Nuclear Magnetic Resonance Volume 4

A Review of the Literature published between June 1973 and May 1974

By R. K. Harris

The Royal Society of Chemistry

Copyright © 1975 The Chemical Society
All rights reserved.
ISBN: 978-0-85186-282-8

Contents

Chapter 1 Nuclear Shielding By R. B. Mallion, 1,
Chapter 2 Nuclear Spin-Spin Coupling By R. Grinter, 67,
Chapter 3 Nuclear Spin Relaxation in Liquids By M. D. Zeidler, 121,
Chapter 4 Bandshape Phenomena in Liquids By T. Drakenberg and H. Wennerström, 141,
Chapter 5 Multiple Resonance By W. McFarlane and D. S. Rycroft, 174,
Chapter 6 Macromolecules By I. D. Robb and G. J. T. Tiddy, 202,
Chapter 7 Liquid Crystals and Micellar Solutions By G. J. T. Tiddy, 233,
Chapter 8 The Solid State By P. S. Allen and W. Derbyshire, 253,
Chapter 9 Medium Effects By M. I. Foreman, 294,
Author Index, 323,

CHAPTER 1

Nuclear Shielding

BY R. B. MALLION

1 Introduction

‘The last thing one discovers in writing … is what to put first.’ This remark of Pascal (Blaise, that is — not the pioneer in the field of magnetic susceptibilities!) is the one which springs most immediately to mind as I take over the chapter on ‘Nuclear Shielding’ from Dr. Raynes, who contributed this chapter for the first three volumes of these Reports. I can, however, start by saying that I shall adhere, as far as possible, to the structure and general format of the Report which he very satisfactorily evolved over the past three years, i.e. it will be what the earlier Reports described as ‘phenomenon-oriented rather than compound-oriented’. No major change has been made for two reasons: firstly, such a policy should be helpful to regular readers who, by now, have become accustomed to Dr. Raynes’ presentation, and secondly, it seemed worthwhile to capitalize, as much as is proper, on the previous Reporter’s very successful experimentation in this regard.

Adopting such a scheme has, however, presented a mild dilemma; whether to repeat, almost verbatim, the basic equations required for each of the separate phenomena discussed, or whether simply to cite the relevant equations detailed explicitly in the earlier Reports. The former policy would ensure that all the requisite information were in one volume, thus obviating the necessity for readers to refer back to the earlier Reports, and the latter policy would save space. Since this Report is meant to be a digest of the literature — in fact, the sort of compendium which (to use F. A. Cotton’s quotation of J. S. Waugh’s description of another eminently assimilable publication) ‘one can read in bed without a pencil’ — I have decided, somewhat reluctantly, to trade brevity for readers’ convenience, and (again following Raynes’ practice) have repeated all the basic equations which are pertinent to the appropriate discussion in the various sections of the Report. There are, however, some minor changes: I prefer to discuss chemical shift anisotropy in the section headed ‘Basic Physical Aspects’; n.m.r. measurements relating to the concept of ‘aromaticity’ are discussed with ‘ring current’ effects (Section 4E), since authors frequently link these two subjects in the literature; and I have not been nearly so conscientious as Dr. Raynes was about distinguishing between the various sorts of ‘transmitted effects’ (e.g. the so-called ‘inductive effects’ and what are often dubbed ‘resonance effects’) discussed in Section 4. Finally, the section on ‘Shieldings of Particular Species’ (Section 5, devoted to 19F, 31P, and ‘other nuclei’) will tend more to be simply lists of papers reporting data, for those works concerning mechanisms or calculation of the shielding of such nuclei will have been dealt with earlier in the appropriate sections.

According to the Reviewer’s ‘brief’, this Report should include articles on nuclear shielding which were published between June 1st 1973 and May 31st 1974. As before, only experimental and theoretical papers relating to nuclear shielding in isolated molecules have been considered [although, actually, this restriction has been somewhat arbitrarily relaxed in Section 2 (‘Basic Aspects’), where I have wanted to discuss the effects of pressure and anisotropic media]; as was the case in the previous Reports, this policy precludes examination of purely experimental techniques of chemical shift measurement (covered in Chapter 4), the quantum-mechanical details of shielding-constant calculations, the mechanisms of intermolecular shielding effects (thus excluding consideration of all solution phenomena as well as the study of contact and pseudo-contact shifts and of weak complex formation, all of which are dealt with in Chapter 10). Studies in the nematic phase have also been excluded, except when such investigations have led explicitly to information concerning chemical shift anisotropy.

Some attempt has been made to include coverage of certain foreign-language journals; I hope, in particular, that the literature in the French language has been covered as adequately as that in English; I must, however, apologize in advance to writers in the German language for a less than satisfactory review of their work; and papers in Russian (unless available in translation or summarized in Chemical Abstracts) have had, regrettably, to be omitted from consideration entirely. Finally, during the period of writing, some journals were unavailable in the Reporter’s library because they were being bound; apologies are, therefore, offered to the authors of any papers which are, thereby, overlooked.

