ReSpect: Relativistic spectroscopy DFT program package

Dynamic and relativistic effects on Pt-Pt indirect spin-spin coupling in aqueous solution studied by ab initio molecular dynamics and two- vs four-component density functional NMR calculations

Treating 195Pt nuclear magnetic resonance parameters in solution remains a considerable challenge from a quantum chemistry point of view, requiring a high level of theory that simultaneously takes into account the relativistic effects, the dynamic treatment of the solvent-solute system, and the dynamic electron correlation. A combination of Car-Parrinello molecular dynamics (CPMD) and relativistic calculations based on two-component zeroth order regular approximation spin-orbit Kohn-Sham (2c-ZKS) and four-component Dirac-Kohn-Sham (4c-DKS) Hamiltonians is performed to address the solvent effect (water) on the conformational changes and JPtPt1 coupling. A series of bridged PtIII dinuclear complexes [L1-Pt2(NH3)4(Am)2-L2]n+ (Am = α-pyrrolidonate and pivalamidate; L = H2O, Cl-, and Br-) are studied. The computed Pt-Pt coupling is strongly dependent on the conformational dynamics of the complexes, which, in turn, is correlated with the trans influence among axial ligands and with the angle N-C-O from the bridging ligands. The J-coupling is decomposed in terms of dynamic contributions. The decomposition reveals that the vibrational and explicit solvation contributions reduce JPtPt1 of diaquo complexes (L1 = L2 = H2O) in comparison to the static gas-phase magnitude, whereas the implicit solvation and bulk contributions correspond to an increase in JPtPt1 in dihalo (L1 = L2 = X-) and aquahalo (L1 = H2O; L2 = X-) complexes. Relativistic treatment combined with CPMD shows that the 2c-ZKS Hamiltonian performs as well as 4c-DKS for the JPtPt1 coupling.

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).

Unified construction of relativistic Hamiltonians

It is shown that the four-component (4C), quasi-four-component (Q4C), and exact two-component (X2C) relativistic Hartree-Fock equations can be implemented in a unified manner by making use of the atomic nature of the small components of molecular 4-spinors. A model density matrix approximation can first be invoked for the small-component charge/current density functions, which gives rise to a static, pre-molecular mean field to be combined with the one-electron term. As a result, only the nonrelativistic-like two-electron term of the 4C/Q4C/X2C Fock matrix needs to be updated during the iterations. A "one-center small-component" approximation can then be invoked in the evaluation of relativistic integrals, that is, all atom-centered small-component basis functions are regarded as extremely localized near the position of the atom to which they belong such that they have vanishing overlaps with all small- or large-component functions centered at other nuclei. Under these approximations, the 4C, Q4C, and X2C mean-field and many-electron Hamiltonians share precisely the same structure and accuracy. Beyond these is the effective quantum electrodynamics Hamiltonian that can be constructed in the same way. Such approximations lead to errors that are orders of magnitude smaller than other sources of errors (e.g., truncation errors in the one- and many-particle bases as well as uncertainties of experimental measurements) and are, hence, safe to use for whatever purposes. The quaternion forms of the 4C, Q4C, and X2C equations are also presented in the most general way, based on which the corresponding Kramers-restricted open-shell variants are formulated for "high-spin" open-shell systems.

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.

Controlling the Activation at NiII -CO2 2- Moieties through Lewis Acid Interactions in the Second Coordination Sphere

Nickel complexes with a two-electron reduced CO2 ligand (CO2 2- , "carbonite") are investigated with regard to the influence alkali metal (AM) ions have as Lewis acids on the activation of the CO2 entity. For this purpose complexes with NiII (CO2 )AM (AM=Li, Na, K) moieties were accessed via deprotonation of nickel-formate compounds with (AM)N(i Pr)2 . It was found that not only the nature of the AM ions in vicinity to CO2 affect the activation, but also the number and the ligation of a given AM. To this end the effects of added (AM)N(R)2 , THF, open and closed polyethers as well as cryptands were systematically studied. In 14 cases the products were characterized by X-ray diffraction and correlations with the situation in solution were made. The more the AM ions get detached from the carbonite ligand, the lower is the degree of aggregation. At the same time the extent of CO2 activation is decreased as indicated by the structural and spectroscopic analysis and reactivity studies. Accompanying DFT studies showed that the coordinating AM Lewis acidic fragment withdraws only a small amount of charge from the carbonite moiety, but it also affects the internal charge equilibration between the LtBu Ni and carbonite moieties.

Scalar Relativistic Effects with Multiwavelets: Implementation and Benchmark

The importance of relativistic effects in quantum chemistry is widely recognized, not only for heavier elements but throughout the periodic table. At the same time, relativistic effects are strongest in the nuclear region, where the description of electrons through a linear combination of atomic orbitals becomes more challenging. Furthermore, the choice of basis sets for heavier elements is limited compared with lighter elements where precise basis sets are available. Thanks to the framework of multiresolution analysis, multiwavelets provide an appealing alternative to overcoming this challenge: they lead to robust error control and adaptive algorithms that automatically refine the basis set description until the desired precision is reached. This allows one to achieve a proper description of the nuclear region. In this work, we extended the multiwavelet-based code MRChem to the scalar zero-order regular approximation framework. We validated our implementation by comparing the total energies for a small set of elements and molecules. To confirm the validity of our implementation, we compared both against a radial numerical code for atoms and the plane-wave-based code EXCITING.

