theoretical modeling of XAS is a challenging task since XAS spectra feature a fine structure due to scalar (SC) and spin-orbit (SO) relativistic effects, in particular near L and M absorption edges. While full four-component (4c) calculations of XAS are nowadays feasible, there is still interest in developing approximate relativistic methods that enable XAS calculations at the two-component (2c) level while maintaining the accuracy of the parent 4c approach. In this article we present theoretical and numerical insights into two simple yet accurate 2c approaches based on an (extended) atomic mean-field exact two-component Hamiltonian framework, (e)amfX2C, for the calculation of XAS using linear eigenvalue and damped response time-dependent density functional theory (TDDFT). In contrast to the commonly used one-electron X2C (1eX2C) Hamiltonian, both amfX2C and eamfX2C account for the SC and SO two-electron and exchange-correlation picture-change (PC) effects that arise from the X2C transformation. As we demonstrate on L- and M-edge XAS spectra of transition metal and actinide compounds, the absence of PC corrections in the 1eX2C approximation results in a substantial overestimation of SO splittings, whereas (e)amfX2C Hamiltonians reproduce all essential spectral features such as shape, position, and SO splitting of the 4c references in excellent agreement, while offering significant computational savings. Therefore, the (e)amfX2C PC correction models presented here constitute reliable relativistic 2c quantum-chemical approaches for modeling XAS.

The four-component DFT method for the calculation of the EPR g-tensor using a restricted magnetically balanced basis and London atomic orbitals.

Four-component relativistic treatments of the electron paramagnetic resonance g-tensor have so far been based on a common gauge origin and a restricted kinetically balanced basis. The results of such calculations are prone to exhibit a dependence on the choice of the gauge origin for the vector potential associated with uniform magnetic field and a related dependence on the basis set quality. In this work, this gauge problem is addressed by a distributed-origin scheme based on the London atomic orbitals, also called gauge-including atomic orbitals (GIAOs), which have proven to be a practical approach for calculations of other magnetic properties. Furthermore, in the four-component relativistic domain, it has previously been shown that a restricted magnetically balanced (RMB) basis for the small component of the four-component wavefunctions is necessary for achieving robust convergence with regard to the basis set size. We present the implementation of a four-component density functional theory (DFT) method for calculating the g-tensor, incorporating both the GIAOs and RMB basis and based on the Dirac-Coulomb Hamiltonian. The approach utilizes the state-of-the-art noncollinear Kramers-unrestricted DFT methodology to achieve rotationally invariant results and inclusion of spin-polarization effects in the calculation. We also show that the gauge dependence of the results obtained is connected to the nonvanishing integral of the current density in a finite basis, explain why the results of cluster calculations exhibit surprisingly low gauge dependence, and demonstrate that the gauge problem disappears for systems with certain point-group symmetries.

Exact two-component Hamiltonians for relativistic quantum chemistry: Two-electron picture-change corrections made simple.

Based on self-consistent field (SCF) atomic mean-field (amf) quantities, we present two simple yet computationally efficient and numerically accurate matrix-algebraic approaches to correct both scalar-relativistic and spin-orbit two-electron picture-change effects (PCEs) arising within an exact two-component (X2C) Hamiltonian framework. Both approaches, dubbed amfX2C and e(xtended)amfX2C, allow us to uniquely tailor PCE corrections to mean-field models, viz. Hartree-Fock or Kohn-Sham DFT, in the latter case also avoiding the need for a point-wise calculation of exchange-correlation PCE corrections. We assess the numerical performance of these PCE correction models on spinor energies of group 18 (closed-shell) and group 16 (open-shell) diatomic molecules, achieving a consistent ≈10-5 Hartree accuracy compared to reference four-component data. Additional tests include SCF calculations of molecular properties such as absolute contact density and contact density shifts in copernicium fluoride compounds (CnFn, n = 2,4,6), as well as equation-of-motion coupled-cluster calculations of x-ray core-ionization energies of 5d- and 6d-containing molecules, where we observe an excellent agreement with reference data. To conclude, we are confident that our (e)amfX2C PCE correction models constitute a fundamental milestone toward a universal and reliable relativistic two-component quantum-chemical approach, maintaining the accuracy of the parent four-component one at a fraction of its computational cost.

