Nuclear Magnetic Resonance: Volume 44

by Krystyna Kamienska-Trela (Editor)

Overview

Applications of nuclear magnetic resonance span a wide range of scientific disciplines, from physics to medicine. This series has provided an essential digest of the NMR literature for more than four decades and each volume provides unrivalled coverage of the literature on this topic. Continuous coverage on some topics such as theoretical and physical aspects of nuclear shielding is balanced by the desire for coverage on newer topics like applications in biological systems and materials science. For those wanting to become rapidly acquainted with NMR or seasoned practitioners, this is an invaluable source of current methods and applications.

Product Details

ISBN-13:	9781782625155
Publisher:	RSC
Publication date:	05/12/2015
Series:	Specialist Periodical Reports - Nuclear Magnetic Resonance , #44
Sold by:	Barnes & Noble
Format:	eBook
Pages:	616
File size:	2 MB

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Nuclear Magnetic Resonance Volume 44

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By Krystyna Kamienska-Trela

The Royal Society of Chemistry

Copyright © 2015 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-1-78262-275-8

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CHAPTER 1

Theoretical and physical aspects of nuclear shielding

Cynthia J. Jameson and Angel C. De Dios

DOI: 10.1039/9781782622758-00046

This chapter is a review of the literature published between June 2013 and May 2014 on physical and theoretical aspects of nuclear magnetic shielding. Theoretical aspects include (a) general theory, for example, newly developed approaches in relativistic theory of nuclear shielding, the relation between the spin-rotation tensor and shielding in relativistic theory, NMR shielding for electronically degenerate states and benchmark studies (b) ab initio and DFT calculations, both relativistic and non-relativistic. Physical aspects include (a) anisotropy of the shielding tensor, (b) shielding surfaces and rovibrational averaging, (c) isotope shifts, (d) intermolecular effects on nuclear shielding, and (e) absolute shielding scales.

1 Theoretical aspects of nuclear shielding

1.1 General theory

Aucar shows that by using the path integral formalism that was developed within quantum field theory, (as opposed to the wave-function-based formalism that is used in quantum chemistry) polarization propagators have the same formal definition within both the relativistic and non-relativistic domains, providing a unified view and new insights on the relationship between spin and time-reversal operators. In this article he explains why the non-relativistic limit is obtained by scaling the velocity of light to infinity, and how within the relativistic regime the negative energy states contribute to electron correlation, and how to consider both quantum electrodynamics (QED) and electron correlation effects on the same ground. Although QED effects are not yet implemented in calculations of NMR quantities, it had been suggested by Pyykko and Zhao that the QED effects on NMR parameters in molecules containing heavy atoms could be of the same size as solvent effects. With the generalization of polarization propagators to the relativistic framework diamagnetic and paramagnetic contributions which arise from completely different electronic mechanisms within the non-relativistic regime have become unified. They are produced by one and the same mechanism which is still not completely understood in our non-relativistic and perturbative way of thinking. In a proper relativistic framework the spin is no longer a good quantum number, so spin–orbit effects cannot be used to explain heavy atom effects in a molecule within the full relativistic formalism. Despite this, we likely will continue to think about and explain NMR parameters in terms of spin–orbit contributions, diamagnetic and paramagnetic terms, and other familiar crutches.