Novel Computational Chemistry Infrastructure for Simulating Astatide in Water: From Basis Sets to Force Fields Using Particle Swarm Optimization

Using the example of astatine, the heaviest naturally occurring halogen whose isotope At-211 has promising medical applications, we propose a new infrastructure for large-scale computational models of heavy elements with strong relativistic effects. In particular, we focus on developing an accurate force field for At- in water based on reliable relativistic density functional theory (DFT) calculations. To ensure the reliability of such calculations, we design novel basis sets for relativistic DFT, via the particle swarm optimization algorithm to optimize the coefficients of the new basis sets and the polarization-consistent basis set idea's extension to heavy elements to eliminate the basis set error from DFT calculations. The resulting basis sets enable the well-grounded evaluation of relativistic DFT against "gold-standard" CCSD(T) results. Accounting for strong relativistic effects, including spin-orbit interaction, via our redesigned infrastructure, we elucidate a noticeable dissimilarity between At- and I- in halide-water force field parameters, radial distribution functions, diffusion coefficients, and hydration energies. This work establishes the framework for the systematic development of polarization-consistent basis sets for relativistic DFT and accurate force fields for molecular dynamics simulations to be used in large-scale models of complex molecular systems with elements from the bottom of the periodic table, including actinides and even superheavy elements.

Relativistic DFT Calculations of Cadmium and Selenium Solid-State NMR Spectra of CdSe Nanocrystal Surfaces

Solid-state NMR spectra have been used to probe the structure of CdSe nanocrystals and propose detailed models of their surface structures. Density functional theory (DFT)-optimized cluster models that represent probable molecular structures of carboxylate-coordinated surface sites have been proposed. However, to the best of our knowledge, 113Cd and 77Se chemical shifts have not been calculated for these surface models. We performed relativistic DFT calculations of cadmium and selenium magnetic shielding tensors on model compounds with previously measured solid-state NMR spectra with (i) the four-component Dirac-Kohn-Sham (DKS) Hamiltonian and (ii) the scalar and (iii) spin-orbit levels within the ZORA Hamiltonian. Molecular clusters with Cd and Se sites in varying bonding environments were used to model CdSe (100) and CdSe(111) surfaces capped with carboxylic acid ligands. Our calculations identify the observed 113Cd isotropic chemical shifts δ(iso) of -465, -318, and -146 ppm arising from CdSeO3, CdSe2O2, and CdSe3O surface groups, respectively, with very good agreement with experimental measurements. The 113Cd chemical shifts linearly decrease with the number of O-neighbors. The calculated spans (δ11 - δ33) encompass the experimental values for CdSe3O and CdSe2O2 clusters but are slightly larger than the measured value for CdSeO3 clusters. Relativistic DFT calculations predicted a one-bond 113Cd-77Se scalar coupling of 258 Hz, which is in good agreement with the experimental values of 250 Hz. With a dense coverage of carboxylic acid ligands, the CdSe (100) surface shows a distribution of Cd-Se bond lengths and J-couplings. Relativistic DFT simulations thus aid in interpretation of NMR spectra of CdSe nanocrystals and related nanomaterials.

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.

Unravelling strong temperature-dependence of JHD in transition metal hydrides: solvation and non-covalent interactions versus temperature-elastic H-H bonds

A number of transition metal hydrides reveal intriguing temperature-dependent JHD in their deuterated derivatives and possibly the temperature dependent hydrogen-hydrogen distance (r(H-H)) as well. Previously, theoretical studies rationalized JHD and r(H-H) changes in such compounds through a "temperature-elastic" structure model with a significant population of vibrational states in an anharmonic potential. Based on the first variable temperature neutron diffraction study of a relevant complex, (p-H-POCOP)IrH2, observation of its elusive counterpart with longer r(H-H), crystallized as an adduct with C6F5I, and thorough spectroscopic and computational study, we argue that the model involving isomeric species in solution at least in some cases is more relevant. The existence of such isomers is enabled or enhanced by solvation and weak non-covalent interactions with solvent, such as halogen or dihydrogen bonds. "Non-classical" hydrides with r(H-H) ≈ 1.0-1.6 Å are especially sensitive to the above-mentioned factors.

Understanding Polaritonic Chemistry from Ab Initio Quantum Electrodynamics

In this review, we present the theoretical foundations and first-principles frameworks to describe quantum matter within quantum electrodynamics (QED) in the low-energy regime, with a focus on polaritonic chemistry. By starting from fundamental physical and mathematical principles, we first review in great detail ab initio nonrelativistic QED. The resulting Pauli-Fierz quantum field theory serves as a cornerstone for the development of (in principle exact but in practice) approximate computational methods such as quantum-electrodynamical density functional theory, QED coupled cluster, or cavity Born-Oppenheimer molecular dynamics. These methods treat light and matter on equal footing and, at the same time, have the same level of accuracy and reliability as established methods of computational chemistry and electronic structure theory. After an overview of the key ideas behind those ab initio QED methods, we highlight their benefits for understanding photon-induced changes of chemical properties and reactions. Based on results obtained by ab initio QED methods, we identify open theoretical questions and how a so far missing detailed understanding of polaritonic chemistry can be established. We finally give an outlook on future directions within polaritonic chemistry and first-principles QED.

Computational NMR spectroscopy of 205 Tl

We have investigated the NMR chemical shift of 205 Tl in several thallium compounds, ranging from small covalent Tl(I) and Tl(III) molecules to supramolecular complexes with large organic ligands and some thallium halides. NMR calculations were run at the ZORA relativistic level, with and without spin-orbit coupling using few selected GGA and hybrid functionals, namely BP86, PBE, B3LYP, and PBE0. We also tested solvent effects both at the optimization level and at the NMR calculation step. At the ZORA-SO-PBE0 (COSMO) level of theory we find a very good performance of the computational protocol that allows to discard or retain possible structures/conformations based on the agreement between the calculated chemical shift and the experimental value.