Accurate X-ray Absorption Spectra near L- and M-Edges from Relativistic Four-Component Damped Response Time-Dependent Density Functional Theory.

The simulation of X-ray absorption spectra requires both scalar and spin-orbit (SO) relativistic effects to be taken into account, particularly near L- and M-edges where the SO splitting of core p and d orbitals dominates. Four-component Dirac-Coulomb Hamiltonian-based linear damped response time-dependent density functional theory (4c-DR-TDDFT) calculates spectra directly for a selected frequency region while including the relativistic effects variationally, making the method well suited for X-ray applications. In this work, we show that accurate X-ray absorption spectra near L2,3- and M4,5-edges of closed-shell transition metal and actinide compounds with different central atoms, ligands, and oxidation states can be obtained by means of 4c-DR-TDDFT. While the main absorption lines do not change noticeably with the basis set and geometry, the exchange-correlation functional has a strong influence with hybrid functionals performing the best. The energy shift compared to the experiment is shown to depend linearly on the amount of Hartee-Fock exchange with the optimal value being 60% for spectral regions above 1000 eV, providing relative errors below 0.2% and 2% for edge energies and SO splittings, respectively. Finally, the methodology calibrated in this work is used to reproduce the experimental L2,3-edge X-ray absorption spectra of [RuCl2(DMSO)2(Im)2] and [WCl4(PMePh2)2], and resolve the broad bands into separated lines, allowing an interpretation based on ligand field theory and double point groups. These results support 4c-DR-TDDFT as a reliable method for calculating and analyzing X-ray absorption spectra of chemically interesting systems, advance the accuracy of state-of-the art relativistic DFT approaches, and provide a reference for benchmarking more approximate techniques.

Supramolecular Coronation of Platinum(II) Complexes by Macrocycles: Structure, Relativistic DFT Calculations, and Biological Effects.

Platinum-based anticancer drugs are actively developed utilizing lipophilic ligands or drug carriers for the efficient penetration of biomembranes, reduction of side effects, and tumor targeting. We report the development of a supramolecular host-guest system built on cationic platinum(II) compounds bearing ligands anchored in the cavity of the macrocyclic host. The host-guest binding and hydrolysis process on the platinum core were investigated in detail by using NMR, MS, X-ray diffraction, and relativistic DFT calculations. The encapsulation process in cucurbit[7]uril unequivocally promotes the stability of hydrolyzed dicationic cis-[PtII(NH3)2(H2O)(NH2-R)]2+ compared to its trans isomer. Biological screening on the ovarian cancer lines A2780 and A2780/CP shows time-dependent toxicity. Notably, the reported complex and its ß-cyclodextrin (ß-CD) assembly achieve the same cellular uptake as cisplatin and cisplatin@ß-CD, respectively, while maintaining a significantly lower toxicity profile.

Implementation of Relativistic Coupled Cluster Theory for Massively Parallel GPU-Accelerated Computing Architectures.

In this paper, we report reimplementation of the core algorithms of relativistic coupled cluster theory aimed at modern heterogeneous high-performance computational infrastructures. The code is designed for parallel execution on many compute nodes with optional GPU coprocessing, accomplished via the new ExaTENSOR back end. The resulting ExaCorr module is primarily intended for calculations of molecules with one or more heavy elements, as relativistic effects on the electronic structure are included from the outset. In the current work, we thereby focus on exact two-component methods and demonstrate the accuracy and performance of the software. The module can be used as a stand-alone program requiring a set of molecular orbital coefficients as the starting point, but it is also interfaced to the DIRAC program that can be used to generate these. We therefore also briefly discuss an improvement of the parallel computing aspects of the relativistic self-consistent field algorithm of the DIRAC program.

Generalization of Intrinsic Orbitals to Kramers-Paired Quaternion Spinors, Molecular Fragments, and Valence Virtual Spinors.

