Theoretical and Physical Aspects of Nuclear Shielding

BY W. T. RAYNES

1 Introduction

The subject of nuclear magnetic shielding occupies something of an anomalous position at the present time. Being a molecular property, it ought to be classed with properties such as the electric dipole moment, the electrical polarizability, the electric field gradient at a nucleus and, more particularly, the magnetizability (magnetic susceptibility) — properties which all depend upon the molecular electronic wavefunctions. However, it is so closely identified with the n.m.r. technique that it is to books on this branch of spectroscopy which one must turn in order to learn something of shielding. At the present time there are very few monographs on molecular properties and in them nuclear shielding can claim little more than a chapter or so, whereas books on quantum chemistry, of which there is no shortage, are primarily concerned with molecular electronic wavefunctions and energies and seldom deal with the wider array of molecular properties. Works on n.m.r cannot possibly do justice, in one or two chapters, to the vast abundance of theoretical discussion, computational results, and experimental data on shielding which reside in the primary literature.

All this suggests the need for a monograph on nuclear shielding which would review the work of the past three decades and unify the disparate elements in a field which is of interest to theoreticians (tensor aspects, electronic wavefunctions, gauge choice), to physical chemists (isotope shifts, intermolecular effects), to inorganic chemists (contact shifts, shielding of 'other nuclei') and to organic chemists (aromaticity, conformational effects). It is hoped that this void will be filled by a forthcoming publication.

The present Chapter covers the same ground as the first Chapter of Volume 7. However, the distinction between theoretical and physical aspects being imprecise, it has been decided to divide the chapter into three (rather than two) parts. The first part deals with calculations of nuclear shielding, the second part with physical aspects, and the third part with experimental data. This approach allows theoretical aspects to be dealt with in either of the first two parts as is considered appropriate and, to a large degree but not totally, separates the experimental results.

The papers to which reference will be made are those published between June 1st 1977, and May 31st 1978. However, a few papers published after the latter date will be noted although not discussed in any detail. Apologies are offered to authors whose contributions have been omitted. In most cases this is due to linguistic incompetence on the part of the present writer or to the absence from the writer's library of certain journals during the period (Nov./Dec. 1978) when the review was finalised.

2 Calculations of Nuclear Shielding

A. General Theory. — Nuclear shielding is a tensor property. In the absence of any kind of symmetry it requires nine components to describe fully the shielding at a given nuclear site. Quantum mechanics provides an expression for the components of the shielding tensor. This was first obtained by Ramsey. In its most general form the Ramsey equation for the shielding tensor component σαβ for a chosen nucleus of a molecule in its ground electronic state, can be written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The two parts of σαβ are referred to as the diamagnetic shielding σdαβ and the paramagnetic shielding σpαβ. The definitions of the principal quantities in equations (2) and (3) can be understood by reference to Fig. 1, in which D denotes the location of the point dipole (i.e., the nucleus of interest) at which the shielding is required, K is the instantaneous position of the kth electron, and G is the origin of the vector potential of the external magnetic field. The vectors rd and rg are defined in Fig. 1; ldk is the orbital angular momentum of the kth electron about the point D (i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where pk is the linear momentum of the kth electron), and lgk is the orbital angular momentum of the kth electron about the point G (i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). The summations are taken over all electrons and over all excited electronic states. W0 is the energy of the ground state and Wn that of the nth electronic state. All other notation bears its usual significance.

In an earlier report an expression for σαβ was given in which various vectors were defined with respect to an origin of co-ordinates O. This introduced vectors R, S, and rk which defined the positions of D, G, and K with respect to O. By means of the substitutions rg = rk – R and rd = rk – S in equations (2) and (3), it is possible to obtain the eight-term expression for σαβ given previously. (The factor RωSγ in the term σpgmαβ given in Vol. 3 should, in fact have been RωSγ. This error was perpetuated in Volumes 4 and 5 of the present series.)

As can be seen from equations (2) and (3), the magnitudes of σdαβ and σpαβ are dependent on the location of G. Since the shielding itself cannot be dependent on this location, the variation in σdαβ which occurs upon a change of G must be cancelled out by the variation in σpαβ which takes place with the same change in G. That this is the case can be easily proved by making use of the identities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

However, in actual calculations approximate wavefunctions formed with limited basis sets must be used so that σdαβ + σpαβ becomes 'gauge dependent' (i.e., depends on the choice of G). Sadlej has demonstrated how it is possible to select...