Relativistic resolution-of-the-identity with Cholesky integral decomposition

In this study, we present an efficient integral decomposition approach called the restricted-kinetic-balance resolution-of-the-identity (RKB-RI) algorithm, which utilizes a tunable RI method based on the Cholesky integral decomposition for in-core relativistic quantum chemistry calculations. The RKB-RI algorithm incorporates the restricted-kinetic-balance condition and offers a versatile framework for accurate computations. Notably, the Cholesky integral decomposition is employed not only to approximate symmetric large-component electron repulsion integrals but also those involving small-component basis functions. In addition to comprehensive error analysis, we investigate crucial conditions, such as the kinetic balance condition and variational stability, which underlie the applicability of Dirac relativistic electronic structure theory. We compare the computational cost of the RKB-RI approach with the full in-core method to assess its efficiency. To evaluate the accuracy and reliability of the RKB-RI method proposed in this work, we employ actinyl oxides as benchmark systems, leveraging their properties for validation purposes. This investigation provides valuable insights into the capabilities and performance of the RKB-RI algorithm and establishes its potential as a powerful tool in the field of relativistic quantum chemistry.

NMR chemical shift of confined 129Xe: coordination number, paramagnetic channels and molecular dynamics in a cryptophane-A biosensor

Advances in hyperpolarisation and indirect detection have enabled the development of xenon nuclear magnetic resonance (NMR) biosensors (XBSs) for molecule-selective sensing in down to picomolar concentration. Cryptophanes (Crs) are popular cages for hosting the Xe "spy". Understanding the microscopic host-guest chemistry has remained a challenge in the XBS field. While early NMR computations of XBSs did not consider the important effects of host dynamics and explicit solvent, here we model the motionally averaged, relativistic NMR chemical shift (CS) of free Xe, Xe in a prototypic CrA cage and Xe in a water-soluble CrA derivative, each in an explicit H2O solvent, over system configurations generated at three different levels of molecular dynamics (MD) simulations. We confirm the "contact-type" character of the Xe CS, arising from the increased availability of paramagnetic channels, magnetic couplings between occupied and virtual orbitals through the short-ranged orbital hyperfine operator, when neighbouring atoms are in contact with Xe. Remarkably, the Xe CS in the present, highly dynamic and conformationally flexible situations is found to depend linearly on the coordination number of the Xe atom. We interpret the high- and low-CS situations in terms of the magnetic absorption spectrum and choose our preference among the used MD methods based on comparison with the experimental CS. We check the role of spin-orbit coupling by comparing with fully relativistic CS calculations. The study outlines the computational workflow required to realistically model the CS of Xe confined in dynamic cavity structures under experimental conditions, and contributes to microscopic understanding of XBSs.

Cleavage of an Aromatic C-C Bond in Ferrocene by Insertion of an Iridium Nitrido Nitrogen Atom

The intermolecular cleavage of C-C bonds is a rare event. Herein, we report on a late transition-metal terminal nitrido complex, which upon oxidation undergoes insertion of the nitrido nitrogen atom into the aromatic C-C bond of ferrocene. This reaction path was confirmed through ¹⁵N and deuterium isotope labeling experiments of the nitrido complex and ferrocenium, respectively. Cyclic voltammetry and UV/vis spectroscopy monitoring of the reaction revealed that oxidation is the initial step, yielding the tentative radical cationic nitrido complex, which is experimentally supported by extended X and Q-band electron paramagnetic resonance (EPR) and ENDOR, UV/vis, vT ¹H NMR, and vibrational spectroscopic data. Density functional theory (DFT) and multireference calculations of this highly reactive intermediate revealed an S = 1/2 ground state. The high reactivity can be traced to the increased electrophilicity in the oxidized complex. Based on high-level PNO-UCCSD(T) calculations and UV/vis kinetic measurements, it is proposed that the reaction proceeds by initial electrophilic exo attack of the nitrido nitrogen atom at the cyclopentadienyl ring and consecutive ring expansion to a pyridine ring.

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.

Solid-State 19F NMR Chemical Shift in Square-Planar Nickel-Fluoride Complexes Linked by Halogen Bonds

The halogen bond (XB) is a highly directional class of noncovalent interactions widely explored by experimental and computational studies. However, the NMR signature of the XB has attracted limited attention. The prediction and analysis of the solid-state NMR (SSNMR) chemical shift tensor provide useful strategies to better understand XB interactions. In this work, we employ a computational protocol for modeling and analyzing the ¹⁹F SSNMR chemical shifts previously measured in a family of square-planar trans Ni^II-L₂-iodoaryl-fluoride (L = PEt₃) complexes capable of forming self-complementary networks held by a NiF···I(C) halogen bond [Thangavadivale, V.; Chem. Sci. 2018, 9, 3767-3781]. To understand how the ¹⁹F NMR resonances of the nickel-bonded fluoride are affected by the XB, we investigate the origin of the shielding in trans-[NiF(2,3,5,6-C₆F₄I)(PEt₃)₂], trans-[NiF(2,3,4,5-C₆F₄I)(PEt₃)₂], and trans-[NiF(C₆F₅)(PEt₃)₂] in the solid state, where a XB is present in the two former systems but not in the last. We perform the ¹⁹F NMR chemical shift calculations both in periodic and molecular models. The results show that the crystal packing has little influence on the NMR signatures of the XB, and the NMR can be modeled successfully with a pair of molecules interacting via the XB. Thus, the observed difference in chemical shift between solid-state and solution NMR can be essentially attributed to the XB interaction. The very high shielding of the fluoride and its driving contributor, the most shielded component of the chemical shift tensor, are well reproduced at the 2c-ZORA level. Analysis of the factors controlling the shielding shows how the highest occupied Ni/F orbitals shield the fluoride in the directions perpendicular to the Ni-F bond and specifically perpendicular to the coordination plane. This shielding arises from the magnetic coupling of the Ni(3d)/F(2p lone pair) orbitals with the vacant σ_Ni-F^* orbital, thereby rationalizing the very highly upfield (shielded) resonance of the component (δ₃₃) along this direction. We show that these features are characteristic of square-planar nickel-fluoride complexes. The deshielding of the fluoride in the halogen-bonded systems is attributed to an increase in the energy gap between the occupied and vacant orbitals that are mostly responsible for the paramagnetic terms, notably along the most shielded direction.