Localization of molecular orbitals finds its importance in the representation of chemical bonding (and antibonding) and in the local correlation treatments beyond mean-field approximation. In this paper, we generalize the intrinsic atomic and bonding orbitals [G. Knizia, J. Chem. Theory Comput. 2013, 9, 11, 4834-4843] to relativistic applications using complex and quaternion spinors, as well as to molecular fragments instead of atomic fragments only. By performing a singular value decomposition, we show how localized valence virtual orbitals can be expressed on this intrinsic minimal basis. We demonstrate our method on systems of increasing complexity, starting from simple cases such as benzene, acrylic acid, and ferrocene molecules, and then demonstrate the use of molecular fragments and inclusion of relativistic effects for complexes containing heavy elements such as tellurium, iridium, and astatine. The aforementioned scheme is implemented into a standalone program interfaced with several different quantum chemistry packages.

First-Principles Calculation of 1H NMR Chemical Shifts of Complex Metal Polyhydrides: The Essential Inclusion of Relativity and Dynamics.

1H NMR spectroscopy has become an important technique for the characterization of transition-metal hydride complexes, whose metal-bound hydrides are often difficult to locate by X-ray diffraction. In this regard, the accurate prediction of 1H NMR chemical shifts provides a useful, but challenging, strategy to help in the interpretation of the experimental spectra. In this work, we establish a density-functional-theory protocol that includes relativistic, solvent, and dynamic effects at a high level of theory, allowing us to report an accurate and reliable interpretation of 1H NMR hydride chemical shifts of iridium polyhydride complexes. In particular, we have studied in detail the hydride chemical shifts of the [Ir6(IMe)8(CO)2H14]2+ complex in order to validate previous assignments. The computed 1H NMR chemical shifts are strongly dependent on the relativistic treatment, the choice of the DFT exchange-correlation functional, and the conformational dynamics. By combining a fully relativistic four-component electronic-structure treatment with ab initio molecular dynamics, we were able to reliably model both the terminal and bridging hydride chemical shifts and to show that two NMR hydride signals were inversely assigned in the experiment.

ReSpect: Relativistic spectroscopy DFT program package.

With the increasing interest in compounds containing heavier elements, the experimental and theoretical community requires computationally efficient approaches capable of simultaneous non-perturbative treatment of relativistic, spin-polarization, and electron correlation effects. The ReSpect program has been designed with this goal in mind and developed to perform relativistic density functional theory (DFT) calculations on molecules and solids at the quasirelativistic two-component (X2C Hamiltonian) and fully relativistic four-component (Dirac-Coulomb Hamiltonian) level of theory, including the effects of spin polarization in open-shell systems at the Kramers-unrestricted self-consistent field level. Through efficient algorithms exploiting time-reversal symmetry, biquaternion algebra, and the locality of atom-centered Gaussian-type orbitals, a significant reduction of the methodological complexity and computational cost has been achieved. This article summarizes the essential theoretical and technical advances made in the program, supplemented by example calculations. ReSpect allows molecules with >100 atoms to be efficiently handled at the four-component level of theory on standard central processing unit-based commodity clusters, at computational costs that rarely exceed a factor of 10 when compared to the non-relativistic realm. In addition to the prediction of band structures in solids, ReSpect offers a growing list of molecular spectroscopic parameters that range from electron paramagnetic resonance parameters (g-tensor, A-tensor, and zero-field splitting), via (p)NMR chemical shifts and nuclear spin-spin couplings, to various linear response properties using either conventional or damped-response time-dependent DFT (TDDFT): excitation energies, frequency-dependent polarizabilities, and natural chiroptical properties (electronic circular dichroism and optical rotatory dispersion). In addition, relativistic real-time TDDFT electron dynamics is another unique feature of the program. Documentation, including user manuals and tutorials, is available at the program's website http://www.respectprogram.org.

NMR Spin-Spin Coupling Constants Derived from Relativistic Four-Component DFT Theory-Analysis and Visualization.

