Relativistic spin-orbit effects on hyperfine coupling tensors by density-functional theory

Density functional restricted-unrestricted approach for nonlinear properties: application to electron paramagnetic resonance parameters of square planar copper complexes

The density functional restricted-unrestricted approach for treatments of spin polarization effects in molecular properties using spin restricted Kohn-Sham theory has been extended from linear to nonlinear properties. It is shown that the spin polarization contribution to a nonlinear property has the form of a quadratic response function that includes the zero-order Kohn-Sham operator, in analogy to the lower order case where the spin polarization correction to an expectation value has the form of a linear response function. The developed approach is used to formulate new schemes for computation of electronic g-tensors and hyperfine coupling constants, which include spin polarization effects within the framework of spin restricted Kohn-Sham theory. The proposed computational schemes are in the present work employed to study the spin polarization effects on electron paramagnetic resonance spin Hamiltonian parameters of square planar copper complexes. The obtained results indicate that spin polarization gives rise to sizable contributions to the hyperfine coupling tensor of copper in all investigated complexes, while the electronic g-tensors of these complexes are only marginally affected by spin polarization and other factors, such as choice of exchange-correlation functional or molecular structures, will have more pronounced impact on the accuracy of the results.

Dynamics and magnetic resonance properties of Sc3C2@C80 and its monoanion

We report density functional theory (DFT) studies on the endohedral scandium carbide fullerene Sc3C2@C80 and its monoanion [Sc3C2@C80](-). The system consisting of a Sc3C2 moiety inside the Ih C80 fullerene has been studied by using first principles molecular dynamics simulations at the DFT level. On the picosecond time scale, the triangle defined by the scandium atoms is seen to jump between orientations along the equatorial six-membered ring belt of the cage. The confined carbide unit, in turn, is engaged in a flipping motion through the Sc3 plane. In contrast to the equilibrium geometry optimisations using large basis sets that predict a trigonal bipyramidal structure, a planar Sc3C2-moiety is preferred during the finite-temperature simulation. In the molecular dynamics picture, Sc3C2@C80 is best described as an equilibrium between the two static minimum structures. Calculations of the vibrational frequencies show that the earlier predicted C2 and C2v symmetric isomers are in fact saddle points, with one imaginary normal mode frequency that is related to the flipping motion of the confined carbon dimer. Reoptimisation revealed two new minimum energy structures where the C2 unit is tilted with respect to its orientation in the earlier suggested higher-symmetry structures. The nature of the bonding in the static structures of the two isomers of Sc3C2@C80 has been investigated using the electron localisation function and natural population analysis. Some increased electron pair localisation is detected on the six-membered rings closest to the Sc atoms. 13C nuclear magnetic resonance (NMR) chemical shifts have been calculated for the closed-shell monoanion of Sc3C2@C80. The 13C shifts were also calculated for Sc2C2@C84, for further comparison to experimentally measured spectra. The confined carbon atoms are strongly deshielded in these metallofullerenes, implying an incorrect earlier interpretation of the experimental 13C NMR spectrum of Sc2C2@C84. The neutral Sc3C2@C80 system with one unpaired electron is further characterised by calculating the hyperfine coupling constants, the g tensor, as well as paramagnetic NMR (pNMR) 13C shifts for both static isomers. The chemical shifts of the confined carbon atoms and the hyperfine coupling constants of all the confined atoms are strongly dependent on the conformation of the Sc3C2 moiety. Consequently, dynamical effects are expected to be important in the modelling of the magnetic properties of endohedral scandium carbide fullerenes. The two low-lying isomers have rather different pNMR 13C shifts, implying the potential of this method in structural assignment.

