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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.

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.

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.

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.

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.

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 [RuCl 2 (DMSO) 2 (Im) 2 ] and [WCl 4 (PMePh 2 ) 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.

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 (NH 3 ) 2 (H 2 O)(NH 2 -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.

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.

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.

1 H 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 1 H 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 1 H NMR hydride chemical shifts of iridium polyhydride complexes. In particular, we have studied in detail the hydride chemical shifts of the [Ir 6 (IMe) 8 (CO) 2 H 14 ] 2+ complex in order to validate previous assignments. The computed 1 H 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.

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.

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.

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.

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, C 4 H 8 X (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.

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 (Sc 3+ -Ac 3+ ) with nondegenerate ground states, as well as for Group 11 atomic systems (Cu-Rg) and octahedral actinide complexes (PaCl 6 2- , UCl 6 - , and NpF 6 ) with effective doublet ground states.

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.

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, C 4 H 8 X (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.

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 β-diketones. A series of complexes with systematically varied substituents was synthesized and analyzed using 1 H and 13 C 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.

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 1 H 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 5d 2 -5d 8 and 6p 4 HA hydrides and deshielded in 4f 0 , 5d 0 , 6s 0 , and 6p 0 HA hydrides. This general and intuitive concept explains periodic trends in the 1 H 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.

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 [ReF 6 ] 2- and [IrF 6 ] 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 [ReF 6 ] 2- by using hybrid functionals. However, none of the methods were able to reproduce the large anisotropy of the hyperfine coupling tensor of complex [ReF 6 ] 2- . For [IrF 6 ] 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., (© 2018 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.)

Relativistic effects significantly affect various spectroscopic properties of compounds containing heavy elements. Particularly in Nuclear Magnetic Resonance (NMR) spectroscopy, the heavy atoms strongly influence the NMR shielding constants of neighboring light atoms. In this account we analyze paramagnetic contributions to NMR shielding constants and their modulation by relativistic spin-orbit effects in a series of transition-metal complexes of Pt(II), Au(I), Au(III), and Hg(II). We show how the paramagnetic NMR shielding and spin-orbit effects relate to the character of the metal-ligand (M-L) bond. A correlation between the (back)-donation character of the M-L bond in d 10 Au(I) complexes and the propagation of the spin-orbit (SO) effects from M to L through the M-L bond influencing the ligand NMR shielding via the Fermi-contact mechanism is found and rationalized by using third-order perturbation theory. The SO effects on the ligand NMR shielding are demonstrated to be driven by both the electronic structure of M and the nature of the trans ligand, sharing the σ-bonding metal orbital with the NMR spectator atom L. The deshielding paramagnetic contribution is linked to the σ-type M-L bonding orbitals, which are notably affected by the trans ligand. The SO deshielding role of σ-type orbitals is enhanced in d 10 Hg(II) complexes with the Hg 6p atomic orbital involved in the M-L bonding. In contrast, in d 8 Pt(II) complexes, occupied π-type orbitals play a dominant role in the SO-altered magnetic couplings due to the accessibility of vacant antibonding σ-type MOs in formally open 5d-shell (d 8 ). This results in a significant SO shielding at the light atom. The energy- and composition-modulation of σ- vs π-type orbitals by spin-orbit coupling is rationalized and supported by visualizing the SO-induced changes in the electron density around the metal and light atoms (spin-orbit electron deformation density, SO-EDD).

We report the Nuclear Magnetic Resonance (NMR) spin-spin coupling constants for diatomic alkali halides MX, where M = Li, Na, K, Rb, or Cs and X = F, Cl, Br, or I. The coupling constants are determined by supplementing the non-relativistic coupled-cluster singles-and-doubles (CCSD) values with relativistic corrections evaluated at the four-component density-functional theory (DFT) level. These corrections are calculated as the differences between relativistic and non-relativistic values determined using the PBE0 functional with 50% exact-exchange admixture. The total coupling constants obtained in this approach are in much better agreement with experiment than the standard relativistic DFT values with 25% exact-exchange, and are also noticeably better than the relativistic PBE0 results obtained with 50% exact-exchange. Further improvement is achieved by adding rovibrational corrections, estimated using literature data.

