Nuclear Shielding

BY R. DITCHFIELD

1 Introduction

Since the phenomenon of n.m.r. was first detected in bulk matter some thirty years ago, the subject has expanded remarkably, with the result that high-resolution n.m.r. spectroscopy is now one of the most important branches of chemical spectroscopy. In measurements of n.m.r. spectra, nuclei are essentially being used to investigate local magnetic effects in a molecular system. The local magnetic field near a particular nucleus depends on the electronic environment of the nucleus and is determined by many factors. These include the electronic polarization of remote parts of the sample, magnetic moments (nuclear and electronic) of neighbouring molecules, and intramolecular effects due to other nuclei and electrons in the same molecule. Consequently, not only has n.m.r. spectroscopy become a powerful tool for elucidating molecular structure, but it also provides a sensitive probe into various aspects of the electronic structure of molecules.

As is well-known, information about molecular electronic structure is extracted from n.m.r. spectra in terms of nuclear magnetic shielding constants, σi, and nuclear spin–spin coupling constants, Jij. Early n.m.r. studies were mainly devoted to protons, and theories were proposed to explain proton shieldings in many types of molecules. On the whole, although such theories were highly approximate, they were successful in explaining the gross trends in the values of proton magnetic shielding constants.

In the early 1960's, theoretical methods were developed to interpret shielding constants for first-row atoms. Again the theories were rather approximate but of qualitative value in rationalizing the main trends observed in the then limited experimental data. Since the late 1960's, developments in instrumentation and experimental techniques have meant that magnetic shielding data are now readily obtained for many nuclei. Although advances have been made, developments in theoretical methods have not kept pace with experimental progress. Thus experimental data are still largely rationalized using empirical relationships developed some ten or fifteen years ago. For example, changes in shielding have been correlated with the changes in charge density, the changes in electric field effects, and the changes in magnetic anisotropy which occur when a substituent is varied. Although such empirical relationships can be valuable, the fact that there is a growing body of experimental data which cannot be explained adequately in this way suggests that an accurate theoretical understanding of the factors which contribute to nuclear magnetic shielding is still lacking.

This chapter deals with articles on nuclear shielding that were published during the twelve months to the end of May 1975. As in previous Reports in this series, the emphasis is on papers containing results which either do, or may, lead to a better understanding of the phenomenon of nuclear shielding in isolated molecules. Therefore, discussion of the following topics has been excluded: experimental methods of chemical-shift measurement, the details of methods for the quantum-mechanical calculation of shielding constants, and the mechanisms by which intermolecular effects alter shielding constants. Solution phenomena, including the study of contact and pseudocontact shifts and of complex formation, are discussed in Chapter 11. During the period of writing a few journals were unavailable in the Reporter's library because they were being bound; apologies are offered to any author whose work is thereby overlooked.

2 Basic Aspects of Nuclear Shielding

A. General Theory. — It has become the practice in this series of Reports to attempt to make them as self-contained as possible. Since readers have probably become accustomed to this kind of approach, the traditions so ably established by Drs. Raynes and Mallion will be followed here. The equations derived by Ramsey relating nuclear magnetic shielding constants to electronic structure are appropriate to the case where the origin of the vector potential describing the uniform external magnetic field and the origin of the co-ordinate system are taken at the nucleus whose shielding is of interest. In this Report, the generalization presented by Raynes in which the origin of co-ordinates, the gauge origin, and the nucleus of interest are located at different points will be followed.

The approach of Raynes is conveniently discussed with reference to Figure 1. Here O is the co-ordinate origin and G is the origin with respect to which the vector potential is referred; μ is a point magnetic dipole placed at the position where the shielding is required. R, S, and rk are vectors representing the position of G, the position of μ, and the position of electron k relative to O, respectively. Using Rayleigh–Schrödinger perturbation theory and the clamped-nuclei approximation the following eight-term expression for the αβ component of the shielding tensor is obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

In equation (1), the superscripts 'd' and 'p' denote diamagnetic and paramagnetic contributions, respectively. A superscript 'g' indicates those contributions which depend on the choice of gauge origin; such terms clearly vanish when R = 0. A superscript 'μ' denotes the point magnetic dipole, and contributions labelled in this way will be zero when S = 0. The expressions derived by Raynes for the contributions presented in equation (1) are given, in SI form, in equations (2) to (9).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[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)

From these equations, it is clear that the paramagnetic contributions require a knowledge of both ground-state |0> and excited-state wavefunctions |n>. In contrast, the diamagnetic contributions depend on the ground-state wavefunction, |0>, only. In equations (2) — (9), μ0, e, and m are the permeability of free space, the electronic charge, and the electronic mass, respectively. The linear momentum for electron k is denoted by pk, and lk(= rk [conjunction] pk) is the orbital angular momentum for this electron. W0 and Wn are the energies of the ground and nth excited states, respectively, and the prime on the summations in equations (6) — (9) denotes a summation over all values of n except n = 0. It should be noted that integration over the continuum of excited states is implicitly included. The Greek subscripts denote Cartesian components x, y, and z; δαβ = 1 if α = β, and 0 if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the alternating tensor and = 1 if (βγδ) is an even permutation of (xyz) = –1 if (βγδ) is an odd permutation of (xyz), and = 0 if any two of (βγδ) are identical. If R and S are both null vectors then equation (1) reduces to the two-term expression (10) obtained by Ramsey.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The only paper in the review period which deals with the theory of nuclear magnetic shielding, as distinct from calculations of magnetic shielding constants, is the comment by Moss in which he criticizes the approach of Weisenthal and de Graaf to the theory of diamagnetism in neutral molecules. Weisenthal and de Graaf have expanded the molecular charge and current densities about a point R in the molecule, thereby obtaining a molecular Hamiltonian that involves the magnetic flux density B and the electric field strength E, rather than the external vector and scalar potentials. Use of this Hamiltonian gave an expression for the diamagnetic susceptibility that does not depend on the vector potential. This led Weisenthal and de Graaf to claim that they had solved the problem of the most suitable gauge for calculations of diamagnetic susceptibilities when only approximate wavefunctions are available. Moss has pointed out that this is incorrect, their choice of R merely being the selection of a particular gauge. In fact, the molecular Hamiltonian may always be written so that it does not contain the potentials explicitly, by first selecting a gauge and then expressing the potentials in terms of B and E. Moss also states that the Weisenthal–de Graaf method is equivalent to the set of gauge transformations A -> A – [nabla] f with f = (1/2c) B [conjunction] R • μ, where μ is the molecular electric dipole moment. These conclusions are similar to those reached by Woolley and Cordle (see Vol. 4, p. 5).