Synthesis, reactivity, and electronic structure of [n]vanadoarenophanes: an experimental and theoretical study

An optimized procedure for the selective dimetalation of [V(eta (6)-C 6H 6) 2] by BuLi/tmeda allowed for the isolation and characterization of [V(eta (6)-C 6H 5Li) 2].tmeda. X-ray diffraction of its thf solvate [V(eta (6)-C 6H 5Li) 2].(thf) 7 revealed an unsymmetrical, dimeric composition in the solid state, in which both subunits are connected by three bridging lithium atoms. Treatment with several element dihalides facilitated the isolation of [ n]vanadoarenophanes ( n = 1, 2) with boron and silicon in the bridging positions. In agreement with the number and covalent radii of the bridging elements, these derivatives exhibit molecular ring strain to a greater or lesser extent. The B-B bond of the [2]bora species [V(eta (6)-C 6H 5) 2B 2(NMe 2) 2] was readily cleaved by [Pt(PEt 3) 3] to afford the corresponding oxidative addition product. Subsequently, [V(eta (6)-C 6H 5) 2B 2(NMe 2) 2] was employed as a diborane(4) precursor in the diboration of 2-butyne under stoichiometric, homogeneous, and heterogeneous catalysis conditions. This transformation is facilitated by the reduction of molecular ring strain, which was confirmed by a decrease of the tilt angle alpha observed in the corresponding solid-state structures. EPR spectroscopy was used to probe the electronic structure of strained [ n]vanadoarenophanes and revealed an obvious correlation between the degree of molecular distortion and the observed hyperfine coupling constant a iso. State-of-the-art DFT calculations were able to reproduce the measured isotropic vanadium hyperfine couplings and the coupling anisotropies. The calculations confirmed the decrease of the absolute isotropic hyperfine couplings with increasing tilt angle. Closer analysis showed that this is mainly due to increased positive contributions to the spin density at the vanadium nucleus from the spin polarization of doubly occupied valence orbitals of vanadium-ligand sigma-antibonding character. The latter are destabilized and thus made more polarizable in the bent structures.

Nuclear magnetic resonance chemical shift in an arbitrary electronic spin state

We present a general and systematic electronic structure theory of the nuclear magnetic resonance shielding tensor and the associated chemical shift for paramagnetic atoms, molecules, and nonmetallic solids. The approach is for the first time rigorous for an arbitrary spin state as well as arbitrary spatial symmetry and is formulated without reference to spin susceptibility. The leading-order magnetic-field dependence of shielding is derived. The theory is demonstrated by first principles calculations of organometallic molecules.

Computational studies of EPR parameters for paramagnetic molybdenum complexes. II. Larger MoV systems relevant to molybdenum enzymes

The careful validation of modern density functional methods for the computation of electron paramagnetic resonance (EPR) parameters in molybdenum complexes has been extended to a number of low-symmetry MoV systems that model molybdoenzyme active sites. Both g and hyperfine tensors tend to be reproduced best by hybrid density functionals with about 30-40% exact-exchange admixture, with no particular spin contamination problems encountered. Spin-orbit corrections to hyperfine tensors are mandatory for quantitative and, in some cases, even for qualitative agreement. The g11 (g||) component of the g tensor tends to come out too positive when spin-orbit coupling is included only to leading order in perturbation theory. Compared to single-crystal experiments, the calculations reproduce both g- and hyperfine-tensor orientations well, both relative to each other and to the molecular framework. This is significant, as simulations of the EPR spectra of natural-abundance frozen-solution samples frequently do not allow a reliable determination of the hyperfine tensors. These may now be extracted based on the quantum-chemically calculated parameters. In a number of cases, revised simulations of the experimental spectra have brought theory and experiment into substantially improved agreement. Systems with two terminal oxo ligands, and to some extent with an oxo and a sulfido ligand, have been confirmed to exhibit particularly large negative Deltag33 shifts and thus large g anisotropies. This is discussed in the context of the experimental data for xanthine oxidase.

