CHAPTER 1
Nuclear Shielding
BY W. T. RAYNES
1 Introduction
'The most important single parameter to be derived from the n.m.r. spectrum is the chemical shift.' This contention, possibly controversial even when published in 1959, became, one suspects, increasingly less acceptable to the majority of n.m.r. spectroscopists during the course of the nineteen-sixties. However, the introduction of new techniques for signal enhancement in the past few years has meant that shielding data, in the form of proton and fluorine chemical shifts from the peripheral regions of molecules, can now be supplemented by phosphorus, nitrogen and, above all, carbon chemical shifts which yield, in principle, a much more intimate knowledge of electronic distributions in the interior of molecules. This, coupled with the enticing prospect of large amounts of new information on nuclear shielding and nuclear shielding components in solids, provides additional support for those who are inclined to support the above quotation.
The structure of the present Report does not differ in essence from that of Volume 1 of the present series. As before the emphasis will be on reported work in the review period (July 1st 1971 to May 31st 1972) which either leads to, or may lead to, an improved understanding of the phenomenon of nuclear shielding in isolated molecules. Therefore, as with last year's Report, no space has been devoted to any of the following topics: 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 can alter shielding constants. This third restriction forces the exclusion of all solution phenomena including the study of contact and pseudocontact shifts and of weak complex formation. These topics, however, are covered in Chapter 10.
The number of papers presenting new carbon chemical shift data during the review period — about eighty — is more than double the number referred to in Volume 1. In addition the number of papers with new phosphorus chemical shift data has increased to twenty-six — one more than that for fluorine — with nitrogen not far behind. With this large increase for C, P, and N it has been decided to limit the number of references to proton chemical shift studies (which total about two hundred and fifty) only to those of particular value for illustrating proton shielding mechanisms or to those of compounds of particular interest such as the annulenes, substituted derivatives of benzene, etc.
The following two conventions, used in last year's Report on nuclear shielding, have been adopted in subsequent sections of this Report. N.m.r. chemical shifts and shielding constants are occasionally given without the appellation p.p.m. (parts per million). Where substituent effects are considered and the chemical shift is referred to the unsubstituted compound, the shift has been given a positive sign if the nucleus under investigation has a higher shielding constant in the substituted compound than in the unsubstituted compound. During the period in which the present article was being written, a small number of journals were inaccessible to this reporter. Therefore, apologies must be offered to some authors for omission of any reference to their work.
2 Basic Aspects of Nuclear Shielding
A. General Theory. — Placing the origin of co-ordinates at the nucleus of interest and the origin of the vector potential of the uniform external magnetic field at a point having position vector ro from the co-ordinate origin, we obtain for the component σαβ of the nuclear shielding tensor the more general form of Ramsey's equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
The terms on the right-hand side of equation (1) are defined as follows:
[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)
In equations (2) — (5) the symbols μo, e, and m denote respectively the permeability of free space, the electronic charge, and the electronic mass; rk is the position vector of the k'th electron from the nucleus of interest; Wn represents the energy of the n'th excited state; pk denotes the linear momentum operator of the k'th electron; lk(=rk × pk) is the orbital angular momentum operator of this electron and the convention of summation over repeated suffices is used. The prime on the summations in equations (4) and (5) denotes a summation over all values of n except n = 0, including the continuum of excited states. σαβ is the substitution tensor (= 1 if α = β, = 0 if α ≠ β) and σβγδ is the alternating tensor [= 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 ro = 0, equation (1) reduces to the familiar two-term expression [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with the 'conventional' diamagnetic and paramagnetic terms. The terms all and all both dependent upon the origin of the vector potential, i.e. upon the gauge choice — hence the superscript g.
During the review period it has been pointed out that the nuclear shielding tensor is not, in general, symmetric. This means that the component σαβ will not, in general, be equal to σβα. From equation (4) the shielding σDαβ is obviously not necessarily equal to σDβα. However, equation (2) makes clear that σdαβ is always symmetric so that the 'conventional' diamagnetic part of the shielding tensor can be fully specified by six components. For a gauge choice away from the nuclear site of interest we see that σdgαβ and σpgαβ also not necessarily equal to σdgβα and σpgβα respectively. Buckingham and Malm divide the shielding into the sum of an isotropic part, a traceless symmetric part, and an antisymmetric part. Thus
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
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