Quadratic Spin-Orbit Mechanism of the Electronic g-Tensor

Understanding how the electronic g-tensor is linked to the electronic structure is desirable for the correct interpretation of electron paramagnetic resonance spectra. For heavy-element compounds with large spin-orbit (SO) effects, this is still not completely clear. We report our investigation of quadratic SO contributions to the g-shift in heavy transition metal complexes. We implemented third-order perturbation theory in order to analyze the contributions arising from frontier molecular spin orbitals (MSOs). We show that the dominant quadratic SO term─spin-Zeeman (SO²/SZ)─generally makes a negative contribution to the g-shift, irrespective of the particular electronic configuration or molecular symmetry. We further analyze how the SO²/SZ contribution adds to or subtracts from the linear orbital-Zeeman (SO/OZ) contribution to the individual principal components of the g-tensor. Our study suggests that the SO²/SZ mechanism decreases the anisotropy of the g-tensor in early transition metal complexes and increases it in late transition metal complexes. Finally, we apply MSO analysis to the investigation of g-tensor trends in a set of closely related Ir and Rh pincer complexes and evaluate the influence of different chemical factors (the nuclear charge of the central atom and the terminal ligand) on the magnitudes of the g-shifts. We expect our conclusions to aid the understanding of spectra in magnetic resonance investigations of heavy transition metal compounds.

Nature of NMR Shifts in Paramagnetic Octahedral Ru(III) Complexes with Axial Pyridine-Based Ligands

In recent decades, transition-metal coordination compounds have been extensively studied for their antitumor and antimetastatic activities. In this work, we synthesized a set of symmetric and asymmetric Ru(III) and Rh(III) coordination compounds of the general structure (Na⁺/K⁺/PPh₄⁺/LH⁺) [trans-M^IIIL(eq)_nL(ax)₂]^- (M = Ru^III or Rh^III; L(eq) = Cl, n = 4; L(eq) = ox, n = 2; L(ax) = 4-R-pyridine, R = CH₃, H, C₆H₅, COOH, CF₃, CN; L(ax) = DMSO-S) and systematically investigated their structure, stability, and NMR properties. ¹H and ¹³C NMR spectra measured at various temperatures were used to break down the total NMR shifts into the orbital (temperature-independent) and hyperfine (temperature-dependent) contributions. The hyperfine NMR shifts for paramagnetic Ru(III) compounds were analyzed in detail using relativistic density functional theory (DFT). The effects of (i) the 4-R substituent of pyridine, (ii) the axial trans ligand L(ax), and (iii) the equatorial ligands L(eq) on the distribution of spin density reflected in the "through-bond" (contact) and the "through-space" (pseudocontact) contributions to the hyperfine NMR shifts of the individual atoms of the pyridine ligands are rationalized. Further, we demonstrate the large effects of the solvent on the hyperfine NMR shifts and discuss our observations in the general context of the paramagnetic NMR spectroscopy of transition-metal complexes.

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-L_2,3-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.

Electron Correlation or Basis Set Quality: How to Obtain Converged and Accurate NMR Shieldings for the Third-Row Elements?

The quality of theoretical NMR shieldings calculated at the quantum-chemical level depends on various theoretical aspects, of which the basis set type and size are among the most important factors. Nevertheless, not much information is available on the basis set effect on theoretical shieldings of the NMR-active nuclei of the third row. Here, we report on the importance of proper basis set selection to obtain accurate and reliable NMR shielding parameters for nuclei from the third row of the periodic table. All calculations were performed on a set of eleven compounds containing the elements Na, Mg, Al, Si, P, S, or Cl. NMR shielding tensors were calculated using the SCF-HF, DFT-B3LYP, and CCSD(T) methods, combined with the Dunning valence aug-cc-pVXZ, core-valence aug-cc-pCVXZ, Jensen polarized-convergent aug-pcSseg-n and Karlsruhe x2c-Def2 basis set families. We also estimated the complete basis set limit (CBS) values of the NMR parameters. Widely scattered nuclear shieldings were observed for the Dunning polarized-valence basis set, which provides irregular convergence. We show that the use of Dunning core-valence or Jensen basis sets effectively reduces the scatter of theoretical NMR results and leads to their exponential-like convergence to CBS. We also assessed the effect of vibrational, temperature, and relativistic corrections on the predicted shieldings. For systems with single bonds, all corrections are relatively small, amounting to less than 4% of the CCSD(T)/CBS value. Vibrational and temperature corrections were less reliable for H₃PO and HSiCH due to the high anharmonicity of the molecules. An abnormally high relativistic correction was observed for phosphorus in PN, reaching ~20% of the CCSD(T)/CBS value, while the correction was less than 7% for other tested molecules.

