Theoretical and Physical Aspects of Nuclear Shielding

BY W. T. RAYNES

1 Introduction

It is appropriate to commence this Report with a statement of the five ends which the author has in mind in writing it. They are as follows: to achieve some glory for science; to proclaim the names and, in a few cases, describe the work of those who have made contributions to the field of nuclear shielding in the recent past; to improve the knowledge of those who are currently working in the field or will enter it in the future; to provide instruction, information, and delight to the general reader; and finally, to earn a little honest profit for himself. The extent to which these ends are reached is for the reader to decide. However, it can be said with certainty of the last that by the time this volume has reached its destination it will have long since disappeared.

As a subject nuclear shielding is characterized by what Appleman and Dailey have termed 'distant theoretical and experimental wings'. Therefore it is fitting to divide a review of the subject into two chapters. This chapter reviews primarily the more theoretical wing and includes discussions of topics such as the definition, the quantum theory, and ab initio calculations of nuclear shielding. However, new and improved experimental results are of great importance made the more so by the excessive respect shown by a minority of theoreticians for old and unchecked literature data. Therefore a second part of this chapter reports on new experimental data for small molecules, the components of shielding tensors in solids, and intermolecular effects in gases.The last topic is of increasing interest to theoreticians.

As the reader will be well aware, no chapter on nuclear shielding is to be found in Volume 6 of this series. Earlier volumes covered the literature up to the end of May 1975. The present chapter aims to redress the situation by including references to work published from that date to the end of May 1977. It is too much to hope that all relevant papers have been found — especially since some journals were inaccessible to the writer during the preparation of the Report. Any omitted papers will be noted in future volumes.

2 Theoretical Aspects of Nuclear Shielding

A. The Nuclear Shielding Tensor. — For a molecule held fixed in a magnetic field B, the magnetic shielding of any one of its nuclei can be described by a second-rank tensor [??] whose components are defined by the expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where B' is the secondary field at the nucleus of interest created by the currents induced in the electronic system by B. The inclusion of the negative sign in equation (1) makes explicit that the phenomenon involved is that of nuclear shielding (i.e. B' is to be regarded as opposing B). The suffices α and β in equation (1) denote the x-, [y-, and z-co-ordinates of a cartesian system of axes and a repeated Greek suffix in any term signifies summation over all three co-ordinates for that term. In general [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so that it requires nine independent coefficients to specify the shielding fully. However, should the molecule possess some symmetry then the number of coefficients needed may be less than nine.

To determine just how many coefficients are required for symmetrical molecules one must first know not the point group of the molecule, but the 'nuclear site symmetry', i.e. the symmetry of the site at which the nucleus of interest is located in relation to the molecule as a whole (assumed here to be in its equilibrium configuration). This nuclear site symmetry is not difficult to discover and some examples are shown in Table 1. The numbers of shielding components for various nuclear site symmetries are presented in Table 2. This information was first given by Buckingham and Malm, although in a form slightly different from that of Table 2.

For an understanding of the nature of nuclear shielding it is necessary to turn to quantum theory. A straightforward application of second-order perturbation theory leads to the Ramsey equation, which will here be expressed in the abbreviated form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where the superscripts d and p denote respectively 'diamagnetic' and 'paramagnetic'. Diamagnetic parts of the shielding depend only on the wave function of the electronic ground state (assumed here to be the state for which the shielding is required) whereas the paramagnetic parts depend on the wavefunctions of excited states including those in the continuum. The superscript g denotes 'gauge' and refers to the need to define an origin for the vector potential of B (the 'gauge origin') when working with the molecular hamiltonian. Terms with a superscript g depend on this choice of origin. If the gauge origin is taken at the nucleus of interest, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Of course, being a molecular property the nuclear shielding must always be invariant to changes of gauge origin (i.e. gauge-independent). However, in numerical calculations in which approximate wavefunctions must of necessity be used the calculated values of σdgαβ and – σpgαβ are not identical when the gauge origin is not at the nucleus of interest. (There is one exception to this last statement. When the nucleus of interest is located at a site possessing a centre of symmetry or Td symmetry then, provided the origin of co-ordinates is taken at the nucleus, the sum of σdgαβ and σpgαβ vanishes for all choices of gauge origin even with an approximate wavefunction.)

From the above it can be seen that the question of gauge origin is one of great importance in any quantum-mechanical calculation of nuclear shielding. This point will be discussed further below. Explicit expressions for each of the four parts of σαβ using arbitrary origins for the co-ordinate system and the vector potential can be found in each of Volumes 3 — 5 of the present series.

B. The Question of Gauge Origin — As indicated above the manner of dealing with the problem of gauge origin is of considerable importance in the calculation of nuclear...