Professor Krystyna Kamienska-Trela is based at the Institute of Organic Chemistry, Polish Academy of Sciences, Warsaw.

Nuclear Magnetic Resonance Volume 40

A Review of the Literature Published Between January 2009 and May 2010

By K. Kamienska-Trela

The Royal Society of Chemistry

Copyright © 2011 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-1-84973-147-8

Contents

Preface K. Kamienska-Trela, v,
Books and reviews W. Schilf, 1,
Theoretical and physical aspects of nuclear shielding Cynthia J. Jameson and Angel C. de Dios, 37,
Applications of nuclear shielding Shigeki Kuroki, Shingo Matsukawa and Hidekazu Yasunaga, 55,
Theoretical aspects of spin-spin couplings Jaroslaw Jazwinski, 134,
Applications of spin-spin couplings Krystyna Kamienska-Trela and Jacek Wójcik, 162,
Nuclear spin relaxation in liquids and gases Jozef Kowalewski, 205,
Solid state NMR spectroscopy A. E. Aliev and R. V. Law, 254,
NMR of proteins and nucleic acids Peter J. Simpson, 311,
NMR of carbohydrates, lipids and membranes Ewa Swiezewska and Jacek Wójcik, 344,
Synthetic macromolecules Hiromichi Kurosu and Takeshi Yamanobe, 391,
NMR of liquid crystals and micellar solutions Gerardino D’Errico and Luigi Paduano, 432,
NMR in living systems M. J. W. Prior, 472,

CHAPTER 1

Theoretical and physical aspects of nuclear shielding

Cynthia J. Jameson and Angel C. de Dios

DOI: 10.1039/9781849732796-00037

1 Theoretical aspects of nuclear shielding

1.1 General theory

Kutzelnigg and Liu present the formulation of a logical and systematic classification of existing methods of calculations of NMR parameters within relativistic quantum chemistry, together with variants not previously proposed, and new methods are also presented. Various methods have been reported separately in this series over several years; the Kutzelnigg and Liu analysis puts all systematically in the proper context. They consider transformations at operator level versus matrix level, the possible formulations of the Dirac equation in a magnetic field, traditional relativistic theory, field-dependent unitary transformation, bispinor decomposition, equivalence of the methods at operator level. They then consider relativistic theory in a matrix representation, expansion in unperturbed eigenstates, expansion in a kinetically-balanced basis, and expansion in an extended balanced basis. The authors explore decomposition of the lower component, decomposition of the full bispinor, unitary transformation at the matrix level. They pay careful attention to singularities. First they discuss methods which are exact in the sense that their accuracy is only dependent on the quality of the chosen basis. In the limit of a complete basis all these methods yield the same results, but the rate of convergence to the limit can be different.

Among these methods, they consider the ones best suited for each of the magnetic properties. For the case of nuclear magnetic shielding where one vector potential is due to an external field and one is due to a nuclear magnetic dipole, we must discard the method based on the untransformed Dirac operator because it does not give the correct non-relativistic limit. We must also discard the method based on a unitary transformation of the full magnetic field (Full-Field Unitary Transformation, FFUT) because it is plagued by singularities. Here a good choice is the method based on a unitary transformation of the external field only (External-Field Unitary Transformation, EFUT) and formalisms equivalent to it, such as what the authors call the Bispinor Decomposition and the decomposition of the small component (Orbital Decomposition Approximation, ODA). These lead to the correct non-relativistic limit and are not plagued by singularities. They also recommend as a further possibility the method called FFUTm (full-field unitary transformation “at matrix level”) by Xiao et al. where we start formally from a unitary transformation at matrix level but evaluate the diamagnetic term exactly. They also suggest to further consider the brute force expansion in an Extended Balance (EB) basis, although this is likely to be numerically unstable. Instead the restricted magnetic balance (RMB) should be used. The performance of the various methods with respect to the basis set requirement has recently been investigated by Cheng et al. The results differ very little, even for a small basis.

Finally, Kutzelnigg and Liu consider various approximations previously proposed which do not give the exact results in the limit of a complete basis, for example, Xiao et al. All of these are based on methods which give the correct non-relativistic limit and use a pseudo sum-over-states formulation with the restriction of the intermediate eigenstates to those with positive energy. This automatically implies errors of O(c-4), i.e., beyond the leading relativistic order. Along the way, various commonly used approximations such as the Douglass-Kroll-Hess approximation and the Zeroth-Order Regular Approximation (ZORA) are discussed in context.

