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 shielding parameters. This remark also applies to the calculation of magnetic susceptibilities. Many calculations make use of the findings of Chan and Das. These authors demonstrated that the diamagnetic part of the susceptibility is at a maximum for a gauge origin at the molecular electronic centroid and consequently that the paramagnetic part must be at a minimum. They also used physical arguments to propose that the paramagnetic part of the shielding of a nucleus of interest should be numerically small for this same gauge origin. This latter idea has been borne out by subsequent calculations on the shielding of protons; calculated proton shielding constants usually improve quite dramatically (see below) when the gauge origin is moved from the proton of interest to the molecular electronic centroid or, as is done more often, to a 'central atom'.

It is understandable therefore why many authors have chosen a gauge origin which makes the numerical value of (σp + σpg) small. However, this is not the same as choosing a gauge origin which makes the error in (σp + σpg) a minimum and it is this which is of greater practical importance. During the review period this question has been examined independently by Sadlej and Yaris and a significant advance has been made. We shall consider the work of Sadlej which, apart from apparently being a little more general than that of Yaris, also includes a specific numerical example.

Sadlej considers the identity of <0|AB|0> and Σ<0|A|K> <K|B|0> where A and B are two operators relevant to some molecular magnetic property and |K> denotes the complete set of molecular eigenfunctions, |0> being that for the state of interest. The summation is taken over the complete set of states. With a finite basis set the identity ceases to hold and, assuming that both A and B are explicitly dependent on the gauge R, Sadlej defines the gauge-dependent quantity,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where the summation is taken over the discrete set of states generated by the chosen basis. The conditions which force the square of Δ(R) to be minimized will correspond to an improvement. These conditions are governed by the size and nature of the basis set and the choice of gauge origin. Furthermore since either one or both of A and B are different for different magnetic properties, it follows that for a given basis set the 'best' gauge origin for one property will not in general be the same as for another.

Sadlej has applied these arguments to the hydrogen fluoride molecule. He placed the origin of co-ordinates at the fluorine nucleus and the proton on the positive z-axis (with z-co-ordinate 1.7329 a.u.). With the criterion given in the preceding paragraph he found the z-co-ordinates Z for the 'best' gauge origins on the z-axis for the proton ct(H) and fluorine shielding σ(F) using two different but fairly large basis sets of contracted gaussian orbitals (CGTO's) as follows:

22 CGTO 29 CGTO

σ (H) Z = –0.028425 a.u. Z = –0.017931 a.u.
σ (F) Z = +0.022700 a.u. Z = –0.020642 a.u.

Thus for these two basis sets the best gauge origin for the proton is relatively remote, being in fact beyond the fluorine nucleus. The values of the shieldings obtained by Sadlej for various choices of gauge origin including the 'best', the fluorine nucleus, the electronic centroid (EC), and the proton are shown in Table 3. It is evident that substantial changes are produced when the 'best' origin is chosen rather than the fluorine nucleus or the electronic centroid.

An alternative approach in calculating shielding constants is to make use of so-called 'gauge-invariant atomic orbitals', GIAO. Some time ago Epstein raised criticism of this method in that the use of GIAO's does not always lead to the current conservation expected. During the review period Dalgaard has attempted to meet this criticism. However, his reasoning appears to show only that GIAO's are the most natural form of field-dependent basis function within the LCAO scheme. The question of current densities in molecules has also been discussed by Atkins and Gomes with application to current densities of the benzene molecule in a magnetic field. Walnut has discussed the question of gauge in relation to calculations of susceptibility and applied his ideas to the paramagnetic part of the susceptibility of molecular hydrogen and the explanation of ring currents in annulenes. The main users of GIAO's have been R. Ditchfield, and M. Zaucer and co-workers. Their recent work is discussed in the following section in the context of applications to specific molecules and molecular systems.

C. Ab Initio Calculations — A very general formulation of multiple self-consistent perturbation theory (coupled Hartree–Fock perturbation theory) has been published during the review period and has been used to derive expressions for the tensors describing the nuclear shielding and magnetic susceptibilities of molecules in closed-shell electronic states. Explicit formulae are given for the nuclear shielding itself, the magnetic field dependence of nuclear shielding, and the linear and quadratic electric field dependence of nuclear shielding. The only specific application of these expressions to be published during the review period was their use to calculate the linear and quadratic electric field dependence of nuclear shielding in the H molecule. This work is discussed in a later section together with the parallel work of Day and Buckingham on the electric field dependence of the 1H and 19F shielding in the HF molecule. The latter authors also give a formulation of self-consistent perturbation theory for nuclear shielding.

A great deal of attention has been given to the HF molecule in recent years and several new calculated values of the 1H and 19F shielding have been published. These have been collected together and are presented in Table 4 together with earlier results including the experimental values. The manner in which the various authors dealt with the question of gauge origin is indicated at the foot of the Table. Some additional results for HF with a gauge origin at the proton are given by Swanstrpm et al. The basis sets employed by various authors differ considerably. In particular Hladnik et al. have employed four different basis sets which produce differing values for both the 1H and the 19F shielding. In their work the field-dependent matrix elements of their finite perturbation theory are evaluated with the aid of analytical expressions they have derived. As the Table makes clear the results for the proton shielding (i.e. σ, σiso, and Δσ) are in rather poor agreement. Direct comparison of the calculated results with the experimental value of 28.51 (±0.20) p.p.m. is not valid as, of course, the latter includes the effects of nuclear motion which are not yet fully determined and may amount to a deshielding of more than 1 p.p.m. A subsidiary calculation performed by Hladnik et al. is that for the HF molecule in the very large field of 105 T. This problem is interesting on account of the possibility of calculating the magnetic field dependence of the shielding. Unfortunately, the result given is for energy only.

Nuclear shielding calculations for several other small molecules have also been carried out but we shall give little more than brief references to most of them. They are F-, F2, FOH, and FHF- by Zaucer et al., H2O by Zaucer et al. and Swanstrøm et al., and BH by Zaucer et al. Sternberg and Haberditzl, employing a power-series expansion to obtain the first-order perturbed wavefunction, have calculated nuclear shielding in H2, LiH, and N2. A very extensive study of water and the water dimer has been performed by Ditchfield and this will be discussed in some detail below. Of the molecules just mentioned apart from H2O, the ion FHF- has provoked the most recent interest and in Table 5 are presented results for this ion. Again agreement between different calculations is not as satisfactory as it might be. The 'hydrogen bond shift', i.e. the proton shielding of FHF- relative to that of HF, is calculated to be – 9.30, – 13.5, and – 15.246 p.p.m. The latter two results are close to the experimental value of – 14 p.p.m., although values calculated for a single molecular geometry can never be expected to agree with experimental values obtained in liquids or solids: intermolecular effects combine with those of nuclear motion to reduce the shielding considerably from the value at the equilibrium geometry.