In 1950, Ramsey developed the general non-relativistic theory of the nuclear magnetic shielding tensor, and in the same paper showed that for linear molecules, the paramagnetic term (for the gauge origin taken at the center of mass of the molecule) is directly related to the experimentally measurable spin-rotational magnetic interaction constant of the molecule. In 1964 Flygare derived equations relating the spin-rotation constants and nuclear magnetic shielding for any type of molecule and showed how the identity could be useful in obtaining shielding values from the spin-rotation constants that arise from high-resolution microwave spectroscopic measurements in the gas phase. Experimental values of 19F spin rotation tensors for many molecules have been reported, thus providing multiple anchor points for the 19F absolute shielding scale. Agreement was excellent (within experimental error bars of the spin-rotation tensors) between the absolute shieldings individually derived from the spin-rotation tensors and those absolute shielding values that arise from using one 19F nuclear site (19F in HF molecule, for example) as a reference to convert the simultaneous measurements of 19F chemical shifts in a large number of molecules in the zero-density limit relative to 19F in SiF4 (for example)." Absolute shielding scales for other nuclei have likewise been established by combining the spin-rotation-derived paramagnetic shielding contribution with theoretically calculated diamagnetic shielding, and refinements are reported annually in Section 2.5 of this chapter. For molecules containing only light nuclei, where non-relativistic theory is expected to be a reasonable description, these absolute shieldings, σ0(300 K), provide stringent tests for validating theoretical methods of calculating nuclear magnetic shielding and shielding surfaces for rovibrational corrections and thermal averaging. For shielding of heavy nuclei or for shielding of even light nuclei in molecules containing heavy atoms, using the experimental (relativistic) spin-rotation and relativistic diamagnetic shielding (in those formulations where diamagnetic shielding can be explicitly separated out) does not fix the problem because the identity relation between the paramagnetic part of the shielding and the spin-rotation derived in non-relativistic theory no longer holds in relativistic theory. What is needed, and recently has become available, is a general relativistic treatment for both shielding and spin-rotation tensors which will provide a direct mapping between the two tensors. Aucar et al. established a theoretical expression for the spin-rotation tensor in the case of relativistic electrons while treating nuclear motion non-relativistically. The authors applied this theory to HX molecules (X = H, F, Cl, Br, I), established a comparison of the relativistic effects on the nuclear magnetic shielding and the spin rotation tensors in these molecules, and examined the validity of Flygare's relation between these two properties for these molecular systems. Flygare's identity relation derived from non-relativistic theory is anticipated to fail for HX with heavy X; the authors find that for H in this series of molecules, the relation is an approximation for molecules containing atoms of the 4th row or heavier, and for X = Br and I the Flygare relation does not hold at all because operators that describe relativistic effects on the magnetic interaction with B in the shielding tensor have no counterpart in the spin-rotation tensor and the spin contribution to the spin–orbit effect is different in the two properties. In addition, Aucar et al. also considered small corrections (Breit interaction effects) in their nuclear spin rotation theory and applied these to the HX series, and Malkin et al. applied the Aucar theory to Sn spin-rotation and nuclear magnetic shielding in SnH4, Sn(Me)4 and SnCl4. and to spin-rotation and nuclear magnetic shielding in the HCl molecule.

With the same objectives, Xiao and Liu developed a relativistic molecular Hamiltonian that describes electrons relativistically and nuclei quasi-relativistically. After transforming into the body-fixed frame of reference, the body-fixed relativistic Hamiltonian is used to formulate, among other molecular electronic property tensors, a relativistic theory of the spin-rotation tensor for semi-rigid non-linear molecules, which includes Aucar's formulation as a special case. Using the formulation by Xiao and Liu, a formal relationship between the two tensors can be written if the shielding tensor is formulated through the external field-dependent unitary transformation (EFUT) ansatz. This treatment is also applied to linear molecules. For linear molecules, there is no molecular rotation about the line of centers and no paramagnetic term in the component of shielding along this direction in non-relativistic theory, so the paramagnetic component of the shielding along this direction has to be calculated by four-component relativistic theory. A relativistic mapping between nuclear magnetic shielding and spin-rotation tensors is proposed as follows: The previous non-relativistic identity relation between paramagnetic shielding tensor vu component at equilibrium molecular geometry for the Kth nucleus and the spin-rotation uv tensor component, where I0vv the principal inertia tensor, gK is the g factor for the Kth nucleus, and μn is the nuclear magneton, is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The relativistic relation between these tensors is designated by Xiao and Liu as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where the correction, i.e., the difference between the electronic paramagnetic contributions to the nuclear shielding and the spin-rotation tensors, which they designate as the 'relativistic effect in the paramagnetic mapping', ΔK, pvu, eq, can be directly evaluated in the EFUT approach and formally written in sum over states form. Therefore, Xiao, Zhang and Liu enumerate three relativistic corrections to the absolute shielding: (a) the relativistic effect in the paramagnetic mapping as described above, (b) the relativistic effect in the nuclear shielding diamagnetism, and (c) the relativistic effect in the spin-rotation diamagnetism.

In our earlier attempts to include relativistic corrections in the absolute shielding (for example, for Se and Te in ref. 19), by using the experimental value of the spin-rotation tensor, we included (c) the relativistic effect in the spin-rotation diamagnetism, and by using a relativistic value for the shielding of the free (Se or Te) atom, we included (b) the relativistic effect in the nuclear shielding diamagnetism approximately, but we failed to include (a) the relativistic effect in the paramagnetic mapping. Unfortunately, it has been recently shown by Xiao and Liu that the latter correction predominates over the other two. Indeed, recent 4-component calculations of the spin-rotation tensor and the absolute isotropic shielding for Se and Te in SeF6 and TeF6 by Ruud et al.21 show that this is also the case for these systems.