It has been recognized recently that the incorporation of the magnetic balance condition between the small and large components of the Dirac spinors is absolutely essential for four-component relativistic theories of magnetic properties. Cheng et al. show that the magnetic balance can be adapted to distributed gauge origins, leading to, e.g., magnetically balanced gauge-including atomic orbitals (MB-GIAOs) in which each magnetically balanced atomic orbital has its own local gauge origin placed on its center. Such a MB-GIAO scheme can be combined with any level of theory for electron correlation. The first implementation is done by the authors at the coupled-perturbed Dirac–Kohn–Sham level. The calculated molecular magnetic shielding tensors are not only independent of the choice of gauge origin but also converge rapidly to the basis set limit. Close inspections reveal that zeroth order negative energy states are only important for the expansion of first order electronic core orbitals. Their contributions to the paramagnetism are therefore transferable from atoms to molecule and are essentially canceled out for chemical shifts. This allows for simplifications of the coupled-perturbed equations. Earlier, Quiney et al. and Ilias et al. had also adopted the GIAO method in their uncoupled-DHF or CP-DHF four-component relativistic treatment of NMR shielding, but they did not take the magnetic balance into account. Contrary to the statement made by Ilias et al. Cheng et al. show that the combination of GIAO and MB brings in no complications. Cheng et al. also provide, in their introduction, critical remarks on the various schemes for relativistic calculations of NMR properties, classifying and comparing, noting where singularities occur and where numerical instabilities could occur, comparing various schemes to recover the relativistic diamagnetic contributions to the nuclear shielding, such as to guarantee the correct non-relativistic limit.

Komorovsky et al. have also incorporated the gauge including atomic orbitals (GIAO) approach in relativistic four-component density functional (DFT) method for calculation of NMR shielding tensors using restricted magnetically balanced basis sets. The authors carried out relativistic calculations for xenon dimer and the HX series (X=F, Cl, Br, I), where spin-orbit effects are known to be very pronounced for hydrogen shieldings. It is not surprising that, when compared to shieldings calculated at the four-component level with a common gauge origin, the results clearly demonstrate that the GIAO approach solves the gauge origin problem in fully relativistic calculations as it does in the non-relativistic case. Finally, what had been routine (use of GIAOs) for non-relativistic calculations of shielding is becoming an integral part of four-component calculations of nuclear shielding.

In addition to the formulation of four-component relativistic theory of NMR parameters described above, Cheng and co-workers also present an exact two-component relativistic theory for nuclear magnetic shielding (and magnetizability and J coupling). This is obtained by first a single block-diagonalization of the matrix representation of the Dirac operator in a magnetic-field-dependent basis and then a magnetic perturbation expansion of the resultant two-component Hamiltonian and transformation matrices. They show that all the problems (singularities, numerical instabilities) associated with the earlier attempts at an exact two-component treatment of NMR parameters can be avoided by going to a matrix formulation. That is, the matrix representation of the full Dirac operator in a magnetic-field-dependent basis can be block-diagonalized in a single step, just like the previous matrix formulation of the exact two-component algebraic Hamiltonians in the absence of magnetic fields. The resulting Hamiltonian and transformation matrices can then be expanded to obtain the expressions for NMR parameters. Such a matrix formulation is not only simple but also general in the sense that the various ways of incorporating the field dependence can be treated in a unified manner. The diamagnetic and paramagnetic terms agree individually with the corresponding four-component ones up to machine accuracy for any basis. The authors suggest that this formulation be adopted in lieu of quasi-relativistic theories.

Polarization propagators have been successfully applied since the 1970s to calculate NMR parameters. They are special theoretical devices from which one can do a deep analysis of the electronic mechanisms that underlie any molecular response property from basic theoretical elements, like molecular orbitals, electronic excitation energies, coupling pathways, entanglement, contributions within different levels of theory, etc. All this is obtained in a natural way in both regimes: relativistic and non-relativistic. In a recent review article, Aucar et al. discuss the new insights on magnetic shielding from relativistic polarization propagators, using model compounds CH3X molecules (X=F, Cl, Br, I) and XHn (X=Xe, I, Te, Sb, Sn; n=0–4) as examples.

Although Gaussian type functions are more commonly used as basis functions, there are some who prefer Slater-type functions because they satisfy the cusp at the origin. Slevinsky et al. show that the Fourier integral transformation can be applied for the analytical development of integrals of the paramagnetic contribution in the relativistic calculation of the shielding tensor using exponential-type functions (ETF, such as Slater-type functions) as a basis set of atomic orbitals.