2 Basic Aspects of Nuclear Shielding

A. General Theory. — Ramsey’s familiar ‘two-term’ expression for the component, σαβ, of the shielding tensor of a given nucleus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where the superscripts ‘d’ and ‘p’ denote ‘diamagnetic’ and ‘paramagnetic’, respectively, is valid only for the very special case in which the gauge origin (i.e. the origin of the vector potential due to the uniform, external magnetic field) and the origin of co-ordinates coincide at the nucleus whose shielding is being calculated; a four-term expression that allows for arbitrary choice of the gauge origin but still requires the position of the nucleus of interest to be the origin of co-ordinates has been detailed by Raynes.’ In last year’s Report, Raynes gave a yet more general version of the Ramsey equation, applicable when the co-ordinate origin, the gauge origin, and the nucleus in question may be different points; for reference purposes these expressions are now repeated here.

The formulae which follow should be studied with reference to Figure 1. The origin of co-ordinates is denoted O, and the gauge origin is at G, related to O by the vector R. An infinitesimal ‘test’ dipole of moment μ is then placed at the pointf at which it is required to calculate the shielding (which point may, or may not, be the position of a nucleus). The kth electron is, at any given time, displaced from O by the vector rk; the nuclei are assumed to be fixed. An eight-term expression then results:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

In this expression, calculation of the diamagnetic terms (with superscript ‘d’) requires knowledge only of the unperturbed ground-state wavefunction, whereas the paramagnetic terms (superscripted ‘p’) are also a function of the excited-state wavefunctions (and are, thereby, that much more difficult to calculate). The terms bearing a superscript ‘g’ are gauge dependent, and will be zero only when the gauge origin and the origin of co-ordinates coincide. The superscript ‘μ’ refers to the point dipole, and terms containing this superscript are zero only when the dipole is located at the origin of co-ordinates. In giving explicit expressions for the terms on the right-hand side of equation (2), we again follow Raynes in using SI, which is now becoming more customary:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

In equations (3) — (10), μo is the permeability of a vacuum, e and m are the charge and mass, respectively, of the electron; pk is the linear momentum operator for the kth electron and lk (= rk [conjunction] pk) denotes the orbital angular momentum operator for this electron. Wn represents the energy of the nth excited state, and the prime on the summations in equations (7) — (10) is meant to indicate a summation over all values of n except n = 0 (but including the continuum of states). The Greek subscripts denote Cartesian components x, y, and z; δαβ is 1 if α = β, and zero if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is 1 if (βγδ) is an even permutation of (xyz), – 1 if (βγδ) is an odd permutation of (xyz), and zero if any two of (βγδ) are identical.

In the conventional formulation of electromagnetic theory, the external magnetic field manifests itself in the expressions obtained as an electromagnetic vector potential. These potentials are, however, defined only to within the space and time derivatives of an arbitrary function and, in two papers which were discussed in a previous Report (and which have also been commented on by Moss and Perry), Weisenthal and de Graaf derived a new Hamiltonian formalism in which the coupling between the charges and the external fields is expressed in terms of multipole moments of the charge distribution and the field variables E(x,t) and B(x,t), rather than the usual expression involving the particle momenta and field potentials, φ(x,t) and A(x,t); they made the exciting (but perhaps somewhat startling) claim that by writing the Hamiltonian explicitly in terms of the field strengths rather than their vector potentials, any question regarding gauge invariance or best choice of gauge would be obviated. During the present review period, however, Woolley and Cordle have pointed out that such gauge difficulties are not avoided by making a unitary transformation to a formalism which is independent of the electromagnetic field potentials, and that neither of the formalisms of Weisenthal and de Graaf possesses the uniqueness that has been attributed to them. By a somewhat different reasoning, J. A. N. F. Gomes (Theoretical Chemistry Department, Oxford) has also shown that, unfortunately, the new Hamiltonian, which is apparently and superficially gauge invariant, does, in fact, implicitly involve a particular choice of gauge for the vector potential, and that it has, therefore, none of the advantages which have been claimed for it.

The general problem of ‘symmetry-enforced’ gauge invariance of nuclear shielding constants has been discussed in an important note by Okninski and Sadlej. Since it can be shown (e.g. ref. 11) that a gauge transformation of the vector potential of the external magnetic field merely changes the relative contributions of the diamagnetic and paramagnetic terms of equation (1) to the total shielding tensor σαβ, it has been thought that an appropriate choice of gauge might avoid completely the need for a calculation of the tiresome paramagnetic term σpαβ. For the very specific gauge transformation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Chan and Das proved the existence of such a gauge origin, which ensures a vanishing paramagnetic contribution to the shielding. In the paper under discussion, Okninski and Sadlej have demonstrated that, for some symmetry species, the proof of Chan and Das loses its general character and that, under certain symmetry conditions, neither term in equation (1) depends on gauge so that the individual terms of the whole shielding tensor become accidentally gauge invariant. Furthermore, since this feature does not depend on the accuracy of the wavefunctions employed, when such a case arises (diagnosed by two formal theorems presented in ref. 10) gauge invariance of the shielding tensor cannot be used as a criterion of completeness of the basis set. In such cases, information about completeness of the basis set can be gleaned only by a study of the origin dependence of the magnetic susceptibility, since gauge-origin independence of σαβ does not guarantee gauge invariance of the induced currents and, hence, of the magnetic susceptibility tensor.