Xiao, Zhang and Liu also find (using four-component DFT with either LDA or PP86 functionals) that the relativistic effect in the paramagnetic mapping leads to a correction of 4.5 ppm for the absolute shielding of 19F in HF. If this is correct, the 19F absolute shieldings would all be shifted by this much. This may not be the final word, however, because Xiao et al. have used DFT in their calculations. Teale et al. have previously demonstrated the inaccuracy of DFT in a systematic non-relativistic study, in which DFT calculations (using LDA and 13 other functionals) lead to mean absolute errors in 19F spin-rotation constants which are 10-30 times as large as CCSD(T) in comparison with experiments and similarly deficient compared to CCSD(T) in shielding calculations.

New two-component relativistic formalisms have been proposed. Yoshizawa and Hada developed a new formalism for Foldy–Wouthuysen transformation including a vector potential which can be used to introduce restricted magnetic balance (RMB) in the Douglas–Kroll–Hess (DKH) method. They also developed a way of using GIAOs in the DKH method by separating the GIAO function into two functions. First applications of this new DKH2 formulation to shielding in noble gas atoms Ne to Xe, HX molecules (X = F to I), and H2X molecules (X = O to Te) appear to provide results that are comparable to 4-component calculations by Manninen and Vaara. An efficient method of treating scalar relativistic effects on shielding based on the spin-free exact-two-component theory (X2C) in its one-electron variant has been proposed by Cheng, Gauss and Stanton. Increased efficiency in comparison to 4-component methods arises from a focus on spin-free contributions and from elimination of the small component. They start from 4-component theory, by separating the spin-free and spin-dependent terms for the matrix representation of the Dirac equation in terms of a RMB GIAO basis. The resulting spin-free 4-component matrix elements required for calculating nuclear magnetic shielding are then used to construct the Hamiltonian for the corresponding spin-free exact-two-component theory in its one-electron variant and its perturbed counterpart using analytic derivative theory. They suggest that for calculations of (a) heavy atom shielding in cases where both spin-free relativistic and spin–orbit effects are generally important, or for calculations of (b) light atom shieldings in cases where the bonding between the light and heavy elements is dominated by p-type orbitals of the light element, the spin-dependent terms are small and a pure spin-free relativistic treatment turns out to be useful; scalar-relativistic effects can be obtained rigorously at a highly correlated level, CCSD(T) for example, while the computationally expensive spin–orbit contributions may be treated in a perturbative manner or via additivity schemes. With this new formalism, they carry out coupled-cluster calculations for 129Xe in xenon fluorides and 17O shielding in [MO4]2- complexes (M = Cr, Mo and W), but this new method is not applicable for systems like 1H shielding in HX (X = F, Cl, Br, I) where spin-orbit contributions dominate. The experimental 17O chemical shifts also include deshielding medium effects, of course, which are not included in the theoretical calculations. When converted to absolute shielding, all experimental values are deshielded compared to CCSD(T) relativistic values. In these systems, electron correlation effects are very large, especially for [CrO4]2- and are greater than relativistic effects. For 129Xe in XeF2, XeF4 and octahedral XeF6, The calculated isotropic chemical shifts relative to free Xe atom compare favorably with the gas phase data for these molecules. Unfortunately the authors did not report their calculated δ[parallel] and δ[perpendicular to]. It would have been interesting to see to what extent δ[parallel] differed from zero and to what extent (δ[parallel] - δ[perpendicular to]) differed from (3/2)δiso in the theoretical results for this case. The individual components δ[parallel] and δ[perpendicular to] had been measured for 129Xe in XeF2 by Wasylishen et al. in the solid state relative to free Xe atom, although there are intermolecular effects in the experiment that could be of the order of 300 ppm (deshielded).

Autschbach reports on a previously neglected term from the response of the exchange–correlation (XC) potential which has recently been included in the relativistic NMR module of the ADF package. The XC response markedly improves calculated proton chemical shifts for hydrogen halides. Mercury chemical shifts for mercury dihalides are also noticeably altered. The term vanishes in the absence of spin–orbit coupling. The new results of fully relativistic calculations are compared with ZORA. While absolute shielding values for Hg are not accurately predicted with ZORA, the ZORA chemical shifts agree well with those from fully relativistic calculations. Autschbach provides a review of relativistic calculations of magnetic resonance parameters, with selected applications of relativistic-DFT.

(Continues...)

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Table of Contents

NMR Books and Reviews;
Theoretical and Physical aspects of Nuclear Shielding;
Applications of Nuclear Shielding;
Theoretical aspects of Spin-Spin Couplings;
Applications of Spin-Spin Couplings;
Nuclear Spin Relaxation in Liquids and Gases;
Solid-state NMR Spectroscopy;
NMR of Proteins and Nucleic Acids;
NMR of Lipids;
NMR of Carbohydrates;
Synthetic Macromolecules;
NMR of Liquid Crystals and Micellar Solutions;
NMR in Living Systems;
Oriented Molecules;
Molecular Simulations and NMR