First-principles calculations of zero-field splitting parameters

In this work, an implementation of an approach to calculate the zero-field splitting (ZFS) constants in the framework of ab initio methods such as complete active space self-consistent field, multireference configuration interaction, or spectroscopy oriented configuration interaction is reported. The spin-orbit coupling (SOC) contribution to ZFSs is computed using an accurate multicenter mean-field approximation for the Breit-Pauli Hamiltonian. The SOC parts of ZFS constants are obtained directly after diagonalization of the SOC operator in the basis of a preselected number of roots of the spin-free Hamiltonian. This corresponds to an infinite order treatment of the SOC in terms of perturbation theory. The spin-spin (SS) part is presently estimated in a mean-field fashion and appears to yield results close to the more complete treatments available in the literature. Test calculations for the first- and second-row atoms as well as first-row transition metal atoms and a set of diatomic molecules show accurate results for the SOC part of ZFSs. SS contributions have been found to be relatively small but not negligible (exceeding 1 cm(-1) for oxygen molecule). At least for the systems studied in this work, it is demonstrated that the presented method provides much more accurate estimations for the SOC part of ZFS constants than the emerging density functional theory approaches.

Density functional study of EPR parameters and spin-density distribution of azurin and other blue copper proteins

Modern density functional methods have been used to study spin-density distribution, g tensors, as well as Cu and ligand hyperfine tensors for azurin models, for two more blue copper proteins plastocyanin and stellacyanin, and for small model complexes. The aim was to establish a consistent computational protocol that provides a realistic description of the EPR parameters as probes of the spin-density distribution between metal and coordinated ligands in copper proteins. In agreement with earlier conclusions for plastocyanin, hybrid functionals with appreciable exact-exchange admixtures, roughly around 50%, provide the best overall agreement with all parameters. Then the bulk of the spin density is almost equally shared by the copper atom and the sulfur atom of the equatorial cysteine ligand, and the best values are obtained for copper, histidine nitrogen, and cysteine beta-proton hyperfine couplings, as well as for g(parallel). Spin-orbit effects on the EPR parameters may be appreciable and have to be treated carefully to obtain agreement with experiment. Most notably, spin-orbit effects on the (65)Cu hyperfine coupling tensors in blue copper sites are unusually large compared to more regularly coordinated Cu(II) complexes with similar spin density on copper. In addition to the often emphasized high covalency of the Cu-S(Cys) bond, the characteristically small A(parallel) component of blue copper proteins is shown to derive to a large part from a near-cancellation between negative first-order (Fermi contact and dipolar) and unusually large positive second-order (spin-orbital) contributions. The large spin-orbit effects relate to the distorted tetrahedral structures. Square planar dithiolene complexes with similar spin density on copper exhibit much more negative A(parallel) values, as the cancellation between nonrelativistic and spin-orbit contributions is less complete. Calculations on a selenocysteine-substituted variant of azurin have provided further insight into the relations between bonding and EPR parameters.

Computational Studies of Electron Paramagnetic Resonance Parameters for Paramagnetic Molybdenum Complexes. 1. Method Validation on Small and Medium-Sized Systems

A variety of density functional methods have been evaluated in the computation of electronic g-tensors and molybdenum hyperfine couplings for systems ranging from the Mo atom through MoIIIN, [MoVOCl4]-, and [MoVOF5]2- to two larger MoV complexes MoXLCl2 (X=O, S; L=tris(3,5-dimethylpyrazolyl)hydroborate anion). In particular, the influence of the molybdenum basis set and of various exchange-correlation functionals with variable admixtures of Hartree-Fock exchange on the computed EPR parameters have been evaluated in detail. Careful basis-set studies have provided a moderate-sized 12s6p5d all-electron basis on molybdenum that gives hyperfine tensors in excellent agreement with much larger basis sets and that will be useful for calculations on larger systems. The best agreement with experimental data for both hyperfine and g-tensors is obtained with hybrid functionals containing approximately 30-40% Hartree-Fock exchange. Only for MoSLCl2 does increasing spin contamination with increasing exact-exchange admixture restrict the achievable computational accuracy. In all cases, spin-orbit corrections to the hyperfine tensors are sizable and have to be included in accurate calculations. Scalar relativistic effects enhance the isotropic Mo hyperfine coupling by approximately 15-20%. Two-component g-tensor calculations with variational inclusion of spin-orbit coupling show that the Deltag parallel components in [MoVOCl4]- and [MoVOF5]2- depend on higher-order spin-orbit contributions and are thus described insufficiently by the usual second-order perturbation approaches. Computed orientations of g- and hyperfine tensors relative to each other and to the molecular framework for the MoXLCl2 complexes provide good agreement between theory and single-crystal electron paramagnetic resonance experiments. In these cases, the hyperfine tensor orientations are influenced only slightly by spin-orbit effects.

