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.     

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.     

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.     

Second-order polarization propagator calculations of indirect nuclear spin-spin coupling tensors in the water molecule

The contribution to indirect nuclear spin-spin coupling tensors provided by the Fermi contact, the spin-dipolar, the Fermi contact/spin-dipolar crossterm, and the paramagnetic spin-orbit interactions are investigated in a zeroth-, first- (the same as the coupled Hartree-Fock method), and second-order polarization propagator approach. Numerical applications to the water molecule show that the second-order results for both the HO and the HH coupling constants are in good agreement with experimental data - especially if vibrational corrections and the diamagnetic spin-orbit contributions are taken into account. We find that the correlation corrections beyond coupled Hartree-Fock are important. We also report how the second-order results are influenced by neglect of some of the most time-consuming steps in the calculation.     

Density functional theory study of indirect nuclear spin-spin coupling constants with spin-orbit corrections

This work outlines the calculation of indirect nuclear spin-spin coupling constants with spin-orbit corrections using density functional response theory. The nonrelativistic indirect nuclear spin-spin couplings are evaluated using the linear response method, whereas the relativistic spin-orbit corrections are computed using quadratic response theory. The formalism is applied to the homologous systems H2X (X=O,S,Se,Te) and XH4 (X=C,Si,Ge,Sn,Pb) to calculate the indirect nuclear spin-spin coupling constants between the protons. The results confirm that spin-orbit corrections are important for compounds of the H2X series, for which the electronic structure allows for an efficient coupling between the nuclei mediated by the spin-orbit interaction, whereas in the case of the XH4 series the opposite situation is encountered and the spin-orbit corrections are negligible for all compounds of this series. In addition we analyze the performance of the density functional theory in the calculations of nonrelativistic indirect nuclear spin-spin coupling constants.     

High-resolution electron spin resonance spectroscopy of XeF* in solid argon. The hyperfine structure constants as a probe of relativistic effects in the chemical bonding properties of a heavy noble gas atom

Xenon fluoride radicals were generated by solid-state chemical reactions of mobile fluorine atoms with xenon atoms trapped in Ar matrix. Highly resolved electron spin resonance spectra of XeF* were obtained in the temperature range of 5-25 K and the anisotropic hyperfine parameters were determined for magnetic nuclei 19F, 129Xe, and 131Xe using naturally occurring and isotopically enriched xenon. Signs of parallel and perpendicular hyperfine components were established from analysis of temperature changes in the spectra and from numerical solutions of the spin Hamiltonian for two nonequivalent magnetic nuclei. Thus, the complete set of components of hyperfine- and g-factor tensors of XeF* were obtained: 19F (Aiso=435, Adip=1249 MHz) and 129Xe (Aiso=-1340, Adip=-485 MHz); g(parallel)=1.9822 and g(perpendicular)=2.0570. Comparison of the measured hyperfine parameters with those predicted by density-functional theory (DFT) calculations indicates, that relativistic DFT gives true electron spin distribution in the 2Sigma+ ground-state, whereas nonrelativistic theory underestimates dramatically the electron-nuclear contact Fermi interaction (Aiso) on the Xe atom. Analysis of the obtained magnetic-dipole interaction constants (Adip) shows that fluorine 2p and xenon 5p atomic orbitals make a major contribution to the spin density distribution in XeF*. Both relativistic and nonrelativistic calculations give close magnetic-dipole interaction constants, which are in agreement with the measured values. The other relativistic feature is considerable anisotropy of g-tensor, which results from spin-orbit interaction. The orbital contribution appears due to mixing of the ionic 2Pi states with the 2Sigma+ ground state, and the spin-orbit interaction plays a significant role in the chemical bonding of XeF*.     

Scalar relativistic calculations of hyperfine coupling tensors using the Douglas-Kroll-Hess method with a finite-size nucleus model

A scalar relativistic method to calculate hyperfine coupling tensors at the Douglas-Kroll-Hess level has been extended to incorporate a finite-size nucleus model using a Gaussian charge and magnetic moment distribution. Density functional calculations at gradient-corrected and hybrid functional levels have been carried out for the group 11 atoms and for a set of small group 12 molecules, comparing nonrelativistic as well as scalar relativistic results at second-order Douglas-Kroll-Hess level with and without finite-size nucleus. While nonrelativistic calculations underestimate isotropic hyperfine couplings increasingly with increasing nuclear charge, scalar relativistic calculations with point nucleus provide somewhat overestimated values. Inclusion of the finite-size nuclear model in the calculation of the wavefunction, and in the transformed hyperfine operators both decrease the magnitude of the hyperfine couplings. The effects, which are cumulative, improve agreement with experiment.     

