Nuclear Magnetic Resonance Volume 35

A Review of the Literature Published between June 2004 and May 2005

By G.A. Webb

The Royal Society of Chemistry

Copyright © 2006 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-0-85404-357-6

Contents

NMR Books and Reviews W. Schilf, 1,
Theoretical and Physical Aspects of Nuclear Shielding Cynthia J. Jameson and Angel C. de Dios, 52,
Application of Nuclear Shielding Shigeki Kuroki, Naoki Asakawa and Hidekazu Yasunaga, 82,
Theoretical Aspects of Spin-Spin Couplings H. Fukui, 130,
Applications of Spin-Spin Couplings Krystyna Kamienska-Trela and Jacek Wójcik, 152,
Nuclear Spin Relaxation in Liquids and Gases R. Ludwig, 199,
Solid State NMR Spectroscopy A.E. Aliev and R.V. Law, 234,
Multiple Pulse NMR I. Barsukov, 313,
NMR of Proteins and Nucleic Acids S.J. Matthews, 336,
NMR of Carbohydrates, Lipids and Membranes Elizabeth F. Hounsell, 362,
Synthetic Macromolecules Hiromichi Kurosu and Takeshi Yamanobe, 389,
NMR in Living Systems Malcolm J. W. Prior, 433,
Nuclear Magnetic Resonance Imaging Tokuko Watanabe, 457,
Oriented Molecules K.V. Ramanathan, G.A. Nagana Gowda and C.L. Khetrapal, 486,
NMR of Liquid Crystals and Micellar Solutions Maura Monduzzi and Sergio Murgia, 533,

CHAPTER 1

Theoretical and Physical Aspects of Nuclear Shielding

BY CYNTHIA J. JAMESON AND ANGEL C. DE DIOS

1 Theoretical Aspects of Nuclear Shielding

1.1 General Theory. – Increasing attention is being paid to non-linear responses of molecules to intense electric and magnetic fields. The possibility of deviations from linear dependence of the resonance frequencies on the strength of the external magnetic field in NMR experiments was first suggested by Ramsey. Earlier computed values indicated that these effects are too small to be concerned with in NMR measurements, and although Bendall and co-workers have attempted measurements for a supposedly favorable case of Co, this latter experimental result may be in error, given the more recent theoretical calculations which indicate the effect for Co in [Co(NH3)6]3+ to be one to two orders of magnitude different from what was suggested by the experiments. Furthermore, with routine use of stronger field magnets in NMR spectrometers, a definitive study of the magnitude of the field dependence of the magnetic shielding and the magnetizability on a test set of molecules has been needed. In a series of papers, Lazzeretti and co-workers explore the fourth rank tensors responsible for the non-linear response in B. For a spatially uniform and time independent magnetic field B and permanent magnetic dipoles ITLμITLI at the Ith nucleus, the energy of a diamagnetic molecules in its electronic reference state a is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where Wa(0) is the energy of the isolated molecule, χαβ is the magnetizability of the molecule, σIαβ is the nuclear magnetic shielding at nucleus I, and the fourth rank tensors Xαβγδ and ΣIαβγδ account for the non-linear response in B, which are measures of the magnetic field dependences of the magnetizability and the nuclear shielding, respectively. The tensor components Σαβγδ of five molecules (H2, HF, H2O, NH3, CH4) have been calculated for two origins of the magnetic vector potential for four basis sets of increasing quality, aug-cc-pCVDZ, aug-cc-pCVTZ, aug-cc-pCQTZ, aug-cc-pCV5Z. The same fourth rank tensors were calculated for the rare gas atoms, He, Ne, Ar, and Kr, using d-aug-cc-pV5Z and t-aug-cc-pV5Z basis sets. The authors also provide the diagonal second rank electron densities for the rare gas atoms as three-dimensional surface maps and contour maps. Since the density of a rare gas atom in the absence of the magnetic field has a spherical shape, the change induced by a strong spatially uniform magnetic field is visually obvious in these maps, that is, a contraction of the electronic charge, which distorts to an elongated shape, having the longer axis parallel to B. The dependence of the nuclear shielding on even powers of an external magnetic field had been previously analyzed by Vaara et al.

