As a spectroscopic method, nuclear magnetic resonance (NMR) has seen spectacular growth, both as a technique and in its applications. Today’s applications of NMR span a wide range of scientific disciplines, from physics to biology to medicine. Each volume of Nuclear Magnetic Resonance comprises a combination of annual and biennial reports which together provide comprehensive coverage of the literature on this topic. This Specialist Periodical Report reflects the growing volume of published work involving NMR techniques and applications, in particular NMR of natural macromolecules, which is covered in two reports: NMR of Proteins and Nucleic Acids; and NMR of Carbohydrates, Lipids and Membranes. For those wanting to become rapidly acquainted with specific areas of NMR, Nuclear Magnetic resonance provides unrivalled scope of coverage. Seasoned practitioners of NMR will find this an invaluable source of current methods and applications.

Nuclear Magnetic Resonance Volume 37

A Review of the Literature Published between June 2006 and May 2007

By G.A. Webb

The Royal Society of Chemistry

Copyright © 2008 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-0-85404-115-2

Contents

Preface G. A. Webb, 7,
NMR books and reviews W. Schilf, 21,
Theoretical and physical aspects of nuclear shielding Cynthia J. Jameson and Angel C. de Dio, 51,
Applications of nuclear shielding Shigeki Kuroki, Tsunenori Kameda and Hidekazu Yasunaga, 68,
Theoretical aspects of spin–spin couplings Hiroyuki Fukui, 24,
Applications of spin–spin coupling Krystyna Kamienska-Trela and Jacek Wójcik, 145,
Nuclear spin relaxation in liquids and gases R. Ludwig, 180,
Solid-state NMR spectroscopy A. E. Aliev and R. V. Law, 208,
NMR of proteins and nucleic acids P. J. Simpson, 257,
NMR of carbohydrates, lipids and membranes Elizabeth Hounsell, 274,
Synthetic macromolecules Hiromichi Kurosu and Takeshi Yamanobe, 293,
NMR in living systems M. J. W. Prior, 327,
Oriented molecules K. V. Ramanathan, Uday R. Prabhu and C. L. Khetrapal, 357,

CHAPTER 1

Theoretical and physical aspects of nuclear shielding

Cynthia J. Jameson and Angel C. de Dios

DOI: 10.1039/b617218k

1 Theoretical aspects of nuclear shielding

1.1 General theory

The symmetry at the nuclear site determines the number of distinct components of the shielding tensor. The symmetry of the nuclear site is related to the symmetry of the whole molecule if the nucleus is located at one of the molecular symmetry axis, center, or plane. Therefore, in many situations, the molecular symmetry has an important bearing on the anisotropy of the shielding tensor. Useful correlations between specific geometrical features of molecules and the shielding tensor are well known. Avnir suggests that a quantitative relation with molecular symmetry can be demonstrated by studying how continuous deviation from exact symmetry around a nucleus affects its shielding. He employs the continuous symmetry measures methodology, which allows one to quantify the degree of content of a given symmetry, a methodology he has applied to many properties such as hyperpolariz-ability. The model case he uses for this purpose is a population of distorted SiH4 structures, for which he follows the Si shielding anisotropy as a function of the degree of tetrahedral symmetry and of square-planar symmetry. Quantitative correlations between the degree of these symmetries and the NMR shielding parameters emerge. This approach may be very useful for cases where large excursions away from the minimum energy geometry can take place, as when part of the shielding surface is explored in conjunction with broad and shallow potential wells.