Electric field effects on the shielding constants of noble gases: a four-component relativistic Hartree-Fock study

Second derivatives of nuclear shielding constants with respect to an electric field, i.e., shielding polarizabilities, have been calculated for the noble gas atoms from helium to xenon. The calculations have been carried out using the four-component relativistic Hartree-Fock method. In order to assess the importance of the individual relativistic corrections, the shielding polarizabilities have also been calculated at the nonrelativistic Hartree-Fock level, with spin-orbit and scalar (Darwin and mass-velocity) effects having been established by perturbative methods. Electron correlation effects have been estimated using the second-order polarization propagator approach. The relativistic effects on the tensor components of the shielding polarizabilities are found to be larger and changing less regularly with the atomic number than for the shielding constant itself. However, there is a partial cancellation of the contributions to the parallel and perpendicular components of the shielding polarizability and as a consequence the mean shielding polarizability is far less affected than the individual components.     

Simplification of complex EPR spectra by cepstral analysis

As the Fourier transform of time-series data is known as the spectrum, the Fourier transform of the logarithm of the time-series data is called the cepstrum of the data. When cepstral analysis is applied to free induction decay signals of free radicals showing first-order EPR spectra, the identification of nuclear hyperfine coupling constants becomes simple. In a systematic manner, we have examined how the technique of cepstral analysis is affected by the presence of aliasing, noise, uncertainty in the time origin of the free induction decay, the presence of second-order hyperfine couplings, and the applications of various apodization methods. This technique was then applied to analyze the EPR spectrum of anthraquinone anion radical, and anion radicals of porphycene and tetrapropyl-porphycene, and the hyperfine coupling constants thus obtained were compared with published data. A good agreement was always found. We make a case for the usefulness of cepstral analysis in determining the hyperfine coupling constants of complex EPR spectra of organic free radicals.     

Investigations of the EPR g factors and hyperfine structure constants for two trigonal Co2+ centers in Al2O3 crystals

The EPR g factors gparallel, gperpendicular and the hyperfine structure constants Aparallel, Aperpendicular for two trigonal Co2+ centers (i.e. Co2+(I) center at the substitutional site and Co2+(II) center at the interstitial site) in Al2O3 crystals are calculated from the second-order perturbation formulas based on the cluster approach. In these formulas, the contributions to EPR parameters from both the spin-orbit coupling parameter of central 3dn ion and that of ligand are included. The calculated results are in reasonable agreement with the observed values. Based on the calculations, the defect structures of both Co2+ centers in Al2O3 crystals are obtained and the negative sign of Aparallel for Co2+(I) center is suggested. The results are discussed.     

Calculations of the indirect nuclear spin–spin coupling constants of PbH4

We report ab initio calculations of the indirect nuclear spin–spin coupling constants of PbH₄ using a basis set which was specially optimized for correlated calculations of spin–spin coupling constants. All nonrelativistic contributions and the most important part of the spin–orbit correction were evaluated at the level of the random phase approximation. Electron correlation corrections to the coupling constants were calculated with the multiconfigurational linear-response method using extended complete and restricted active space wavefunctions as well as with the second-order polarization propagator approximation and the second-order polarization propagator approximation with coupled-cluster singles and doubles amplitudes. The effects of nuclear motion were investigated by calculating the coupling constants as a function of the totally symmetric stretching coordinate. We find that the Fermi contact term dominates the Pb‐H coupling, whereas for the H‐H coupling it is not more important than the orbital paramagnetic and diamagnetic contributions. Correlation affects mainly the Fermi contact term. Its contribution to the one-bond coupling constant is reduced by correlation, independent of the method used; however, the different correlated methods give ambiguous results for the Fermi contact contribution to the H‐H couplings. The dependence of both coupling constants on the Pb‐H bond length is dominated by the change in the Fermi contact term. The geometry dependence is, however, overestimated in the random phase approximation. Received: 16 November 1998 / Accepted: 30 March 1999 / Published online: 14 July 1999     

A comparison of two-component and four-component approaches for calculations of spin-spin coupling constants and NMR shielding constants of transition metal cyanides

Relativistic density functional theory (DFT) calculations of nuclear spin-spin coupling constants and shielding constants have been performed for selected transition metal (11th and 12th group of periodic table) and thallium cyanides. The calculations have been carried out using zeroth-order regular approximation (ZORA) Hamiltonian and four-component Dirac-Kohn-Sham (DKS) theory with different nonrelativistic exchange-correlation functionals. Two recent approaches for representing the magnetic balance (MB) between the large and small components of four-component spinors, namely, mDKS-RMB and sMB, have been employed for shielding tensor calculations and their results have been compared. Relativistic effects have also been analysed in terms of scalar and spin-orbit contributions at the two-component level of theory, including discussion of heavy-atom-on-light-atom effects for (1)J(CN), σ(C), and σ(N). The results for molecules containing metals from 4th row of periodic table show that relativistic effects for them are small (especially for spin-spin coupling constants). The biggest effects are observed for the 6th row where nonrelativistic theory reproduces only about 50%-70% of the two-component ZORA results for (1)J(MeC) and about 75% for heavy metal shielding constants. It is important to employ a full Dirac picture for calculations of heavy metal shielding constants, since ZORA reproduces only 75%-90% of the DKS results. Smaller discrepancies between ZORA-DFT and DKS are observed for nuclear spin-spin coupling constants. No significant differences are observed between the results obtained using mDKS-RMB and sMB approaches for magnetic balance in four-component calculations of the shielding constants.     