Electric field effects on the nuclear shielding of rare gas atoms can be treated analogously as in equation (1), except that the expansion of the electronic energy is in terms of powers of the electric field. Early contributors to this theoretical aspect of shielding include Buckingham, Raynes, Bishop, and Dykstra, who, among others, also considered molecular systems. Recently, relativistic contributions to the electric field derivatives of the nuclear shielding of He, Ne, Ar, Kr, and Xe have been calculated using four component relativistic Dirac-Coulomb Hamiltonian in comparison to non-relativistic Hartree-Fock. The relativistic contributions to the isotropic shielding polarizability are small, except in the case of Xe, and opposite in sign to the non-relativistic value (except for Ne). For Xe, Pecul et al. found 2415 ppm au-2 compared to the non-relativistic 4000 ppm au-2, using the same basis set as was used by Bishop and Cybulski for the non-relativistic calculations in 1993. For convenience we adopt their notation (PCB, for Partridge and Faegri, Cybulski and Bishop) for this basis set. Contributions to the parallel and perpendicular components are large and canceling. The relativistic contributions to the isotropic shielding polarizability are -0.14, 5.60, -5.81, -178.51, and -1584.27 ppm au-2 for He to Xe.

The relativistic corrections to the isotropic shielding scale with atomic number roughly as Z. (See earlier calculations by Vaara and Pyykko.) The relativistic effect on the shielding polarizability is found to be even larger, and changing less regularly with atomic number (perhaps an indication that some significant contributing term has been left out of the relativistic calculations). The effect of correlation on the shielding polarizabilities of the rare gases is substantial, as indicated by the comparison of the non-relativistic HF and SOPPA results. The electric field breaks the spherical symmetry of the atom, leading, in the non-relativistic formulation, to the appearance of the paramagnetic components that are usually more sensitive to electron correlation effects than the diamagnetic terms. Another important conclusion is that, in Xe atom, electron correlation contributions to the shielding polarizability goes in the opposite direction with respect to the relativistic correction. Since the presence of a neighbor atom likewise breaks the spherical symmetry of the Xe atom, the appearance of paramagnetic components when Xe suffers intermolecular interactions also leads to electron correlation contributions. This has been demonstrated by Jameson et al. for Xe2 in the comparison between the DFT-B3LYP vs. Hartree-Fock using the PCB basis set. More recently, Hanni et al. carried out coupled cluster calculations and arrived at the same conclusion based on the comparison between their CCSD(T) and Hartree-Fock calculations. Although the relativistic calculations have not been carried out yet for Xe2, the electron correlation contributions and relativistic corrections to the intermolecular shielding in Xe2 are, like the shielding polarizability of Xe atom, likely to be opposite in sign.

The misconception of simple parameterized models attributing intermolecular Xe shifts to purely electrostatic effects has been particularly difficult to eradicate and is perpetuated in the literature by careless citation. That the shielding response of a rare gas atom to electric fields is too small to account for the observed intermolecular shifts of Xe had been established in 1993 by the non-relativistic calculations of Bishop and Cybulski of the second derivative of the Xe shielding with respect to electric field using the PCB basis set and MP2-level calculations; and the more recent calculations of Pecul et al. including both relativistic and electron correlation effects verify these earlier findings. Furthermore, the major contribution to intermolecular shielding of Xe and the other rare gases had already been established as largely accountable by Hartree Fock calculations on the rare gas atom plus neighbor atom cluster, attributing the major portion of intermolecular shielding to the overlap and exchange contributions that are fully accounted for in a Hartree Fock calculation on the cluster. In fact, estimates of overlap and exchange contributions to intermolecular Xe shielding were made as early as 1964 by Adrian. More recent calculations, mentioned in the above two paragraphs, verify these findings. Electron correlation provides additional contributions. Including the effects of electron correlation using either DFT (B3LYP) or ab initio (CCSD) methods lead to somewhat larger values for intermolecular shielding of Xe-rare gas pairs in comparison to the Hartree Fock values; the DFT method uniformly overestimates for all rare gas partners, as recently shown by the more accurate CCSD results.

Relativistic effects on shielding tensors for the H2X (X = O, S, Se, Te, Po) and HX (X = F, Cl, Br, I, At) molecules have been evaluated using the relativistic and magnetic operators as perturbations on an equal footing, calculated using analytical linear and quadratic response theory applied on top of a non-relativistic reference state provided by self-consistent field calculations. The nuclear spin dependent active contributions that had not been included in an earlier paper on these molecules, have recently been calculated, and added to the previous results, thus, completing the leading order one-electron perturbational relativistic effects on the shielding tensors in these molecules. The results are compared with four-component Dirac-Hartree-Fock calculations that include positronic excitations. The perturbational approach breaks down for the heaviest elements; some of the relativistic corrections are of similar magnitude to the non-relativistic values; the errors encountered in the isotropic shielding exceed 20% in these cases. The errors in the shielding anisotropy are worse, especially for the fifth row elements. Nevertheless, this perturbational approach is comparable in accuracy to other quasi-relativistic treatments based on the zeroth order Hamiltonian, or the Douglas-Kroll-Hess method, according to the results cited in Ref. 22.