B. Basic Physical Aspects. — Following tradition, this sub-section contains the results of a number of calculations and experiments which add to the more fundamental knowledge of nuclear shielding phenomena.

Jameson et al. have extended their studies of the temperature and density dependence of the 129Xe chemical shift in rare-gas mixtures to the 3 — 28 amagat density range. As in previous work, they write a virial expansion of shielding in ascending powers of density

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

The ultra-low densities used allowed the elimination of three-body and higher-order interactions, with the result that the present data give more accurate values for σ1 than did the earlier measurements. The main result of these studies is the precise determination of σ1 (T) for 129Xe in xenon gas and in mixtures with krypton and argon.

A knowledge of an accurate value of the proton shielding constant in molecular hydrogen would be particularly useful for the establishment of an absolute scale for proton magnetic shielding. In this regard, the theoretical studies of Reid, who has attempted to obtain a precise value for the proton shielding constant in H2, are especially important. Using Ramsey's theory in the adiabatic approximation, and with the origin of the vector potential and the origin of the co-ordinates taken at the nucleus of interest, the shielding in a state characterized by vibrational and rotational quantum numbers υ and J is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

The diamagnetic contribution, σd, to the shielding was calculated using the 87-term James–Coolidge-type electronic wavefunction used by Reid and Vaida previously.

The required averages over vibrational motion were performed using wavefunctions obtained numerically from the adiabatic potential of Kolos and Wolniewicz. For the ground vibrational state of H2 Reid has calculated values of 32.022 (J = 0) and 31.999 (J = 1) p.p.m. The corresponding values in HD are 32.072 (J = 0) p.p.m. and 32.055 (J = 1) p.p.m. Reid has used these calculated values of σd, together with experimental spin-rotation data and the experimental value of the isotope shift σ (HD) – σ (ortho-H2) at 295 K, to determine σp. The values of σp thus obtained were – 5.622, – 5.637, – 5.654, and – 5.650 p.p.m., for H2 (J = 1), HD (J = 1), D2 (J = 1), and D2 (J = 2), respectively. These values are in good agreement with the results reported by Cade and Ramsey discussed previously.

The total shielding constants at 295 K obtained by Reid are presented in Table 1, where they are compared with the values obtained by other workers. One standard-deviation error in σ(ortho-H2) at 295 K was found to be 0.014 p.p.m. Reid, however, has reported an error of ±0.070 p.p.m. in order to allow for the contributions which relativistic and non-adiabatic effects make to σ (ortho- H2). The calculated value is in good agreement with the experimental value quoted in Table 1 and with the experimental value estimated by Raynes recently. Reid has also calculated the temperature dependence of the shielding in H2 and in various isotopomers, and has found that the gross features of such temperature dependence are similar to those obtained by Raynes et al.

New experimental data for the A1F molecule have been presented during the review period by Honerjaeger and Tischer. The Zeeman effect in the microwave rotational spectrum of 27Al 19F has been measured in the ground vibrational state. The spectrum obtained in the magnetic field exhibited resolvable hyperfine structure from the 27Al nucleus, which enabled the determination of a shielding anisotropy value of – 672 (±71) p.p.m. This value was compared with an anisotropy of – 333 (±22) p.p.m. obtained from an interrelation with the spin–rotation interoction constant. No explanation of the discrepancy between these two values was affered.

Several experimental investigations have been published during the review period establishing absolute scales of shielding constants for cadmium, zinc, and potassium. Krueger et al. have measured Cd chemical shifts in aqueous solutions of CdCl2, Cd(NO3), CdSO4, and Cd(ClO4)2 as a function of concentration. From such data they have obtained a magnetogyric ratio for the cadmium ion in solution. A comparison of this ratio with the magnetogyric ratio for the free cadmium atom gives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using a similar approach, Lutz and co-workers have reported FT n.m.r. studies of 67Zn. The shielding constant for the hydrated zinc ion relative to the shielding constant for the free zinc atom was evaluated to be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The trend of increasing magnitude for such shielding differences for Group IIB elements has been further substantiated by measurements on 199Hg.

Sahm and Schwenk have reported n.m.r. studies of 39K, 40K, and 41K. From the results of studies of n.m.r. lines of 39K and 41K in solutions of many potassium salts in H2O and D2O and the results of atomic beam magnetic resonance experiments they find

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This result implicitly indicates that the experimental data revealed no significant primary isotope effects.

All the studies referred to above have examined the effect of the isotopic composition of the solvent on the measured magnetic shielding constants. A discussion of such effects is deferred to Section 4H.