Due to large contributions from electrons moving at relativistic velocities near the nuclei, the effects of relativity have to be taken into account in theoretical calculations of NMR observables, in particular the nuclear magnetic shielding tensor. The relativistic corrections can be significant even for shielding of light nuclei in molecules containing heavy atoms. The relativistic heavy atom effects on the shielding tensor of the heavy atom itself (HAHA effects) are particularly large. Fully relativistic four-component Dirac-Hartree-Fock methods can be applied to the calculations of nuclear shielding; however, four-component methods have not yet been extended to the correlated level for this purpose. Thus, for highly accurate studies of systems containing moderately heavy atoms, four-component theory is not yet competitive to methods based on a non-relativistic reference wave function. Transformed two-component Hamiltonians in which the positronic degrees of freedom have been eliminated from the Dirac equation but spin orbit coupling is still included variationally offer the next lower level of theory. At this level, self-consistent variationally stable two-component approaches such as the Douglas-Kroll-Hess to second order (DKH2), and the zeroth order regular approximation (ZORA) have been used. The ZORA has been implemented for both Hartree-Fock and density functional approximations, whereas DKH2 has been implemented only for Hartree-Fock wave functions, until recently. It has been shown that for molecules containing elements from the first five rows of the periodic table, relativistic corrections to the shielding tensor can be successfully described using perturbation theory. In this approach, the relativistic corrections to the diamagnetic and paramagnetic contributions of the non-relativistic theory can partly be obtained by perturbing the non-relativistic reference wave function by the various leading-order relativistic operators. These relativistic corrections to the shielding tensor can in turn be calculated as higher-order response functions. In addition, new interaction mechanisms may also appear from higher order interactions in the Breit-Pauli Hamiltonian. The advantages of this approach are that both the electron correlation and the gauge-origin problems can be handled via existing techniques that have been developed for non-relativistic calculations. That is, existing methods for calculating response properties with correlated wavefunctions can be used and GIAOs can be used. The formulations and applications of this perturbation approach” have been reviewed in previous volumes of this series. More recently, linear response elimination of small component method has been established which has been shown to give the same final theoretical expressions.

In the present review period, particular attention has been paid to the heavy atom effects on the shielding of the heavy atom itself, using the Breit-Pauli perturbation treatment of relativistic effects described above. Lantto et al. considered small molecular systems of the type XH2, XH3 and XF3, in addition to some charged systems such as XH3-, and monatomic ions, all closed-shell systems, for × = group-14 and group-15 from Si to Pb, P to Bi, while Jaszunski and Ruud considered XH4 molecules, where XQC to Pb. The dependence of the HAHA effects on the chemical environment of the heavy atom, the relative magnitudes of the various relativistic correction terms and electron correlations on shielding, and on chemical shifts, are compared. Most of the discussion is based on augmented basis sets. Both Hartree-Fock self consistent field (SCF) and complete active space (CAS) multi-configurational self consistent field (MCSCF) levels of theory were used, except for the fluorides and the closed shell monatomic ions for which only SCF levels were used. Fully relativistic four-component Dirac-Hartree-Fock shielding tensor calculations were also carried out for comparison. Gauge-including atomic orbitals (GIAO) were used, except that Lantto et al. used common gauge origin (CGO) and coordinate origin at the heavy nucleus for the relativistic corrections. No difference had been found between using GIAOs or CGOs for the non-relativistic shieldings at these sizes of basis sets. The important conclusions from these two works are as follows: (a) The relativistic corrections to the heavy atom nuclear shieldings are large and scale as very nearly Z3 (3.04, as reported by ref. 26). This has been reported previously in other works. Thus, for the heaviest nuclei the relativistic effects have to be taken into account to obtain a reliable value of the absolute total shielding. (b) In comparison to the four-component calculations, the complete Breit-Pauli perturbational approach underestimates the relativistic correction for the heaviest nuclei (6th row). (c) All the relativistic correction terms provide significant contributions to the absolute shielding of the heavy atom. (d) The dominant contribution to the HAHA effect on shielding is the FC/SZ-KE contribution, the second order cross term between the Fermi contact (FC) hyperfine Hamiltonian and the relativistically modified electronic spin-Zeeman (SZ-KE) Hamiltonian. This contribution is almost completely produced in the s orbitals of the heavy atom, the values diminishing with the principal quantum number. (e) Whereas the non-relativistic shielding terms are sensitive to both electron correlation and the chemical surroundings of the heavy atom, the dependence of the FC/SZ-KE relativistic terms on these factors is much smaller. This reflects the predominantly core nature of these relativistic terms and implies small contributions of the FC/SZ-KE relativistic corrections of the HAHA type to chemical shifts between different molecules. (f) For the same reasons, the FC/SZ-KE relativistic corrections of the HAHA type to the anisotropy of the chemical shift tensor are small. (g) However, the spin-orbit effects have several terms and of these, the SO-I terms are sensitive to both electron correlation and chemical surroundings, thus require correlated wavefunctions and have significant contributions to HA chemical shifts between different molecules. (h) The SO contributions become more and more important for the anisotropy of the chemical shift tensor of the heavier nuclei. For heavier X, there is an increasing positive relativistic contribution to the shielding anisotropy, e.g., (σ[parallel] -σ[perpendicular to]) in axial HA environments.