AB initio MO LCAO UHF calculations of magnetic hyperfine interactions in π-radicals. Isotropic and anisotropic couplings of ·CH3, ·NH3+, ·C2H5 and ·N2H4+

A series of π-electron radicals have been investigated with ab initio methods, using the unrestricted Hartree-Fock approximation. The geometries have been partially optimized, and magnetic hyperfine coupling constants have been evaluated. For the dipolar hyperfine couplings, the results obtained are in fair agreement with experiment, whereas the isotropic coupling constants show greater deviations. These values are also shown to be sensitive to annihilation of the quartet contamination of the wavefunction.     

High-order electron-correlation methods with scalar relativistic and spin-orbit corrections

An assortment of computer-generated, parallel-executable programs of ab initio electron-correlation methods has been fitted with the ability to use relativistic reference wave functions. This has been done on the basis of scalar relativistic and spin-orbit effective potentials and by allowing the computer-generated programs to handle complex-valued, spinless orbitals determined by these potentials. The electron-correlation methods that benefit from this extension are high-order coupled-cluster methods (up to quadruple excitation operators) for closed- and open-shell species, coupled-cluster methods for excited and ionized states (up to quadruples), second-order perturbation corrections to coupled-cluster methods (up to triples), high-order perturbation corrections to configuration-interaction singles, and active-space (multireference) coupled-cluster methods for the ground, excited, and ionized states (up to active-space quadruples). A subset of these methods is used jointly such that the dynamical correlation energies and scalar relativistic effects are computed by a lower-order electron-correlation method with more extensive basis sets and all-electron relativistic treatment, whereas the nondynamical correlation energies and spin-orbit effects are treated by a higher-order electron-correlation method with smaller basis sets and relativistic effective potentials. The authors demonstrate the utility and efficiency of this composite scheme in chemical simulation wherein the consideration of spin-orbit effects is essential: ionization energies of rare gases, spectroscopic constants of protonated rare gases, and photoelectron spectra of hydrogen halides.     

Toward spectroscopic accuracy for open-shell systems: molecular structure and hyperfine coupling constants of H2CN, H2CP, NH2, and PH2 as test cases

In the present paper, we investigate the molecular structure and hyperfine couplings of a series of σ radicals containing first- and second-row atoms (H(2)CN,H(2)CP,NH(2),PH(2)) for which accurate gas-phase microwave results are available. The presence of α- and, especially, β-hydrogen atoms makes the evaluation of magnetic properties of these radicals particularly challenging. Geometrical parameters have been computed by the coupled-cluster ansatz in conjunction with hierarchical series of basis sets, thus accounting for extrapolation to the complete basis-set limit. Core correlation as well as higher excitations in the electronic-correlation treatment have also been taken into account. An analogous approach has been employed for evaluating hyperfine coupling constants with particular emphasis given to basis-set, correlation, and geometrical effects. The corresponding vibrational corrections, required for a meaningful comparison to experimental data, have also been investigated. The remarkable agreement with experiment confirms the reliability of the present computational approach, already validated for π radicals, thus establishing the way for setting up a benchmark database for magnetic properties.     

Renner-Teller and spin-orbit vibronic coupling effects in linear triatomic molecules with a half-filled pi shell

The vibronic and spin-orbit-induced interactions among the (3)Sigma(-), (1)Delta, and (1)Sigma(+) electronic states arising from a half-filled pi orbital of a linear triatomic molecule are considered, employing the microscopic (Breit-Pauli) spin-orbit coupling operator. The 6 x 6 Hamiltonian matrix is derived in a diabatic spin-orbital electronic basis set, including terms up to fourth order in the expansion of the molecular Hamiltonian in the bending normal coordinate about the linear geometry. The symmetry properties of the Hamiltonian are analyzed. Aside from the nonrelativistic fourth-order Renner-Teller vibronic coupling within the (1)Delta state and the second-order nonrelativistic vibronic coupling between the (1)Sigma(+) and (1)Delta states, there exist zeroth-order, first-order, as well as third-order vibronic coupling terms of spin-orbit origin. The latter are absent when the phenomenological expression for the spin-orbit coupling operator is used instead of the microscopic form. The effects of the nonrelativistic and spin-orbit-induced vibronic coupling mechanisms on the (3)Sigma(-), (1)Delta, and (1)Sigma(+) adiabatic potential energy surfaces as well as on the spin-vibronic energy levels are discussed for selected parameter values.     

AB initio calculation of the spin dipole-dipole interaction in ground-state triplets

Ab initio results are presented for the spin dipole-dipole interaction in the ground-state linear triplet radicals NCN, CNN and CCO. The relative contributions from the first-order spin dipole-dipole interaction and the second-order spin-orbit interaction, to the observed spin coupling constant in these systems, are discussed.