The role of tridentate ligands on the redox stability of anticancer gold(III) complexes

Gold(III) complexes are promising compounds for cancer chemotherapy, whose action depends on their redox stability. In this context, the choice of ligands is crucial to adjust their reactivity and biological response. The present study addressed the effect of the gold coordination sphere on the reduction potential (E^o) for ten gold(III) complexes containing five or six-membered rings tridentate ligands - [Au^III(trident)Cl]³⁺ⁿ (trident = N^N^N, C^N^N, C^C^N, C^N^C, and N^C^N). The calculated E^o covered a broad range of 2500 mV with the most stable complexes containing two AuC bonds (E^o = -1.85 V for [Au^III(C^C^N)Cl] - f). For complexes with one AuC bond, the N^C^N ligands stabilize the gold(III) complex more efficiently than N^N^C; however, the inclusion of the non-innocent ligand bipy (2,2'-bipyridine) in N^N portion provides an extra stabilization effect. Among the derivatives with one AuC bond, [Au^III(N^N^C)Cl]⁺ (N^N = bipy) (a) showed E^o = -1.20 V. For the complexes with N^N^N ligands, E^o was positive and almost constant (+0.60 V). Furthermore, the kinetics for ligand exchange reactions (Cl^-/H₂O, H₂O/Cys and Cl^-/Cys) were monitored for the most stable compounds and the energy profiles compared to the reduction pathways.

Frozen-Density Embedding for Including Environmental Effects in the Dirac-Kohn-Sham Theory: An Implementation Based on Density Fitting and Prototyping Techniques

Frozen density embedding (FDE) represents an embedding scheme in which environmental effects are included from first-principles calculations by considering the surrounding system explicitly by means of its electron density. In the present paper, we extend the full four-component relativistic Dirac-Kohn-Sham (DKS) method, as implemented in the BERTHA code, to include environmental and confinement effects with the FDE scheme (DKS-in-DFT FDE). The implementation, based on the auxiliary density fitting techniques, has been enormously facilitated by BERTHA's python API (PyBERTHA), which facilitates the interoperability with other FDE implementations available through the PyADF framework. The accuracy and numerical stability of this new implementation, also using different auxiliary fitting basis sets, has been demonstrated on the simple NH₃-H₂O system, in comparison with a reference nonrelativistic implementation. The computational performance has been evaluated on a series of gold clusters (Au_n, with n = 2, 4, 8) embedded into an increasing number of water molecules (5, 10, 20, 40, and 80 water molecules). We found that the procedure scales approximately linearly both with the size of the frozen surrounding environment (consistent with the underpinnings of the FDE approach) and with the size of the active system (in line with the use of density fitting). Finally, we applied the code to a series of heavy (Rn) and super-heavy elements (Cn, Fl, Og) embedded in a C₆₀ cage to explore the confinement effect induced by C₆₀ on their electronic structure. We compare the results from our simulations, with respect to more-approximate models employed in the atomic physics literature. Our results indicate that the specific interactions described by FDE are able to improve upon the cruder approximations currently employed, and, thus, they provide a basis from which to generate more-realistic radial potentials for confined atoms.

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 (CnF_n, 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.

Comparison of Length, Velocity, and Symmetric Gauges for the Calculation of Absorption and Electric Circular Dichroism Spectra with Real-Time Time-Dependent Density Functional Theory

A velocity and symmetric gauge implementation for real-time time-dependent density functional theory (RT-TDDFT) in the CP2K package using a Gaussian and plane wave approach is presented, including the explicit gauge-transformed contributions due to the nonlocal part of pseudopotentials. Absorption spectra of gas-phase α-pinene are calculated in length and velocity gauges in the long-wavelength approximation for the application of a δ pulse in linear and full order. The velocity gauge implementation is also applied to a solvated uracil molecule to showcase its use within periodic boundary conditions (PBC). For the calculation of the expectation value of the electric dipole moment in PBC, both the velocity representation and the modern theory of polarization give equivalent absorption spectra if a distributed reference point is used for the nonlocal term of the velocity operator. The discussion of linear response theory takes place in a unified framework in terms of linear response functions in propagator notation, distinguishing the parts of the linear response functions associated with perturbation and response. To further investigate gauge dependence, electric circular dichroism (ECD) spectra of α-pinene were calculated either as magnetic response to an electric field perturbation, in length or velocity gauge, or as electric response to a magnetic field perturbation in the symmetric gauge. Both approaches, electric and magnetic perturbations, have been found to yield equivalent ECD spectra.

Computational Prediction of Tc-99 NMR Chemical Shifts in Technetium Complexes with Radiopharmaceutical Applications

The Tc-99m nucleus is the most used nuclide in radiopharmaceuticals designed for imaging diagnosis. The metal can exist in nine distinct oxidation states and forms distinct coordination complexes with a variety of chelating agents and geometries. These complexes are usually characterized through Tc-99 NMR that is very sensitive to the Tc coordination sphere. Therefore, predicting Tc-99 NMR might be useful to assist experimentalists in structural characterization. In the present study, we propose three computational protocols for predicting Tc-99 NMR chemical shifts based on density functional theory calculations using relativistic and nonrelativistic Hamiltonians: the relativistic Model 1, the nonrelativistic Model 2, and the empirical nonrelativistic Model 3. In Models 2 and 3, the NMR-DKH basis set was used for all atoms, including the Tc, for which it was developed here. All models were applied for a set of 41 Tc-complexes with metal oxidation states 0, I, and V, for which the Tc-99 chemical shift was available experimentally. The mean absolute deviation and the mean relative deviation were 67 ppm and 4.8% (Model 1), 92 ppm and 6.2% (Model 2), and 65 ppm and 4.9% (Model 3), respectively. Last, the effect of the explicit solvent was evaluated for the [TcO₂(en)₂]⁺─Tc(V) complex. The calculated results for the Tc-99 NMR chemical shift at SO-ZORA-SSB-D/TZ2P-ZORA/COSMO//TPSS/def2-SVP/IEF-PCM(UFF) show that the inclusion of 14 water molecules (first solvation shell) together with the implicit solvation model leads to an absolute deviation of only 7 ppm (0.3%) from the experimental value, indicating that the solvent effects play a key role in predicting Tc-99 NMR.