During the review period there has also been some discussion of relativistic and radiative interactions of many-electron atoms in an external magnetic field, and the derivatives of the eigenvalues of the Hamiltonian to fifth order have been examined in the context of molecular properties in general (including nuclear magnetic shielding).

B. Basic Physical Aspects. — This subsection will be devoted to the results of a number of experiments, and some calculations, which have a direct bearing on the more fundamental aspects of nuclear shielding.

(i) Pressure and Temperature Dependences. A particularly extensive study of the temperature and pressure dependence of 129Xe chemical shift in xenon gas has been made by Jameson et al. Previous 1H, 19F, and 129Xe studies on gaseous systems showed that the chemical shift has an essentially linear dependence on density up to densities of ca. 100 amagat; in the range 30 — 250 amagat, the curve is of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where, at room temperature,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

The work of Jameson et al. concerns experimental determinations of σ1(T) for pure xenon, over the temperature range 240 — 440 K. An accurate temperature dependence of σ1 is essential for determining the functional form of the shielding of two interacting molecules. It is found that the experimental values can be fitted by the following fourth-degree polynomial:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where τ = (T/K) – 300.

The 1H chemical shift of hydrogen chloride has been studied as a function of temperature and density by Trappeniers and Ahlers. The quantity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] was found to fit the following relation, below 293 K:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

(t/°C). The effects of temperature on the 13C shifts of benzene, cyclohexane, pyridine, neopentane, hexane, and heptane, in the range 283 — 343 K, as well as the temperature shift of 2H in D2O, have been studied by Litchman and McLaughlin. All shifts were found to be linear functions of temperature, within the experimental error. The pressure dependence of the 1H chemical shift of chloroform has been studied by Yamada et al. Although again slightly outside our somewhat arbitrarily restricted scope, two other papers are merely mentioned here, one concerning the temperature dependence of isotropic dipolar shieldings in oxygen and silicon, and the other concerned with 1H and 19F shifts as a function of temperature and concentration.

Raynes et al. have previously reported ab initio calculations of bond-length dependences of nuclear shielding in the hydrogen molecule and, by means of the method of Herman and Short, obtained an average over the nuclear motion which gave the shielding of H2 and its isotopomers (HD, HT, D2, DT, and T2) in their lower rotational and vibrational states: they thus deduced temperature variation of the shielding constants at low gas densities, in addition to the various isotope shifts. Unfortunately, however, no nuclear shielding data are available of sufficient accuracy to enable these calculations to be tested critically. Accordingly, in order to check their choice of wavefunction, the form of perturbation theory applied, and the Herman–Short procedure, Raynes et al. have been forced to consider very accurate calculations of some other property of hydrogen, magnetic susceptibility being selected. They have, therefore, calculated the magnetic susceptibility of hydrogen and compared their results with those of Kolos and Wolniewicz. Good agreement is obtained. They also tested the Herman–Short procedure for averaging over the nuclear motion by comparing the predicted susceptibilities of H2, HD, and D2 with those obtained by Jain and Salmi, once more with gratifying results. Meanwhile, Zeroka has also independently made a theoretical study of the variation of the diamagnetic properties — both susceptibility and shielding — of the hydrogen molecule, with internuclear separation. Studying the range of internuclear separations R, 1.0 — 12.0 a.u., he has applied his previously described variation-perturbation approach, using the wavefunction of Fraga and Ransil, accurate over this range of R. The shielding at Re = 1.4 a.u. is 27.095 p.p.m. For normal H2, the average over nuclear motion, where zero-point vibration and centrifugal distortion are taken into account, is = 26.760 p.p.m. at 300 K and varies little in the range 20 — 300 K. Zeroka’s results for the magnetic shielding of H as a function of internuclear separation are shown in Table 1. The dependence of the paramagnetic shielding in diatomic molecules on the interatomic distance has also been investigated by Volodicheva and Rebane, who have obtained vibrational-rotational corrections of shieldings calculated by the ‘vector potential variation method’ for HF, DF, HC1, DCI, HBr, and DBr. Dixon et al., in continuing their study of one-electron properties, have generated LCAO-MO SCF wavefunctions for the methane molecule using a minimal basis set of SCF orbitals, and have used them to investigate the dependence of the diamagnetic shielding (and diamagnetic susceptibility) of methane on internuclear distance. The results are shown in Table 2, where the diamagnetic shielding of the proton is given in p.p.m., the C — H bond length, RCH, is varied from 1.3 to 1.5 a.u., and pH, the orbital exponent, ranges from 1.95 to 2.15.

(Continues…)Excerpted from Nuclear Magnetic Resonance Volume 4 by R. K. Harris. Copyright © 1975 The Chemical Society. Excerpted by permission of The Royal Society of Chemistry.
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