Density functional calculations of NMR shielding tensors for paramagnetic systems with arbitrary spin multiplicity: validation on 3d metallocenes

The calculation of nuclear shieldings for paramagnetic molecules has been implemented in the ReSpect program, which allows the use of modern density functional methods with accurate treatments of spin-orbit effects for all relevant terms up to order Omicron(alpha4) in the fine structure constant. Compared to previous implementations, the methodology has been extended to compounds of arbitrary spin multiplicity. Effects of zero-field splittings in high-spin systems are approximately accounted for. Validation of the new implementation is carried out for the 13C and 1H NMR signal shifts of the 3d metallocenes 4VCp2, 3CrCp2, 2MnCp2, 6MnCp2, 2CoCp2, and 3NiCp2. Zero-field splitting effects on isotropic shifts tend to be small or negligible. Agreement with experimental isotropic shifts is already good with the BP86 gradient-corrected functional and is further improved by admixture of Hartree-Fock exchange in hybrid functionals. Decomposition of the shieldings confirms the dominant importance of the Fermi-contact shifts, but contributions from spin-orbit dependent terms are frequently also non-negligible. Agreement with 13C NMR shift tensors from solid-state experiments is of similar quality as for isotropic shifts.

Density Functional Study of Electron Paramagnetic Resonance Parameters and Spin Density Distributions of Dicopper(I) Complexes with Bridging Azo and Tetrazine Radical-Anion Ligands

There exists a growing class of dinuclear complexes with bridging radical-anion ligands that is of interest both for bioinorganic and for supermolecular chemistry. Their bonding situation as well as chemical and spectroscopic properties are not described adequately by standard models such as the ligand-field theory. For rational design of complexes with desired properties, it is thus necessary to understand better the interrelations between electronic structure, spin density, and electron paramagnetic resonance (EPR) parameters in dinuclear systems with redox-active bridging ligands and to evaluate the performance of density functional methods in their description. As particularly suitable, experimentally well-characterized representatives, a series of dinuclear copper(I) complexes with azo or tetrazine bridge ligands have been studied here by different density functional methods. To reproduce the available experimental metal hyperfine couplings, the inclusion of spin-orbit effects into the calculations is necessary. An unusual direction of the dependence of computed hyperfine couplings on an exact-exchange admixture into the exchange-correlation functional may be understood from a McConnell-type spin polarization of the sigma-framework of the bridge. Ligand nitrogen hyperfine couplings are also compared with experiment where available. Electronic g-tensors are reproduced well by the calculations and have been analyzed in detail in terms of atomic spin-orbit contributions and electronic excitations.

Density-functional calculations of relativistic spin-orbit effects on nuclear magnetic shielding in paramagnetic molecules

Terms arising from the relativistic spin-orbit effect on both hyperfine and Zeeman interactions are introduced to density-functional theory calculation of nuclear magnetic shielding in paramagnetic molecules. The theory is a generalization of the former nonrelativistic formulation for doublet systems and is consistent to O(alpha4), the fourth power of the fine structure constant, for the spin-orbit terms. The new temperature-dependent terms arise from the deviation of the electronic g tensor from the free-electron g value as well as spin-orbit corrections to hyperfine coupling tensor A, the latter introduced in the present work. In particular, the new contributions include a redefined isotropic pseudocontact contribution that consists of effects due to both the g tensor and spin-orbit corrections to hyperfine coupling. The implementation of the spin-orbit terms makes use of all-electron atomic mean-field operators and/or spin-orbit pseudopotentials. Sample results are given for group-9 metallocenes and a nitroxide radical. The new O(alpha4) corrections are found significant for the metallocene systems while they obtain small values for the nitroxide radical. For the isotropic shifts, none of the three beyond-leading-order hyperfine contributions are negligible.