The great majority of shielding calculations reported in the literature these days use gauge-including atomic orbitals (GIAO). Alternative approaches for gauge invariant calculations have also been investigated. Of these, the continuous transformation of origin of the current density (CTOCD) approach, based on the continuous set of gauge transformations suggested by Keith and Bader, has been extensively explored by Lazzeretti and co-workers. In particular, they have done calculations in the scheme where the diamagnetic contribution of the current density is set to zero (CTOCD-DZ). In this approach, the point where the current density is calculated is also the origin of the reference system. For this reason, this approach is also called the ipsocentric method of plotting current density maps., The method is intrinsically independent of the origin of the coordinate system used in the calculations, irrespective of other approximations retained, that is, at any basis set level. In a recent investigation, Lazzeretti and co-workers extend the CTOCD-DZ calculations to density functional theory (DFT) using a variety of functionals, and coupled-cluster -singles-and-doubles linear response theory. When the aug-cc-pCVTZ-CTOCD-uc basis sets are used, the fulfillment of the hypervirial relations is almost as good at the correlated level as at the SCF level. That is, the agreement between CTOCD-DZ DFT and common origin or GIAO DFT results is good. They found that the KT3 functional of Keal and Tozer performs on average much better than the popular B3LYP functional. In particular, for the difficult shieldings of multiply-bonded molecules where electron correlation effects are very large, KT3 gives results in close agreement with CCSD calculations.

Since precise experimental values of NMR chemical shifts are available for a wide variety of chemical environments, they have become a popular test for density functional theory research. Cohen et al. examined two new methods proposed by Yang and Wu which are based on the Kohn-Sham potential by calculating shieldings in small highly correlated molecules containing main group atoms. The first is a method which reproduces an accurate input density (WY) and the second is an implementation of the optimized effective potential (OEP) method. They found that these methods give results which are very similar to each other, and when the methods are applied to a hybrid functional (e.g.B3LYP) they obtain good agreement with experiment. Recently, Tozer et al. compared various functionals using the Yang-Wu implementation of the optimized effective potential approach in density functional theory in calculations of fourth row transition metal chemical shifts. They examined nine transition metal complexes using several GGA functionals and found that expanding the potential in the primary orbital basis leads to reasonably good accuracy, providing a physically appropriate reference potential is used. Further improvement requires the use of a much more extensive potential expansion which is appropriately balanced between the atom types. They also considered hybrid functionals. When the orbital basis is used for the potential expansion, the OEP B3LYP calculations produce significant improvements over conventional B3LYP; mean absolute and rms errors are reduced by a factor of 2. Similar improvements are obtained using the PBE0 functional. In line with the findings of Cohen et al. for chemical shifts in molecules containing main group atoms, Tozer et al. find that the use of the uncoupled OEP procedure for calculating transition metal chemical shifts using hybrid functionals do give improvement over conventional calculations using hybrid functionals, but are computationally more complicated.

Ramsey considered the possibility of field-dependent nuclear magnetic shielding, that is, deviations from linear dependence of the resonance frequencies on the external magnetic field when the field strengths are significantly higher than those in conventional high-field spectrometers. There have been attempts to calculate the magnitudes of the field dependent terms in the nuclear magnetic shielding for the Co nucleus, for which Bendall and Doddrell had reported some measurements. Manninen and Vaara presented an analytical response theory formulation for the dependence of the nuclear magnetic resonance shielding tensor of molecules on the external magnetic field. First-principles calculations for 59Co in Co(acetylacetonate)3 and Co(NH3)63+ were carried out using the Hartree-Fock self-consistent field method as well as density-functional theory. They reported magnetic-field dependence of the 59Co shielding constant of -6 × 10-3 ppm/T2 in Co(acetylacetonate), which is at the limit of being experimentally observable. Lazzeretti and co-workers developed a general computational scheme for a fourth-rank magnetic hypershielding tensor at a nucleus in a molecule in the presence of an external spatially uniform time-independent magnetic field. They calculated the values at the SCF level using a common origin for H2, HF, H2O, NH3, and CH4. For these molecules, all the values are very small, about 20–30 times as small as was calculated for 59Co: -0.3 × 10-3 ppm/T2 for C in CH4, and -0.2 × 10-3 ppm/T2 for for O in H2O.