The Liouville-von Neumann equation based on the four-component matrix Dirac-Kohn-Sham Hamiltonian is transformed to a quasirelativistic exact two-component (X2C) form and then used to solve the time evolution of the electronic states only. By this means, a significant acceleration by a factor of 7 or more has been achieved. The transformation of the original four-component equation of motion is formulated entirely in matrix algebra, following closely the X2C decoupling procedure of Ilias and Saue [ J. Chem. Phys. 2007 , 126 , 064102 ] proposed earlier for a static (time-independent) case. In a dynamic (time-dependent) regime, however, an adiabatic approximation must in addition be introduced in order to preserve the block-diagonal form of the time-dependent Dirac-Fock operator during the time evolution. The resulting X2C Liouville-von Neumann electron dynamics (X2C-LvNED) is easy to implement as it does not require an explicit form of the picture-change transformed operators responsible for the (higher-order) relativistic corrections and/or interactions with external fields. To illustrate the accuracy and performance of the method, numerical results and computational timings for nonlinear optical properties are presented. All of the time domain X2C-LvNED results show excellent agreement with the reference four-component calculations as well as with the results obtained from frequency domain response theory.

We present a formulation of Laplace-transformed atomic orbital-based second-order Møller-Plesset perturbation theory (MP2) energies for two-component Hamiltonians in the Kramers-restricted formalism. This low-order scaling technique can be used to enable correlated relativistic calculations for large molecular systems. We show that the working equations to compute the relativistic MP2 energy differ by merely a change of algebra (quaternion instead of real) from their non-relativistic counterparts. With a proof-of-principle implementation we study the effect of the nuclear charge on the magnitude of half-transformed integrals and show that for light elements spin-free and spin-orbit MP2 energies are almost identical. Furthermore, we investigate the effect of separation of charge distributions on the Coulomb and exchange energy contributions, which show the same long-range decay with the inter-electronic/atomic distance as for non-relativistic MP2. A linearly scaling implementation is possible if the proper distance behavior is introduced to the quaternion Schwarz-type estimates as for non-relativistic MP2.

We have calculated the (13)C NMR chemical shifts of a large ensemble of halogenated organic molecules (81 molecules for a total of 250 experimental (13)C NMR data at four different levels of theory), ranging from small rigid organic compounds, used to benchmark the performance of various levels of theory, to natural substances of marine origin with conformational degrees of freedom. Carbon atoms bonded to heavy halogen atoms, particularly bromine and iodine, are known to be rather challenging when it comes to the prediction of their chemical shifts by quantum methods, due to relativistic effects. In this paper, we have applied the state-of-the-art four-component relativistic density functional theory for the prediction of such NMR properties and compared the performance with two-component and nonrelativistic methods. Our results highlight the necessity to include relativistic corrections within a four-component description for the most accurate prediction of the NMR properties of halogenated organic substances., (© 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

The PSiP pincer-supported complex ((Cy)PSiP)PdH [(Cy)PSiP = Si(Me)(2-PCy2-C6H4)2] has been implicated as a crucial intermediate in carboxylation of both allenes and boranes. At this stage, however, there is uncertainty regarding the exact structure of ((Cy)PSiP)PdH, especially in solution. Previously, both a Pd(II) structure with a terminal Pd hydride and a Pd(0) structure featuring an η(2)-silane have been proposed. In this contribution, a range of techniques were used to establish that ((Cy)PSiP)PdH and the related Pt species, ((Cy)PSiP)PtH, are true M(II) hydrides in both the solid state and solution. The single-crystal X-ray structures of ((Cy)PSiP)MH (M = Pd and Pt) and the related species ((iPr)PSiP)PdH [(iPr)PSiP = Si(Me)(2-P(i)Pr2-C6H4)2] are in agreement with the presence of a terminal metal hydride, and the exact geometry of ((Cy)PSiP)PtH was confirmed using neutron diffraction. The (1)H and (29)Si{(1)H}NMR chemical shifts of ((Cy)PSiP)MH (M = Pd and Pt) are consistent with a structure containing a terminal hydride, especially when compared to the chemical shifts of related pincer-supported complexes. In fact, in this work, two general trends relating to the (1)H NMR chemical shifts of group 10 pincer-supported terminal hydrides were elucidated: (i) the hydride shift moves downfield from Ni to Pd to Pt and (ii) the hydride shift moves downfield with more trans-influencing pincer central donors. DFT calculations indicate that structures containing a M(II) hydride are lower in energy than the corresponding η(2)-silane isomers. Furthermore, the calculated NMR chemical shifts of the M(II) hydrides using a relativistic four-component methodology incorporating all significant scalar and spin-orbit corrections are consistent with those observed experimentally. Finally, in situ X-ray absorption spectroscopy (XAS) was used to provide further support that ((Cy)PSiP)MH exist as M(II) hydrides in solution.