Comparative Density-Functional Study of the Electron Paramagnetic Resonance Parameters of Amavadin

Electronic g tensors and hyperfine coupling tensors have been calculated for amavadin, an unusual eight-coordinate vanadium(IV) complex isolated from Amanita muscaria mushrooms. Different density-functional methods have been compared, ranging from local via gradient-corrected to hybrid functionals with a variable Hartree-Fock exchange admixture. For both electron paramagnetic resonance (EPR) properties, hybrid functionals with an appreciable exact-exchange admixture provide the closest agreement with experimental data. Second-order spin-orbit corrections provide non-negligible contributions to the 51V hyperfine tensor. The orientation of g and A tensors relative to each other also depends on spin-orbit corrections to the A tensor. A rationalization for the close resemblance of the EPR parameters of amavadin to those of the structurally rather different vanadyl complexes is provided, based on the nature of the relevant frontier orbitals.

Performance of Nonrelativistic and Quasi-Relativistic Hybrid DFT for the Prediction of Electric and Magnetic Hyperfine Parameters in 57Fe Mössbauer Spectra

57Fe electric and magnetic hyperfine parameters were calculated for a series of 10 iron model complexes, covering a wide range of oxidation and spin states. Employing the B3LYP hybrid method, results from nonrelativistic density functional theory (DFT) and quasi-relativistic DFT within the zero-order regular approximation (ZORA) were compared. Electron densities at the iron nuclei were calculated and correlated with experimental isomer shifts. It was shown that the fit parameters do not depend on a specific training set of iron complexes and are, therefore, more universal than might be expected. The nonrelativistic and quasi-relativistic electron densities gave fit parameters of similar quality; the ZORA densities are only shifted by a factor of 1.32, upward in the direction of the four-component Dirac-Fock value. From a correlation of calculated electric field gradients and experimental quadrupole splittings, the value of the 57Fe nuclear quadrupole moment was redetermined to a value of 0.16 barn, in good agreement with other studies. The ZORA approach gave no additional improvement of the calculated quadrupole splittings in comparison to the nonrelativistic approach. The comparison of the calculated and measured 57Fe isotropic hyperfine coupling constants (hfcc's) revealed that both the ZORA approach and the inclusion of spin-orbit contributions lead to better agreement between theory and experiment in comparison to the nonrelativistic results. For all iron complexes with small spin-orbit contributions (high-spin ferric and ferryl systems), a distinct underestimation of the isotropic hfcc's was found. Scaling factors of 1.81 (nonrelativistic DFT) and 1.69 (ZORA) are suggested. The calculated 57Fe isotropic hfcc's of the remaining model systems (low-spin ferric and high-spin ferrous systems) contain 10-50% second-order contributions and were found to be in reasonable agreement with the experimental results. This is assumed to be the consequence of error cancellation because g-tensor calculations for these systems are of poor quality with the existing DFT approaches. Excellent agreement between theory and experiment was found for the 57Fe anisotropic hfcc's. Finally, all of the obtained fit parameters were used for an application study of the [Fe(H2O)6]3+ ion. The calculated spectroscopic data are in good agreement with the Mossbauer and electron paramagnetic resonance results discussed in detail in a forthcoming paper.