Parity non-conservation (PNC) contributions to the nuclear magnetic shielding of chiral molecules are small and the treatment of electron correlation as well as basis set size strongly influence the results for H2X2 (X = 17O, 33S, 77Se). A systematic four- component relativistic study of the constants of chiral molecules has been presented for the P enantiomers of the series H2X2 (X = 17O, 33S, 77Se, 125Te, 209Po). The PNC contributions are obtained within a linear response approach at the Hartree-Fock level. The magnitude of the parity nonconservation (PNC) contribution to the (isotropic) NMR shielding is found to scale with the charge of the nucleus as Z2.4 in non-relativistic theory, in agreement with previous results. The calculations show that the overall scaling is significantly modified by relativistic effects. The scalar relativistic effect scales as Z4.7 for the selected set of molecules, whereas the spin-orbit effect, of opposite sign, scales better than Z6 and completely dominates the PNC contribution for the heaviest elements. This opens up the intriguing possibility of the experimental observation of PNC effects on NMR parameters of molecules containing heavy atoms.

1.2 Ab initio and DFT calculations

Calculations of shielding tensors for heavy nuclei in metal complexes in solution are fraught with difficulties. First among these is the problem of taking into account the true solution environment with specific as well as non-specific solvent effects which not only provide absolute and differential contributions to the shielding (which do not subtract out in taking chemical shifts) but also determine the geometry of the (usually charged) complex in solution. The latter is very important since shielding is extremely sensitive to bond distances to the nucleus in question. Continuum methods of including solvent effects, such as COSMO (conductor-like screening model), are not adequate. Second, there is the problem of choosing a reasonable reference substance. There is usually no gas phase sample against which all other complexes could be referenced. Typically one highly symmetric charged complex among the series is picked. Because of the first problem, the choice of reference compound could skew the comparisons with experiment. On the positive side, as basis sets are being developed and extended, the choice of atomic base sets poses more of a computational time vs. size compromise than a real problem. Electron correlation has been found to be important. But these complexes are usually not small enough to permit the use of the CCSD(T) method, so DFT is often used. The question still remains as to whether using hybrid functionals are intrinsically better for transition metal shielding or not. For the lighter transition metal nuclei, there is the question of whether relativistic effects contribute sufficiently to be necessarily included when the other above-mentioned problems still persist. Unfortunately, when various functionals are tested against each other while the first and second problems have not been taken care of, comparisons against experimental chemical shifts do not provide a test of functionals at all. Geometry, solvation, and reference problems inevitably confuse the interpretation of the results. The situation is very different from the light main group atoms, where there are comparisons against experimental gas phase data, where the geometries are well determined, and there are the CCSD(T) calculations for a set of benchmark molecules against which other computational methods can be assessed. For 183W a possible gas phase reference is WF6, but so far no one has tried to determine the W chemical shifts against this reference. Relativistic DFT calculations of 183W shielding in polyoxotungstates of different shapes and charges have been reported by Bagno et al. The authors adopted the BP86 functional (Becke 88 exchange plus the Perdew 86 correlation),and chose to report chemical shifts with reference to [W6O19]2- which is 58.6 ppm relative to the usual experimental reference, [WO4]2-. They used a continuum model (COSMO) to approximate the solvent effects in geometry optimization and NMR calculations, and they included relativistic effects by using zero-order regular approximation (ZORA) including spin–orbit terms. The linear correlation of the calculated chemical shifts against the experimental chemical shifts was used as an indication of the quality of the results. They found that this depended systematically on the charge density as expressed by the charge to surface area of the complex. Complexes with low ratios of charge to surface area display the best agreement with experiments. At the highest level approach adopted as described above (although they also tested many other levels), the correlation line relative to their chosen reference, [W6O19]2-, has slope 0.905 and an intercept which is -7 ppm over a 500 ppm range of chemical shifts, and the mean average error is 35 ppm. In the case of α-[PW11TiO40]5-, the six signals are ranked computationally so as to almost reproduce the experimental ordering even though the signals are spaced by as little as 5 ppm. Of course, Bagno et al. have only explored a small section of the 8000 ppm range of W chemical shifts. The remaining unsolved problems include the effect of counter ions which were completely neglected here and the true solvent effects, in addition to the truly dynamic structure in solution which is not reflected by having done the calculations in a single geometry fixed at that found by geometry optimization in a continuum description of the medium.

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