An unambiguous assignment of coupling pathways plays an important role in the description and rationalization of NMR indirect spin-spin coupling constants (SSCCs). Unfortunately, the SSCC analysis and visualization tools currently available to quantum chemists are restricted to nonrelativistic theory. Here, we present the theoretical foundation for novel relativistic SSCC visualization techniques based on analysis of the SSCC densities and the first-order current densities induced by the nuclear magnetic dipole moments. Details of the implementation of these techniques in the ReSpect program package are discussed. Numerical assessments are performed on through-space SSCCs, and we choose as our examples the heavy-atom Se-Se, Se-Te, and Te-Te coupling constants in three similar molecules for which experimental data are available. SSCCs were calculated at the nonrelativistic, scalar relativistic, and four-component relativistic density functional levels of theory. Furthermore, with the aid of different visualization methods, we discuss the interpretation of the relativistic effects, which are sizable for Se-Se, very significant for Se-Te, and cannot be neglected for Te-Te couplings. A substantial improvement of the theoretical SSCC values is obtained by also considering the molecular properties of a second conformation.

NMR absolute shielding scales and nuclear magnetic dipole moments of transition metal nuclei.

This work reports new, accurate nuclear magnetic dipole moments for NMR-active transition metal nuclei where the long-standing systematic error due to obsolete diamagnetic correction has been eliminated by ab initio calculations of absolute NMR shielding constants. The error of the diamagnetic correction reaches ≈ -14 000 ppm for rhenium, which results in magnetic dipole moment corrections of about -3 × 10-2µN for the 185Re and 187Re nuclei. Such extreme values are one to two orders of magnitude larger than the corrections reported in the literature so far. These findings may help to resolve discrepancies in hyperfine splitting experiments involving rhenium ions. To obtain the corrected transition metal nuclear magnetic dipole moments, NMR shielding constants for a series of transition metal complexes defined as NMR standards [Harris et al., Pure Appl. Chem., 2001, 73, 1795] were calculated using the non-relativistic coupled cluster method and four-component relativistic density functional theory. To reproduce the experimental conditions of the NMR standards, the solvent effects were incorporated by explicit and implicit solvent models.

Four-component relativistic time-dependent density-functional theory using a stable noncollinear DFT ansatz applicable to both closed- and open-shell systems.

We present a formulation of relativistic linear response time-dependent density functional theory for the calculation of electronic excitation energies in the framework of the four-component Dirac-Coulomb Hamiltonian. This approach is based on the noncollinear ansatz originally developed by Scalmani and Frisch [J. Chem. Theory Comput. 8, 2193 (2012)] and improves upon the past treatment of the limit cases in which the spin density approaches zero. As a result of these improvements, the presented approach is capable of treating both closed- and open-shell reference states. Robust convergence of the Davidson-Olsen eigenproblem algorithm for open-shell reference states was achieved through the use of a solver which considers both left and right eigenvectors. The applicability of the present methodology on both closed- and open-shell reference states is demonstrated on calculations of low-lying excitation energies for Group 3 atomic systems (Sc3+-Ac3+) with nondegenerate ground states, as well as for Group 11 atomic systems (Cu-Rg) and octahedral actinide complexes (PaCl6 2-, UCl6 -, and NpF6) with effective doublet ground states.

Relativistic four-component linear damped response TDDFT for electronic absorption and circular dichroism calculations.

We present a detailed theory, implementation, and a benchmark study of a linear damped response time-dependent density functional theory (TDDFT) based on the relativistic four-component (4c) Dirac-Kohn-Sham formalism using the restricted kinetic balance condition for the small-component basis and a noncollinear exchange-correlation kernel. The damped response equations are solved by means of a multifrequency iterative subspace solver utilizing decomposition of the equations according to Hermitian and time-reversal symmetry. This partitioning leads to robust convergence, and the detailed algorithm of the solver for relativistic multicomponent wavefunctions is also presented. The solutions are then used to calculate the linear electric- and magnetic-dipole responses of molecular systems to an electric perturbation, leading to frequency-dependent dipole polarizabilities, electronic absorption, circular dichroism (ECD), and optical rotatory dispersion (ORD) spectra. The methodology has been implemented in the relativistic spectroscopy DFT program ReSpect, and its performance was assessed on a model series of dimethylchalcogeniranes, C4H8X (X = O, S, Se, Te, Po, Lv), and on larger transition metal complexes that had been studied experimentally, [M(phen)3]3+ (M = Fe, Ru, Os). These are the first 4c damped linear response TDDFT calculations of ECD and ORD presented in the literature.