1.2 Ab initio and DFT calculations

Relativistic calculations of NMR shielding for heavy nuclei in this reporting period include 183W in polyoxometalates of W and Au, 183W in poly-oxotungstates and a family of Keggin anions, 195Pt in two complexes, 195Pt in 24 Pt(II) square-planar complexes, and 187Os in osmium phosphines, Using large basis sets of QZ4P quality and taking into account the conductor-like screening model (COSMO) to account for solvent effects (in aqueous and organic solutions), reasonable geometries were found for the polyoxotungstate anions. Anions studied included α-[XW12O40]q- (X=B, Al, Si, P, Ga, Ge, As, Zn), β- and γ-[SiW12O40]4- geometric isomers, [P2W18O62]6-, [W6O19]2- , and [W10O32]4-. From these optimal geometries the 183W nuclear shieldings were computed with standard basis sets of triple zeta with polarization (TZP) quality and including spin-orbit corrections inside the zero-order regular approximation (ZORA). Bagno et al. investigated 183W nuclear shielding in the anions [MATHEMATICAL EXPRESSION OMITTED] and [P2 W21O71(OH2)3]6- using DFT calculations including relativistic effects by means of the twocomponent zero-order regular approximation (ZORA), at the scalar (ZSC), or spin-orbit (ZSO) levels, with the Becke 88-Perdew 86 (BP) functional. The basis sets were of double- and triple-zeta quality, singly or doubly polarized Slater functions, with flexible (all electron) or frozen cores. 195Pt nuclear shieldings in [H2PtV9O28]5- were calculated using the same method. Twenty four Pt(II) square planar complexes were investigated by Ziegler and co-workers, calculating the 195Pt NMR shielding tensor with gauge-including Slater type orbital basis functions and the ZORA Hamiltonian with spin-orbital coupling. The application of the B3LYP functional yields smaller deviations from experiment for the 195Pt NMR chemical shifts in comparison to BLYP values. For the majority of studied Pt(II) square complexes, local functionals (BLYP and BP86) underestimate (in absolute value) the NMR chemical shifts. The same methods applied to 187Os nuclear shielding in only 3 osmium complexes leads to less clear preference between BLYP and B3LYP.

Relativistic calculations of light atom nuclear shieldings in molecules bearing heavy atoms have also been reported, for example 17O in Np complexes, 17O in complexes of Pt, W, and Au, 13C in carbonyl complexes of Hf, Ta, W, Re, Os, Ir, and Hg, 13C in monohalo (F, Cl, Br, I) organic compounds, and 13C and 15N in 6-halo(Cl, Br, I) purines. The experimentally observed 300 ppm shift in 17O chemical shifts between the known [UO2(OH)4]2- and the Np(VII) solution is shown to be partly a function of the central metal, that is, Np(VII) vs. U(VI), and not of the coordination environment (tetraoxo vs. dioxo). Here, calculations were carried out with relativistic DFT-ZORA using the PBE functional, and all-electron Slater-type basis sets of triple- and quadruple-z polarized quality (ZORA-TZ2P, ZORA-QZ4P) were used, with bulk solvation effects modeled with the COSMO model. In a study of the heavy atom effect on 13C in monohalo organic compounds, the spin orbit-Fermi contact SO/FC contribution to 13C substituent chemical shifts were calculated within the scalar ZORA, SO-ZORA, scalar PAULI, and SOPAULI approaches, and the results are compared. It is observed that the SO and FC parts of the SO/FC term are sensitive enough to show observable differences for both equatorial and axial cyclohexane conformers. Heavy atom effects on 13C and 14N shielding in 6-halopurines are most significant for the carbon and nitrogen atoms in the six-membered pyrimidine ring of the purine molecule.

Calculations of nuclear shielding in biomolecules are becoming more commonplace. In a study of one of the phospholipids abundant in mammalian membranes, sphingomyelin or N-acyl-sphingosine-1- phos-phorylcholine, the 31P nuclear shielding in the phosphate head group was calculated using DFT/B3LYP-GIAO. Both the geometry optimization and the shielding calculations were carried out using the polarizable continuum model (PCM).with dielectric constant values ranging from 0 (gas phase) to 78. The 31P shielding was found to be 280.0 to 280.7 ppm for dielectric constants 0 to 8. The Na+ binding site in a calix[4]areneguanosine conjugate dimer serves as a model for the Na+ coordination environment in the guanosine G quadruplex DNA structures where the alkali ion resides in plane or above the plane of the G quartet. By constructing a series of theoretical models for the conjugate dimer and comparing with 23Na NMR experiments, the experimental binding site is theoretically described by having agreement in the electric field gradient and the nuclear shielding simultaneously. This study yields benchmark 23Na NMR parameters for penta-coordinated (in-plane) Na+ ions that can be used in searching for similar Na+ binding sites in G-quadruplex DNA and in other supramolecular assemblies containing G-quartets. Bactericides and other drugs continue to be the subject of NMR studies and nuclear shielding computations. For example, the 1H and 13C nuclear shielding in 2,4-difluorobenzaldehyde isonicotinoylhydrazone and 2,3-dichlorobenzaldehyde isonicotinoylhydrazone have been investigated with GIAO, IGAIM, and CSGT models using DFT. Calculation of fluorine chemical shift tensors was found very useful in the interpretation of oriented 19F-NMR spectra of gramicidin A in membranes. Calculations of 15N and 13C nuclear shieldings in halopurine nucleosides, in particular, 6-(fluoro, chloro, bromo, and iodo)purine 2′-deoxynucleosides, were useful in the interpretation of the NMR spectra. The use of B3LYP functional and DGDZVP basis set for the GIAO/DFT calculation was found to give reasonably good agreement between experiments and calculations, whereas the larger 6-311+G(3df,2d) basis gives better agreement. As expected, inclusion of solvation effects improves the results, particularly for 15N. Solvation effects were included by the conductor-like polarizable continuum model (COSMO).

(Continues…)Excerpted from Nuclear Magnetic Resonance Volume 40 by K. Kamienska-Trela. Copyright © 2011 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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