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 the 'best' position of G. This method shows that for a given basis set the 'best' position of G varies with D (i.e., it depends at which nucleus the shielding is required). During the review period Braun and Rebane have considered the question of 'optimal gauge'. Their treatment appears to be a generalisation of that of Sadlej in that it allows for not only a variation in the choice of G but also an additional flexibility in the choice of basis functions. They cite several papers in which they have varied the vector potentials in the calculation of molecular magnetic properties. We note here a recent paper by Parker describing a gauge-invariant method which proceeds by calculating the electric current density induced in molecules by an external magnetic field.

B. Ab initio Calculations. — The molecules H2, HF, and H2O are those which have been most frequently chosen in carrying out accurate ab initio calculations of nuclear shielding. In Volume 7 of the present series a tabular summary of recent ab initio calculations of the shielding of the 1H and 19F nuclei in the HF molecule was given. In this section we give a similar summary for the 1H and 17O shielding in the H2O molecule. Table 1 lists values of the oxygen shielding obtained in recent calculations and the results are given in order of increasing mean shielding constant σ(= 1/3σαα). Basis set sizes (using standard notation for the process of contraction of gaussian atomic orbitals) are shown together with some indication of how the various authors dealt with the gauge problem. All calculations were performed within the Hartree–Fock limit and at equilibrium geometry, although slightly different choices of O — H bond length and HOH angle were adopted by different authors.

It is an unfortunate fact that there is no reliable experimental value for the 17O shielding of the water molecule at the present time. (Further discussion of this may be found in Section 3C).The most recent theoretical values — using large basis sets — cluster around 325p.p.m. As illustrated in Table 1, minimal basis sets are quite incapable of accounting for shielding and even when fairly large basis sets are used there are still substantial differences. The method of GIAO ('gauge invariant atomic orbitals') in which gauge dependent atomic orbitals are employed, has been criticised in principle and, as yet, no adequate defence has been made. However, it does seem to give good values of shielding even when rather small basis sets are used. The results of the GIAO method involve no choice of gauge. All the other results in Table 1 would be different for locations of the origin of the vector potential which are different from those used. The gauge location denoted by 'optim' in Table 1 for entry No. 4 was found using the criterion of Sadlej (see also Vol. 7) as giving the 'best' gauge choice for the chosen basis set.

Ab initio results for the mean value of the proton shielding constant of the water molecule are listed in Table 2 in order of increasing value. Most authors choose the oxygen nucleus or the molecular electronic centroid as the origin of the vector potential when calculating the proton shielding. The reason for this can be seen by noting the results when the origin is taken at the proton (entries Nos. 14 and 15). Even with the very large basis set of 47 Slater-type orbitals, the calculated proton shielding is still too high with this gauge choice. The best experimental value for the proton shielding is 30.052 (±0.015) p.p.m. [see Vol. 7, Chapter 1, equation (17)]. However, this includes the effects of rotational and vibrational motion which are estimated to reduce the shielding by about 1.6 p.p.m.

The two papers on water to be published during the review period are those of Lamanna et al. and Lazzeretti and Zanasi. The former authors used the 'equations-of-motion' method, taken from nuclear physics, to calculate the paramagnetic part of the shielding. As well as water they calculated the shielding in the N2 molecule obtaining a value of – 110.6 p.p.m. for the mean nitrogen shielding as compared with the experimental value of – 101(±20)p.p.m. Their work shows that the negative shielding ('antishielding') can be largely explained without the inclusion of electron correlation. Lazzeretti and Zanasi used a combination of uncoupled and coupled Hartree–Fock perturbation theory together with a wide variety of basis sets. With their best basis set they obtained the results for water given in Tables 1 and 2. They also calculated results for the 1H and 14N shielding in ammonia and the 1H and 13C shielding in methane. In all cases their results demonstrate the sensitivity of the computed shielding to the choice of basis set. We note here a calculation of the diamagnetic shielding constants of the water molecule performed by Woodruff and Wolfsberg using the SCF Xα SW/MT method.

For larger molecular systems Henry and Fliszar have made an ab initio study of the relation between the 13C shielding change and the change in charge density at the carbon nucleus for ethylenic carbon atoms. Using a STO-3G basis they found that the shielding increased with increase of charge density. This involves, of course, the adoption of a specific choice to obtain charge densities. This result for ethylenic carbon nuclei is in contrast to the situation predicted for the carbon nuclei of saturated hydrocarbons.

Electron correlation is seldom taken into account when calculating values of one-electron properties such as shielding. For atoms this would certainly appear to be justified by the results in two older papers, not previously mentioned in these reports. They calculated the correlation contributions, σc, to the shielding in several series of isoelectronic atoms. For the isoelectronic series with four and with ten electrons the results are given in Table 3, where σHF is the calculated Hartree–Fock shielding. Negative values of σc were found for a number of neutral atoms. Amos has shown for hydrogen fluoride by the multiconfiguration SCF method that electron correlation in the valence shell has little effect on the diamagnetic shielding at the proton or the fluorine nucleus.

Sterk and Suschnigg have performed a systematic study of the effect on <r-1> of increasing the number of gaussian orbitals in calculations on the acetylene molecule. They find that σd is little affected by the number of gaussians and that there is no general trend of σd with increasing number of gaussians. This is confirmed to a certain extent by the results in Tables 1 and 2 for water. However, it is to be noted from Table 2 that (for a gauge choice at the oxygen nucleus) the value of σd(H) appears to fall with extension of the basis set. Dixon et al. have performed ab initio calculations to obtain the dependence on nuclear geometry of the diamagnetic parts of the nuclear shielding in the bihalide ions FHF-, FHCl-, and ClHCl-. For FHF- a large basis set was used and σd(H) was found to be about 17O p.p.m. and very sensitive to the FH bond length, although less so to the FHF angle. Only minimal basis sets were used for FHCl- and C1HCl-.

As far as the present writer is aware there has been no ab initio calculation of the effect on a nuclear shielding of the presence of a perturbing molecule prior to the review period. However, in the last year two papers on this topic have appeared. Both studies are within the Hartree–Fock limit so that electron correlation is ignored. This simplification probably renders the results unsuitable for comparison with experiment since the dispersion forces are expected to play the largest part in determining the shielding change. For the CO/He system Sadlej et al. show that the valence shell repulsion leads to a deshielding of both carbon and oxygen nuclei. These shielding changes are larger for the linear arrangement of CO/He than for the perpendicular one. The helium shielding was found to increase for both linear configurations and to decrease slightly for the perpendicular one. The calculations of Jackowski et al. are for the systems CH4/He and CH4/CH4. Here the shieldings were found to vary 'non-monotonically' and to be quite sensitive to the choice of basis set.

An accurate calculation of the coefficients describing the linear and quadratic electric field dependence of the nuclear shielding in the H2 molecule was described in Vol. 7, although the results were published during the review period. General formulae for the evaluation of expectation values of shielding using Gaussian functions modified by appropriate Hermite polynomials have been given.