Four-component relativistic 31P NMR calculations for trans-platinum(ii) complexes: importance of the solvent and dynamics in spectral simulations.

We report a combined experimental-theoretical study on the 31P NMR chemical shift for a number of trans-platinum(ii) complexes. Validity and reliability of the 31P NMR chemical shift calculations are examined by comparing with the experimental data. A successful computational protocol for the accurate prediction of the 31P NMR chemical shifts was established for trans-[PtCl2(dma)PPh3] (dma = dimethylamine) complexes. The reliability of the computed values is shown to be critically dependent on the level of relativistic effects (two-component vs. four component), choice of density functionals, dynamical averaging, and solvation effects. Snapshots obtained from ab initio molecular dynamics simulations were used to identify those solvent molecules which show the largest interactions with the platinum complex, through inspection by using the non-covalent interaction program. We observe satisfactory accuracy from the full four-component matrix Dirac-Kohn-Sham method (mDKS) based on the Dirac-Coulomb Hamiltonian, in conjunction with the KT2 density functional, and dynamical averaging with explicit solvent molecules.

Resolution-of-identity accelerated relativistic two- and four-component electron dynamics approach to chiroptical spectroscopies.

We present an implementation and application of electron dynamics based on real-time time-dependent density functional theory (RT-TDDFT) and relativistic 2-component X2C and 4-component Dirac-Coulomb (4c) Hamiltonians to the calculation of electron circular dichroism and optical rotatory dispersion spectra. In addition, the resolution-of-identity approximation for the Coulomb term (RI-J) is introduced into RT-TDDFT and formulated entirely in terms of complex quaternion algebra. The proposed methodology was assessed on the dimethylchalcogenirane series, C4H8X (X = O, S, Se, Te, Po, Lv), and the spectra obtained by non-relativistic and relativistic methods start to disagree for Se and Te, while dramatic differences are observed for Po and Lv. The X2C approach, even in its simplest one-particle form, reproduces the reference 4c results surprisingly well across the entire series while offering an 8-fold speed-up of the simulations. An overall acceleration of RT-TDDFT by means of X2C and RI-J increases with system size and approaches a factor of almost 25 when compared to the full 4c treatment, without compromising the accuracy of the final spectra. These results suggest that one-particle X2C electron dynamics with RI-J acceleration is an attractive method for the calculation of chiroptical spectra in the valence region.

Interplay of Through-Bond Hyperfine and Substituent Effects on the NMR Chemical Shifts in Ru(III) Complexes.

The links between the molecular structure and nuclear magnetic resonance (NMR) parameters of paramagnetic transition-metal complexes are still relatively unexplored. This applies particularly to the contact term of the hyperfine contribution to the NMR chemical shift. We report combining experimental NMR with relativistic density functional theory (DFT) to study a series of Ru(III) complexes with 2-substituted ß-diketones. A series of complexes with systematically varied substituents was synthesized and analyzed using 1H and 13C NMR spectroscopy. The NMR spectra recorded at several temperatures were used to construct Curie plots and estimate the temperature-independent (orbital) and temperature-dependent (hyperfine) contributions to the NMR shift. Relativistic DFT calculations of electron paramagnetic resonance and NMR parameters were performed to interpret the experimental observations. The effects of individual factors such as basis set, density functional, exact-exchange admixture, and relativity are analyzed and discussed. Based on the calibration study in this work, the fully relativistic Dirac-Kohn-Sham (DKS) method, the GIAO approach (orbital shift), the PBE0 functional with the triple-ζ valence basis sets, and the polarizable continuum model for describing solvent effects were selected to calculate the NMR parameters. The hyperfine contribution to the total paramagnetic NMR (pNMR) chemical shift is shown to be governed by the Fermi-contact (FC) term, and the substituent effect (H vs Br) on the through-bond FC shifts is analyzed, interpreted, and discussed in terms of spin-density distribution, atomic spin populations, and molecular-orbital theory. In contrast to the closed-shell systems of Rh(III), the presence of a single unpaired electron in the open-shell Ru(III) analogs significantly alters the NMR resonances of the ligand atoms distant from the metal center in synergy with the substituent effect.