Efficient and accurate approximations to the molecular spin-orbit coupling operator and their use in molecular g-tensor calculations

Approximations to the Breit-Pauli form of the spin-orbit coupling (SOC) operator are examined. The focus is on approximations that lead to an effective quasi-one-electron operator which leads to efficient property evaluations. In particular, the accurate spin-orbit mean-field (SOMF) method developed by Hess, Marian, Wahlgren, and Gropen is examined in detail. It is compared in detail with the "effective potential" spin-orbit operator commonly used in density functional theory (DFT) and which has been criticized for not including the spin-other orbit (SOO) contribution. Both operators contain identical one-electron and Coulomb terms since the SOO contribution to the Coulomb term vanishes exactly in the SOMF treatment. Since the DFT correlation functional only contributes negligibly to the SOC the only difference between the two operators is in the exchange part. In the SOMF approximation, the SOO part is equal to two times the spin-same orbit contribution. The DFT exchange contribution is of the wrong sign and numerically shown to be in error by a factor of 2-2.5 in magnitude. The simplest possible improvement in the DFT-SOC treatment [Veff(-2X)-SOC] is to multiply the exchange contribution to the Veff operator by -2. This is verified numerically in calculations of molecular g-tensors and one-electron SOC constants of atoms and ions. Four different ways of handling the computationally critical Coulomb part of the SOMF and Veff operators are discussed and implemented. The resolution of the identity approximation is virtually exact for the SOC with standard auxiliary basis sets which need to be slightly augmented by steep s functions for heavier elements. An almost as efficient seminumerical approximation is equally accurate. The effective nuclear charge model gives results within approximately 10% (on average) of the SOMF treatment. The one-center approximation to the Coulomb and one-electron SOC terms leads to errors on the order of approximately 5%. Small absolute errors are obtained for the one-center approximation to the exchange term which is consequently the method of choice [SOMF(1X)] for large molecules.

Carboxylate Binding in Copper Histidine Complexes in Solution and in Zeolite Y: X- and W-band Pulsed EPR/ENDOR Combined with DFT Calculations

The complexes of copper with histidine exhibit a wide variety of coordination modes in aqueous solution. This stems from the three potential coordination sites of the histidine molecule and the existence of mono- and bis-complexes. The present work concentrates on the determination of the carboxylate binding mode, via the (13)C hyperfine coupling of the carboxyl, in a number of copper complexes in frozen solutions. These are then used as references for the determination of the coordination mode of two zeolite encapsulated complexes. The (13)C hyperfine coupling (sign and magnitude) was determined by a variety of advanced pulsed EPR and electron-nuclear double resonance (ENDOR) techniques carried out at conventional and high magnetic fields. These showed that while the carboxyl (13)C isotropic hyperfine coupling of an equatorially coordinated carboxylate is negative with a magnitude of 3-4 MHz, that of a free carboxylate is small ( approximately 1 MHz) and positive. To rationalize the experimentally determined ligand hyperfine couplings ((1)H and (13)C) and further understand their dependence on the coordination mode and degree of protonation, density functional theory (DFT) calculations were carried out on a number of model complexes, representing the different Cu-histidine complexes studied experimentally. The exchange-correlation functional used for the calculation of the EPR parameters was B3LYP with triple-zeta plus polarization (TZP) quality basis sets. While the polarization agreement between the magnitudes of the calculated and experimental values varied among the various nuclei, sometimes exhibiting deviations of up to 40%, an excellent agreement was found for the sign prediction. This shows the unique advantage of combining high field ENDOR, by which the sign of the hyperfine can often be determined, with DFT predictions for structure determination.

Density functional theory for hyperfine coupling constants with the restricted-unrestricted approach

This work presents derivation, implementation, and the first applications of the restricted-unrestricted approach based on restricted Kohn-Sham formalism for evaluation of hyperfine coupling constants. By using the spin-restricted Kohn-Sham method the well-known spin contamination problem existing in the unrestricted Kohn-Sham formalism is avoided and a proper description of spin polarization is achieved via the restricted-unrestricted approach without introducing spin contamination into the evaluation of the hyperfine coupling constants. The performance of the proposed formalism is evaluated for a set of organic radicals and transition metal compounds. The results of this investigation indicate promising accuracy of the restricted-unrestricted approach for calculation of the isotropic hyperfine coupling constants in organic radicals as well as transition metal compounds.