Efficient Evaluation of the Breit Operator in the Pauli Spinor Basis

The frequency-independent Coulomb-Breit operator gives rise to the most accurate treatment of two-electron interaction in the non-quantum-electrodynamics regime. The Breit interaction in the Coulomb gauge consists of magnetic and gauge contributions. The high computational cost of the gauge term limits the application of the Breit interaction in relativistic molecular calculations. In this work, we apply the Pauli component integral-density matrix contraction scheme for gauge interaction with a maximum spin- and component separation scheme. We also present two different computational algorithms for evaluating gauge integrals. One is the generalized Obara-Saika algorithm, where the Laplace transformation is used to transform the gauge operator into Gaussian functions and the Obara-Saika recursion is used for reducing the angular momentum. The other algorithm is the second derivative of inverse Coulomb interaction evaluated with Rys-quadrature. This work improves the efficiency of performing Dirac-Hartree-Fock with variational treatment of Breit interaction for molecular systems. We use this formalism to examine relativistic trends in the periodic table, and analyze the relativistic two-electron interaction contributions in heavy-element complexes.

"In Vitro" and "In Vivo" Diagnostic Check for the Thermochemistry of Metal-Organic Compounds

Volatile metal β-diketonates are of interest from both practical and theoretical perspectives (manufacturing of film materials, catalysis, and the nature of metal-ligand bonding). Knowledge of their reliable thermochemical properties is essential for effective applications. However, there is an unacceptable scattering of the available data on the enthalpies of formation. In this work, we proposed "in vitro" and "in vivo" diagnostic tools to verify the available enthalpies of formation in both the crystalline and gaseous states for metal tris-β-diketonates. The "in vitro" procedure involved high-level quantum-chemical calculations and was applied to define a consistent data set on the enthalpies of formation for iron(III) β-diketonates. This data set has provided the basis for "in vivo" structure-property-based diagnostics to evaluate the robustness of the thermochemical data for β-diketonate tris-complexes with metals other than iron.

Assessing MP2 frozen natural orbitals in relativistic correlated electronic structure calculations

The high computational scaling with the basis set size and the number of correlated electrons is a bottleneck limiting applications of coupled cluster algorithms, in particular for calculations based on two- or four-component relativistic Hamiltonians, which often employ uncontracted basis sets. This problem may be alleviated by replacing canonical Hartree-Fock virtual orbitals by natural orbitals (NOs). In this paper, we describe the implementation of a module for generating NOs for correlated wavefunctions and, in particular, second order Møller-Plesset perturbation frozen natural orbitals (MP2FNOs) as a component of our novel implementation of relativistic coupled cluster theory for massively parallel architectures [Pototschnig et al. J. Chem. Theory Comput. 17, 5509, (2021)]. Our implementation can manipulate complex or quaternion density matrices, thus allowing for the generation of both Kramers-restricted and Kramers-unrestricted MP2FNOs. Furthermore, NOs are re-expressed in the parent atomic orbital (AO) basis, allowing for generating coupled cluster singles and doubles NOs in the AO basis for further analysis. By investigating the truncation errors of MP2FNOs for both the correlation energy and molecular properties-electric field gradients at the nuclei, electric dipole and quadrupole moments for hydrogen halides HX (X = F-Ts), and parity-violating energy differences for H₂Z₂ (Z = O-Se)-we find MP2FNOs accelerate the convergence of the correlation energy in a roughly uniform manner across the Periodic Table. It is possible to obtain reliable estimates for both energies and the molecular properties considered with virtual molecular orbital spaces truncated to about half the size of the full spaces.

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

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

Extending NMR Quantum Computation Systems by Employing Compounds with Several Heavy Metals as Qubits

Nuclear magnetic resonance (NMR) is a spectroscopic method that can be applied to several areas. Currently, this technique is also being used as an experimental quantum simulator, where nuclear spins are employed as quantum bits or qubits. The present work is devoted to studying heavy metal complexes as possible candidates to act as qubit molecules. Nuclei such 113Cd, 199Hg, 125Te, and 77Se assembled with the most common employed nuclei in NMR-QIP implementations (1H, 13C, 19F, 29Si, and 31P) could potentially be used in heteronuclear systems for NMR-QIP implementations. Hence, aiming to contribute to the development of future scalable heteronuclear spin systems, we specially designed four complexes, based on the auspicious qubit systems proposed in our previous work, which will be explored by quantum chemical calculations of their NMR parameters and proposed as suitable qubit molecules. Chemical shifts and spin–spin coupling constants in four complexes were examined using the spin–orbit zeroth-order regular approximation (ZORA) at the density functional theory (DFT) level, as well as the relaxation parameters (T1 and T2). Examining the required spectral properties of NMR-QIP, all the designed complexes were found to be promising candidates for qubit molecules.

Quasi-Relativistic Calculation of EPR g Tensors with Derivatives of the Decoupling Transformation, Gauge-Including Atomic Orbitals, and Magnetic Balance

We present an exact two-component (X2C) ansatz for the EPR g tensor using gauge-including atomic orbitals (GIAOs) and a magnetically balanced basis set expansion. In contrast to previous X2C and four-component relativistic ansätze for the g tensor, this implementation results in a gauge-origin-invariant formalism. Furthermore, the derivatives of the relativistic decoupling matrix are incorporated to form the complete analytical derivative of the X2C Hamiltonian. To reduce the associated computational costs, we apply the diagonal local approximation to the unitary decoupling transformation (DLU). The quasi-relativistic X2C and DLU-X2C Hamiltonians accurately reproduce the results of the parent four-component relativistic theory when accounting for two-electron picture-change effects with the modified screened nuclear spin-orbit approximation in the respective one-electron integrals and integral derivatives. According to our benchmark studies, the uncontracted Dyall and segmented-contracted Karlsruhe x2c-type basis sets perform well when compared to large even-tempered basis sets. Moreover, (range-separated) hybrid density functional approximations such as LC-ωPBE and ωB97X-D are needed to match the experimental findings. The impact of the GIAOs depends on the distribution of the spin density, and their use may change the Δg shifts by 10-50% as shown for [(C₅Me₅)₂Y(μ-S)₂Mo(μ-S)₂Y(C₅Me₅)₂]^-. Routine calculations of large molecules are possible with widely available and comparably low-cost hardware as demonstrated for [Pt(C₆Cl₅)₄]^- with 3003 basis functions and three spin-(1/2) La(II) and Lu(II) compounds, for which we observe good agreement with the experimental findings.