Electron correlation at the CCSD and CCSD(T) level had been included by Visscher et al. in a relativistic theory using four-component wavefunctions. A more recent attempt has been reported by Nakatsuji and co-workers. In this treatment, they start with the no-pair Dirac-Coulomb-Breit Hamiltonian. Molecular Dirac Fock spinors are obtained by solving the Dirac-Fock equations in which the nuclear magnetic moment is included explicitly. The zero-order wavefunction is approximated by a single Slater determinant of the molecular DF spinors belonging to the electronic state, that is, only the positive energy solutions of the DF equation. For the electron correlations, they use the CCSD method and for efficiency, they report an algorithm for 2-electron integrals in a direct CI method for singles and doubles. This CCSD-four component relativistic method is applied to HX and CH3X molecules (X = F, Cl, Br, I). For the 1H in HX, the results are within 0.8 ppm of the experimental data; where the uncorrelated calculations overestimated the relativistic effect, including electron correlation brings the values down closer to experiment. These results are in good agreement with the results of Visscher et al. For the 13C in CH3X the results are less satisfactory at both correlated and uncorrelated levels because of the insufficient size of the basis sets used.

In a much larger system, a direct CI relativistic method such as described above is presently out of reach. Thus, DFT was the means used for introducing electron correlation, and effective core potentials (ECP) were used with various relativistic methods to take relativistic effects into account in the 19F and 235U shieldings in the UF6-nCln series of molecules. In this series, there is an interesting chemical effect: the experimental F shifts are strongly dependent on the nature of the ligand (F or Cl) trans to the F in question. It was found that large core ECPs fail completely for calculations of the shieldings of the ligands, but small core ECPs could be used. By comparing different relativistic methods, including a new scalar SC-ECP method, it was discovered that relativistic approximations were not largely responsible for the discrepancies relative to experiments. Different functionals were tried, but none of these brought results into agreement with experiments. The approximate XC functional and solvent effects remain as possible sources of error.

Tozer and co-workers continue their investigation of improving the functionals used in DFT calculations of shielding. They report an investigation into the representation of the exchange potential in hybrid functionals. In a new approach, designated multiplicative Kohn-Sham (MKS), the multiplicative exchange-correlation potential associated with the hybrid density is determined from the electron density using the procedure of Zhao, Morrison and Parr. The corresponding Kohn-Sham orbitals and eigenvalues are then used to determine the shielding tensor in an uncoupled rather than coupled manner. In this recent report, the authors investigate whether MKS-quality results could be obtained without requiring such procedures. An alternative localized Hartree-Fock exchange potential from Della Sala and Görling was tried, with limited success. Instead the authors continue to recommend their generalized gradient approximation (GGA) potentials KT1 and KT2, reviewed previously in this series. They have recently improved KT1 and KT2 by introduction of additional gradient-corrected exchange and correlation terms, leading to a new functional KT3. The form of the functional can be interpreted as a flexible version of the OLYP functional introduced by Handy et al. While KT3 has the same weaknesses as all other GGA functionals in that it cannot compete with hybrids for accuracy of thermochemical predictions, and is unable to describe long-range dispersion interactions, it is said that KT3 provides shielding values of light main group nuclei that are more accurate than those from other GGA functionals and hybrids.

Various functionals have been compared with each other in DFT calculations and with Hartree Fock calculations to gauge their relative ability to reproduce the full shielding tensor information for 13C nuclei in two types of compounds, aromatics and sugars, for which accurate single crystal measurements have been carried out over several years in the laboratory of D. M. Grant, and at the same time accurate atomic positions are available from diffraction data. The data set is an excellent one for testing calculations, in that the chemical shift referencing to TMS is uniformly done, the quality of the data is uniform (precision better than 0.5 ppm), and several inequivalent carbons are available in most cases. Altogether 35 aromatic and 65 saccharide carbons were used in the comparison. In general, six tensor components are used in the icosahedral representation that permits both the axis orientation information and the principal values to be used on an equal footing. The conclusions are that Becke’s three-parameter exchange functional with either the Lee, Yang, and Parr correlation functional or the Perdew-Wang-1991 generally provide the best predictions and outperform Hartree-Fock. This is not surprising because one needs to include electron correlation to properly describe carbon shielding in aromatic systems. The aromatic nuclear sites are a better test than the saccharide sites, since hydrogen bonding was not taken into account in the calculations. It is well-known that hydrogen bonding partners have to be included in the molecular system used for calculations in order to describe carbon shielding in hydrogen-bonded molecules. Unfortunately calculations were carried out only in single isolated molecules in this study. The authors note that the diminished ability of the methods to predict the carbon tensors in the saccharides fails to show universal consistency. It is very likely that with the proper hydrogen-bonded clusters representing the saccharide systems, clearer, more consistent results would have emerged for the carbon tensors in the saccharides as well.

(Continues…)Excerpted from Nuclear Magnetic Resonance Volume 35 by G.A. Webb. Copyright © 2006 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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