We present nuclear spin-rotation constants, absolute nuclear magnetic resonance (NMR) shielding constants, and shielding spans of all the nuclei in (175)LuX and (197)AuX (X = (19)F, (35)Cl, (79)Br, (127)I), calculated using coupled-cluster singles-and-doubles with a perturbative triples (CCSD(T)) correction theory, four-component relativistic density functional theory (relativistic DFT), and non-relativistic DFT. The total nuclear spin-rotation constants determined by adding the relativistic corrections obtained from DFT calculations to the CCSD(T) values are in general in agreement with available experimental data, indicating that the computational approach followed in this study allows us to predict reliable results for the unknown spin-rotation constants in these molecules. The total NMR absolute shielding constants are determined for all the nuclei following the same approach as that applied for the nuclear spin-rotation constants. In most of the molecules, relativistic effects significantly change the computed shielding constants, demonstrating that straightforward application of the non-relativistic formula relating the electronic contribution to the nuclear spin-rotation constants and the paramagnetic contribution to the shielding constants does not yield correct results. We also analyze the origin of the unusually large absolute shielding constant and its relativistic correction of gold in AuF compared to the other gold monohalides.

The role of various factors (structure, solvent, and relativistic treatment) was evaluated for square-planar 4d and 5d transition-metal complexes. The DFT method for calculating the structures was calibrated using a cluster approach and compared to X-ray geometries, with the PBE0 functional (def2-TZVPP basis set) providing the best results, followed closely by the hybrid TPSSH and the MN12SX functionals. Calculations of the NMR chemical shifts using the two-component (2c, Zeroth-Order Regular Approximation as implemented in the ADF package) and four-component (4c, Dirac-Coulomb as implemented in the ReSpect code) relativistic approaches were performed to analyze and demonstrate the importance of solvent corrections (2c) as well as a proper treatment of relativistic effects (4c). The importance of increased exact-exchange admixture in the functional (here PBE0) for reproducing the experimental data using the current implementation of the 2c approach is partly rationalized as a compensation for the missing exchange-correlation response kernel. The kernel contribution was identified to be about 15-20% of the spin-orbit-induced NMR chemical shift, ΔδSO, which roughly corresponds to an increase in ΔδSO introduced by the artificially increased exact-exchange admixture in the functional. Finally, the role of individual effects (geometry, solvent, relativity) in the NMR chemical shift is discussed in selected complexes. Although a fully relativistic DFT approach is still awaiting the implementation of GIAOs for hybrid functionals and an implicit solvent model, it nevertheless provides reliable NMR chemical shift data at an affordable computational cost. It is expected to outperform the 2c approach, in particular for the calculation of NMR parameters in heavy-element compounds.

The solution of the Liouville-von Neumann equation in the relativistic Dirac-Kohn-Sham density matrix formalism is presented and used to calculate X-ray absorption cross sections. Both dynamical relaxation effects and spin-orbit corrections are included, as demonstrated by calculations of the X-ray absorption of SF6 near the sulfur L2,3-edges. We also propose an analysis facilitating the interpretation of spectral transitions from real-time simulations, and a selective perturbation that eliminates nonphysical excitations that are artifacts of the finite basis representation.

We present an implementation of the nuclear spin-rotation (SR) constants based on the relativistic four-component Dirac-Coulomb Hamiltonian. This formalism has been implemented in the framework of the Hartree-Fock and Kohn-Sham theory, allowing assessment of both pure and hybrid exchange-correlation functionals. In the density-functional theory (DFT) implementation of the response equations, a noncollinear generalized gradient approximation (GGA) has been used. The present approach enforces a restricted kinetic balance condition for the small-component basis at the integral level, leading to very efficient calculations of the property. We apply the methodology to study relativistic effects on the spin-rotation constants by performing calculations on XHn (n = 1-4) for all elements X in the p-block of the periodic table and comparing the effects of relativity on the nuclear SR tensors to that observed for the nuclear magnetic shielding tensors. Correlation effects as described by the density-functional theory are shown to be significant for the spin-rotation constants, whereas the differences between the use of GGA and hybrid density functionals are much smaller. Our calculated relativistic spin-rotation constants at the DFT level of theory are only in fair agreement with available experimental data. It is shown that the scaling of the relativistic effects for the spin-rotation constants (varying between Z(3.8) and Z(4.5)) is as strong as for the chemical shieldings but with a much smaller prefactor.