Environment Effects on X-Ray Absorption Spectra With Quantum Embedded Real-Time Time-Dependent Density Functional Theory Approaches

In this work we implement the real-time time-dependent block-orthogonalized Manby-Miller embedding (rt-BOMME) approach alongside our previously developed real-time frozen density embedding time-dependent density functional theory (rt-TDDFT-in-DFT FDE) code, and investigate these methods’ performance in reproducing X-ray absorption spectra (XAS) obtained with standard rt-TDDFT simulations, for model systems comprised of solvated fluoride and chloride ions ([X@(H2O)8−, X = F, Cl). We observe that for ground-state quantities such as core orbital energies, the BOMME approach shows significantly better agreement with supermolecular results than FDE for the strongly interacting fluoride system, while for chloride the two embedding approaches show more similar results. For the excited states, we see that while FDE (constrained not to have the environment densities relaxed in the ground state) is in good agreement with the reference calculations for the region around the K and L1 edges, and is capable of reproducing the splitting of the 1s1 (n + 1)p1 final states (n + 1 being the lowest virtual p orbital of the halides), it by and large fails to properly reproduce the 1s1 (n + 2)p1 states and misses the electronic states arising from excitation to orbitals with important contributions from the solvent. The BOMME results, on the other hand, provide a faithful qualitative representation of the spectra in all energy regions considered, though its intrinsic approximation of employing a lower-accuracy exchange-correlation functional for the environment induces non-negligible shifts in peak positions for the excitations from the halide to the environment. Our results thus confirm that QM/QM embedding approaches are viable alternatives to standard real-time simulations of X-ray absorption spectra of species in complex or confined environments.

Anodic Activity of Hydrated and Anhydrous Iron (II) Oxalate in Li-Ion Batteries

We discuss the applicability of the naturally occurring compound Ferrous Oxalate Dihydrate (FOD) (FeC2O4·2H2O) as an anode material in Li-ion batteries. Using first-principles modeling, we evaluate the electrochemical activity of FOD and demonstrate how its structural water content affects the intercalation reaction and contributes to its performance. We show that both Li0 and Li+ intercalation in FOD yields similar results. Our analysis indicates that fully dehydrated ferrous oxalate is a more promising anodic material with higher electrochemical stability: it carries 20% higher theoretical Li storage capacity and a lower voltage (0.68 V at the PBE/cc-pVDZ level), compared to its hydrated (2.29 V) or partially hydrated (1.43 V) counterparts.

A new double-reference correction scheme for accurate and efficient computation of NMR chemical shieldings

Our novel correction procedure yields high-accuracy DFT predictions of absolute NMR shieldings and enables outliers due to relativistic effects or manifestly inadequate modelling of electron correlation to be easily and unambiguously identified.

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 L_2,3- and M_4,5-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 L_2,3-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.

The paper demonstrates that relativistic four-component TDDFT theory can reproduce and analyze experimental X-ray absorption spectra near L_2,3- and M_4,5-edges of transition metal and actinide compounds with different central atoms, ligands, and oxidation states. With variational inclusion of scalar and spin−orbit relativistic effects and hybrid functionals with an optimized amount of Hartee−Fock exchange (60%), it achieves relative errors below 0.2% and 2% for edge energies and spin−orbit (SO) splittings.

Hyperfine Coupling Constants in Local Exact Two-Component Theory

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

Extended Benchmark Set of Main-Group Nuclear Shielding Constants and NMR Chemical Shifts and Its Use to Evaluate Modern DFT Methods

An extended theoretical benchmark set, NS372, for light main-group nuclear shieldings and NMR shifts has been constructed based on high-level GIAO-CCSD(T)/pcSseg-3//CCSD(T)/cc-pVQZ reference data. After removal of the large static-correlation cases O₃, F₃^-, and BH from the statistical evaluations for the ¹⁷O, ¹⁹F, and ¹¹B subsets, the benchmark comprises overall 372 shielding values in 117 molecules with a wide range of electronic-structure situations, containing 124 ¹H, 14 ¹¹B, 93 ¹³C, 43 ¹⁵N, 31 ¹⁷O, 47 ¹⁹F, 14 ³¹P, and 6 ³³S shielding constants. The CCSD(T)/pcSseg-3 data are shown to be close to the basis-set and method limit and thus provide an excellent benchmark to evaluate more approximate methods, such as density functional approaches. This dataset has been used to evaluate Hartree-Fock (HF) and MP2, and a wide range of exchange-correlation functionals from local density approximation (LDA) to generalized gradient approximations (GGAs) and meta-GGAs (focusing on their current-density functional implementations), as well as global hybrid, range-separated hybrid, local hybrid, and double-hybrid functionals. Starting with absolute shielding constants, the DSD-PBEP86 double hybrid is confirmed to provide the highest accuracy, with an aggregate relative mean absolute error (rel. MAE) of only 0.9%, followed by MP2 (1.1%). MP2 and double hybrids only show larger errors for a few systems with the largest static-correlation effects. The double-hybrid B2GP-PLYP, the two local hybrids cLH12ct-SsirPW92 and cLH12ct-SsifPW92, and the current-density functional meta-GGA cB97M-V follow closely behind (all 1.5%), as do some further functionals, cLH20t and cMN15-L (both 1.6%), as well as B2PLYP and KT3 (both 2.0%). Functionals on the lower rungs of the usual ladder offer the advantage of lower computational cost and access to larger molecules. Closer examination also reveals the best-performing methods for individual nuclei in the test set. Different ways of treating τ-dependent functionals are evaluated. When moving from absolute shielding constants to chemical shifts, some of the methods can benefit from systematic error compensation, and the overall error range somewhat narrows. Further methods now achieve the 2% threshold of relative MAEs, including functionals based on TPSS (TPSSh, cmPSTS).