Relativistic Spin-Orbit Heavy Atom on the Light Atom NMR Chemical Shifts: General Trends Across the Periodic Table Explained.

The importance of relativistic effects on the NMR parameters in heavy-atom (HA) compounds, particularly the SO-HALA (Spin-Orbit Heavy Atom on the Light Atom) effect on NMR chemical shifts, has been known for about 40 years. Yet, a general correlation between the electronic structure and SO-HALA effect has been missing. By analyzing 1H NMR chemical shifts of the sixth-period hydrides (Cs-At), we discovered general electronic-structure principles and mechanisms that dictate the size and sign of the SO-HALA NMR chemical shifts. In brief, partially occupied HA valence shells induce relativistic shielding at the light atom (LA) nuclei, while empty HA valence shells induce relativistic deshielding. In particular, the LA nucleus is relativistically shielded in 5d2-5d8 and 6p4 HA hydrides and deshielded in 4f0, 5d0, 6s0, and 6p0 HA hydrides. This general and intuitive concept explains periodic trends in the 1H NMR chemical shifts along the sixth-period hydrides (Cs-At) studied in this work. We present substantial evidence that the introduced principles have a general validity across the periodic table and can be extended to nonhydride LAs. The decades-old question of why compounds with occupied frontier π molecular orbitals (MOs) cause SO-HALA shielding at the LA nuclei, while the frontier σ MOs cause deshielding is answered. We further derive connection between the SO-HALA NMR chemical shifts and Spin-Orbit-induced Electron Deformation Density (SO-EDD), a property that can be obtained easily from differential electron densities and can be represented graphically. SO-EDD provides an intuitive understanding of the SO-HALA effect in terms of the depletion/concentration of the electron density at LA nuclei caused by spin-orbit coupling due to HA in the presence of a magnetic field. Using an analogy between the SO-EDD concept and arguments from classic NMR theory, the complex question of the SO-HALA NMR chemical shifts becomes easily understandable for a wide chemical audience.

Relativistic DFT Calculations of Hyperfine Coupling Constants in 5d Hexafluorido Complexes: [ReF6 ]2- and [IrF6 ]2.

The performance of relativistic density functional theory (DFT) methods has been investigated for the calculation of the recently measured hyperfine coupling constants of hexafluorido complexes [ReF6 ]2- and [IrF6 ]2- . Three relativistic methods were employed at the DFT level of theory: the 2-component zeroth-order regular approximation (ZORA) method, in which the spin-orbit coupling was treated either variationally (EV ZORA) or as a perturbation (LR ZORA), and the 4-component Dirac-Kohn-Sham (DKS) method. The dependence of the results on the basis set and the choice of exchange-correlation functional was studied. Furthermore, the effect of varying the amount of Hartree-Fock exchange in the hybrid functionals was investigated. The LR ZORA and DKS methods combined with DFT led to very similar deviations (about 20 %) from the experimental values for the coupling constant of complex [ReF6 ]2- by using hybrid functionals. However, none of the methods were able to reproduce the large anisotropy of the hyperfine coupling tensor of complex [ReF6 ]2- . For [IrF6 ]2- , the EV ZORA and DKS methods reproduced the experimental tensor components with deviations of ≈10 and ≈5 % for the hybrid functionals, whereas the LR ZORA method predicted the coupling constant to be around one order of magnitude too large owing to the combination of large spin-orbit coupling and very low excitation energies.