In this work, we report on the development and implementation of a new scheme for efficient calculation of indirect nuclear spin-spin couplings in the framework of four-component matrix Dirac-Kohn-Sham approach termed matrix Dirac-Kohn-Sham restricted magnetic balance resolution of identity for J and K, which takes advantage of the previous restricted magnetic balance formalism and the density fitting approach for the rapid evaluation of density functional theory exchange-correlation response kernels. The new approach is aimed to speedup the bottleneck in the solution of the coupled perturbed equations: evaluation of the matrix elements of the kernel of the exchange-correlation potential. The performance of the new scheme has been tested on a representative set of indirect nuclear spin-spin couplings. The obtained results have been compared with the corresponding results of the reference method with traditional evaluation of the exchange-correlation kernel, i.e., without employing the fitted electron densities. Overall good agreement between both methods was observed, though the new approach tends to give values by about 4%-5% higher than the reference method. On the average, the solution of the coupled perturbed equations with the new scheme is about 8.5 times faster compared to the reference method.

We report the first implementation of real-time time-dependent density functional theory (RT-TDDFT) at the relativistic four-component level of theory. In contrast to the perturbative linear-response TDDFT approach (LR-TDDFT), the RT-TDDFT approach performs an explicit time propagation of the Dirac-Kohn-Sham density matrix, offering the possibility to simulate molecular spectroscopies involving strong electromagnetic fields while, at the same time, treating relativistic scalar and spin-orbit corrections variationally. The implementation is based on the matrix representation of the Dirac-Coulomb Hamiltonian in the basis of restricted kinetically balanced Gaussian-type functions, exploiting the noncollinear Kramers unrestricted formalism implemented in the program ReSpect. We also present an analytic form for the delta-type impulse commonly used in RT-TDDFT calculations, as well as a dipole-weighted transition matrix analysis, facilitating the interpretation of spectral transitions in terms of ground-state molecular orbitals. The possibilities offered by the methodology are illustrated by investigating vertical excitation energies and oscillator strengths for ground-state to excited-state transitions in the Group 12 atoms and in heavy-element hydrides. The accuracy of the method is assessed by comparing the excitation energies obtained with earlier relativistic linear response TDDFT results and available experimental data.

We present an updated semi-experimental absolute shielding scale for the (17)O and (33)S nuclei. These new shielding scales are based on accurate rotational microwave data for the spin-rotation constants of H2(17)O [Puzzarini et al., J. Chem. Phys. 131, 234304 (2009)], C(17)O [Cazzoli et al., Phys. Chem. Chem. Phys. 4, 3575 (2002)], and H2(33)S [Helgaker et al., J. Chem. Phys. 139, 244308 (2013)] corrected both for vibrational and temperature effects estimated at the CCSD(T) level of theory as well as for the relativistic corrections to the relation between the spin-rotation constant and the absolute shielding constant. Our best estimate for the oxygen shielding constants of H2(17)O is 328.4(3) ppm and for C(17)O -59.05(59) ppm. The relativistic correction for the sulfur shielding of H2(33)S amounts to 3.3%, and the new sulfur shielding constant for this molecule is 742.9(4.6) ppm.

Relativistic density functional theory (DFT) has been applied to explore electron paramagnetic resonance (EPR) parameters as well as ground-state spin properties of cob(II)alamin. Cob(II)alamin is an intermediate which participates in many reactions catalyzed by derivatives of vitamin B12 and that can be detected by EPR spectroscopy due to the presence of the paramagnetic Co(II)(d(7)) center. The full structure of cob(II)alamin and its truncated analogues were examined. Three different DFT functionals, B3LYP, BP86, and PBE, have been applied to obtain the g- and A-tensors. Both tensors are axially symmetric and can provide useful insight into specific axial ligand interactions. Of the functionals tested, the hybrid B3LYP functional, was found to overestimate the axial bond length, whereas the GGA-type functionals, BP86 and PBE, produced geometries consistent with experimental data. The reliability of nonrelativistic and approximate relativistic methods for the calculation of EPR parameters has also been tested against a fully relativistic four-component approach. Since the EPR parameters are very sensitive to the local environment surrounding Co(II), a theoretical (DFT-BP86) estimate of the dependence of the g- and A-tensors on the metal-to-axial ligand interatomic distance can be directly correlated with EPR measurements. The usefulness of such an approach has been demonstrated for the methionine synthase enzyme where the reduction of cob(II)alamin takes place during the reactivation cycle.