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-[Pt^II(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 β-cyclodextrin (β-CD) assembly achieve the same cellular uptake as cisplatin and cisplatin@β-CD, respectively, while maintaining a significantly lower toxicity profile.

Current density and molecular magnetic properties

We give an overview of the molecular response to an external magnetic field perturbing quantum mechanical systems. We present state-of-the-art methods for calculating magnetically-induced current-density susceptibilities. We discuss the essence and properties of current-density susceptibilities and how molecular magnetic properties can be calculated from them. We also review the theory of spin-current densities, how relativity affects current densities and magnetic properties. An overview of the magnetic ring-current criterion for aromaticity is given, which has implications on theoretical and experimental research. The recently reported theory of antiaromaticity and how molecular symmetry affects the magnetic response are discussed and applied to closed-shell paramagnetic molecules. The topology of magnetically induced current densities and its consequences for molecular magnetic properties are also presented with twisted and toroidal molecules as examples.

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.

Direct enantiomeric discrimination through antisymmetric hyperfine coupling

Chiral open-shell molecules possessing permanent electric dipole moments have an EPR signal at the difference frequency of the electron and nuclear resonances, allowing direct enantiomeric discrimination by signal phase. The effect depends on the vector antisymmetry of the hyperfine coupling. Quantum chemistry suggests chiral bisfluorene methyl radical derivatives as promising for experiments.

Perturbation Theory Treatment of Spin-Orbit Coupling II: A Coupled Perturbed Kohn-Sham Method

A noncanonical coupled perturbed Kohn-Sham density functional theory (KS-DFT)/Hartree-Fock (HF) treatment of spin-orbit coupling (SOC) is provided. We take the scalar-relativistic KS-DFT/HF solution, obtained with a relativistic effective core potential, as the zeroth-order approximation. Explicit expressions are given for the total energy through the 4th order, which satisfy the 2n + 1 rule. Second-order expressions are provided for orbital energies and density variables of spin-current DFT. Test calculations are carried out on the halogen homonuclear diatomic and hydride molecules, including 6p and 7p elements, as well as open-shell negative ions. The computed properties through second or third order match well with those from reference two-component self-consistent field calculations for total and orbital energies as well as spin-current densities. In only one case (At₂^-) did a significant deviation occur for the remaining density variables. Our coupled perturbation theory approach provides an efficient way of adding the effect of SOC to a scalar-relativistic single-reference KS-DFT/HF treatment, in particular because it does not require diagonalization in the two-component spinor basis, leading to saving factors on the number of required floating-point operations that may exceed one order of magnitude.

Crystal and Substituent Effects on Paramagnetic NMR Shifts in Transition-Metal Complexes

Nuclear magnetic resonance (NMR) spectroscopy of paramagnetic molecules provides detailed information about their molecular and electron-spin structure. The paramagnetic NMR spectrum is a very rich source of information about the hyperfine interaction between the atomic nuclei and the unpaired electron density. The Fermi-contact contribution to ligand hyperfine NMR shifts is particularly informative about the nature of the metal–ligand bonding and the structural arrangements of the ligands coordinated to the metal center. In this account, we provide a detailed experimental and theoretical NMR study of compounds of Cr(III) and Cu(II) coordinated with substituted acetylacetonate (acac) ligands in the solid state. For the first time, we report the experimental observation of extremely paramagnetically deshielded ¹³C NMR resonances for these compounds in the range of 900–1200 ppm. We demonstrate an excellent agreement between the experimental NMR shifts and those calculated using relativistic density-functional theory. Crystal packing is shown to significantly influence the NMR shifts in the solid state, as demonstrated by theoretical calculations of various supramolecular clusters. The resonances are assigned to individual atoms in octahedral Cr(acac)₃ and square-planar Cu(acac)₂ compounds and interpreted by different electron configurations and magnetizations at the central metal atoms resulting in different spin delocalizations and polarizations of the ligand atoms. Further, effects of substituents on the ¹³C NMR resonance of the ipso carbon atom reaching almost 700 ppm for Cr(acac)₃ compounds are interpreted based on the analysis of Fermi-contact hyperfine contributions.

The ligand NMR shifts in paramagnetic acetylacetonato Cr(III) and Cu(II) complexes have been predicted and measured in the solid state and interpreted by relativistic DFT calculations. The effects of the metal atom, ligand, and crystal packing on the spin delocalization and polarization reflected in the Fermi-contact contribution to the hyperfine interaction are rationalized.

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

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

Electrochemical Potential of the Metal Organic Framework MIL-101(Fe) as Cathode Material in Li-Ion Batteries

We discuss the characteristic factors that determine the electrochemical potentials in a metal-organic framework used as cathode for Li-ion batteries via density functional theory-based simulations. Our focus is on MIL-101(Fe) cathode material. Our study gives insight into the role of local atomic environment and structural deformations in generating electrochemical potential.