Aromatic species with heavier Group 14 elements show remarkable differences in terms of stability, structure, and reactivity. Herein we report our experimental and theoretical investigations regarding isomers of germanium- and tin-containing benzene analogues E2Si4R6 (E=Ge, Sn). The germanium-substituted dismutational isomer with a tricyclic six-membered scaffold is isolable, but unlike the homonuclear Si6 analogue slowly rearranges even at room temperature to give the propellane-type global minimum isomer. In case of E=Sn the dismutational isomer may be an intermediate on the pathway to the propellane-type species obtained, but cannot be detected even at low temperature. Unprecedentedly large chemical shift anisotropies in the (29)Si NMR spectra that increase from the Si6 species through Ge2Si4 to Sn2Si4 are rationalized by progressively larger paramagnetic-term contributions to the chemical shift tensor as a result of diminishing HOMO-LUMO gaps, which are also reflected in the absorption spectra, as well as by appearance and symmetry of these frontier orbitals., (© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

A four-component relativistic method for the calculation of NMR shielding constants of paramagnetic doublet systems has been developed and implemented in the ReSpect program package. The method uses a Kramer unrestricted noncollinear formulation of density functional theory (DFT), providing the best DFT framework for property calculations of open-shell species. The evaluation of paramagnetic nuclear magnetic resonance (pNMR) tensors reduces to the calculation of electronic g tensors, hyperfine coupling tensors, and NMR shielding tensors. For all properties, modern four-component formulations were adopted. The use of both restricted kinetically and magnetically balanced basis sets along with gauge-including atomic orbitals ensures rapid basis-set convergence. These approaches are exact in the framework of the Dirac-Coulomb Hamiltonian, thus providing useful reference data for more approximate methods. Benchmark calculations on Ru(III) complexes demonstrate good performance of the method in reproducing experimental data and also its applicability to chemically relevant medium-sized systems. Decomposition of the temperature-dependent part of the pNMR tensor into the traditional contact and pseudocontact terms is proposed.

The spin-rotation and nuclear magnetic shielding constants are analysed for both nuclei in the HCl molecule. Nonrelativistic ab initio calculations at the CCSD(T) level of approximation show that it is essential to include relativistic effects to obtain spin-rotation constants consistent with accurate experimental data. Our best estimates for the spin-rotation constants of (1)H(35)Cl are CCl = -53.914 kHz and C(H) = 42.672 kHz (for the lowest rovibrational level). For the chlorine shielding constant, the ab initio value computed including the relativistic corrections, σ(Cl) = 976.202 ppm, provides a new absolute shielding scale; for hydrogen we find σ(H) = 31.403 ppm (both at 300 K). Combining the theoretical results with our new gas-phase NMR experimental data allows us to improve the accuracy of the magnetic dipole moments of both chlorine isotopes. For the hydrogen shielding constant, including relativistic effects yields better agreement between experimental and computed values.

We demonstrate that the apparent disagreement between experimental determinations and four-component relativistic calculations of the absolute shielding constants of heavy nuclei is due to the breakdown of the commonly assumed relation between the electronic contribution to the nuclear spin-rotation constants and the paramagnetic contribution to the NMR shielding constants. We demonstrate that this breakdown has significant consequences for the absolute shielding constant of (119)Sn, leading to errors of about 1000 ppm. As a consequence, we expect that many absolute shielding constants of heavy nuclei will be in need of revision.

A study of polymorphism using a range of solid-state NMR techniques is presented. We demonstrate the existence of at least six polymorphs in a sample of N-benzoyl-L-phenylalanine. We also present methodology for the characterization of the protonation state, hydrogen bonding, and molecular conformation for the polymorphs, together with results of such a characterization for one of the polymorphs present in our sample. DFT modeling is used to investigate the separate effects hydrogen bonding and molecular conformation have on the chemical shift tensor.