Theoretical and Physical Aspects of Nuclear Shielding

BY CYNTHIA J. JAMESON

BY CYNTHIA J. JAMESON

1 Theoretical Aspects of Nuclear Shielding

A. General Theory — The relativistic analog of Ramsey's theory of nuclear magnetic shielding has been derived by Pyper and Pyykkö but no computed results based on these expressions have been pubUshed until recently. In the latter paper the relativistic corrections are determined by finding the nonrelativisitic limits of various matrix elements of the operators in the relativistic theory. The results show a shielding relativistic effect for the component perpendicular to the bond to X for 1H in H-X, and for C* in H-C [equivalent to] C*-X, and C*H3-X. The 'heavy atom shift' appears to be connected with a spin-orbit induced carbon spin density in mainly halogen (X)π1/2 MOs and a spin-orbit induced pi character in the highest occupied sigma MO.

General relationships between second order quantities and sum rules have been described by considering the electric and magnetic moments induced in a molecule in the presence of a radiation field. In the global approach taken by Lazzeretti and Zanasi all the tensors of the second order properties are obtained from the same set of transition amplitudes and excitation energies calculated by some method (such as random phase approximation (RPA) to the equations of motion methods (EOM). Results are given for shielding in the CH4 molecule:

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to be compared with experiment for the ground vibrational state

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and for the total shielding σe(13C) = 193.991, to be compared with experiment σe(13C) = 198.7.5

Another version of the paper (reviewed in this chapter, vol. 17 of this series) giving the parity nonconservation contribution to nuclear magnetic shielding has appeared, in which the left- and right-handed enantiomers have additional shielding terms ±σPNC, resulting in a splitting in each Une of a racemic mixture in an achiral solvent. Expected magnitudes of this contribution are very small even in the most favorable cases.

A theory of magnetic properties has been presented which is shown to reduce to other known approximate theories under certain conditions. The perturbed state which depends explicitly on the strength of the external magnetic field is determined variationally as a linear combination of Slater determinants associated with a gauge transformation factor. The gauge factor is taken to be a linear function of the position of the electrons with the coefficients being variational parameters. A global gauge function is used, not a different local gauge factor for each AO (Ditchfield's GIAO) or each MO (Schindler and Kutzelnigg's IGLO or Hansen and Bouman's LORG). Also, the gauge function does not depend solely on the external magnetic field. This method is said to ensure current conservation as well as gauge-invariance. As in other methods, when large basis sets of AOs are used, large systems of linear equations have to be solved. Some results are shown in Table 1 for H2O and CO.

B. Ab Initio Calculations — A large number of ab initio calculations of nuclear shielding tensors for heavy atoms have been reported in this review period, in molecular systems with many more heavy atoms than previously attempted. Coupled Hartree Fock methods used include standard CHF (common gauge origin) method, individual gauge for localized molecular orbitals (IGLO) method, and gauge-including atomic orbitals (GIAO) method. Sum-over-states calculations have also been reported. The very large number of systems studied preclude a report of the values obtained for the components of the shielding tensor. We list only the systems studied (in Tables 2, 3 and 4) and some comparisons of the calculated isotropic average shielding with experiments at the zero-pressure limit (Table 5) or with other calculations in a selected number of molecules (Tables 6 and 7).

In addition IGLO calculations of 13C chemical shift tensors in all the molecules shown in Figure 1 have been reported.

Some of the important general conclusions are as follows:

1. Local origin calculations such as IGLO invariably give superior agreement with experiment compared to standard CHF methods using a common gauge origin. The latter tend to underestimate the paramagnetic terms.

2. Basis sets including at least two sets of polarization functions for the second row atoms are necessary to obtain satisfactory agreement with experiments. Larger basis sets are required for 14,15N, 17O, 19F, than for 13C, for 31P than for 29Si. d functions are mandatory for 31P shielding: without a d function on P, calculations are off by almost 1000 ppm.

3. Lone pair contributions have a dominating role, they seem to determine the orientation of σ(33S) for example, and the lone pair contributions of the pyridine-like N in azines are so strongly anisotropic that their anisotropy governs the direction of the principal components of σ(14,15N).

4. There is some indication in some molecules that correlation effects may be important, i.e., the gap between experiment and calculation remains large even when the basis sets are expanded. Multiple bonds are usually involved. Examples are NNO (center 1BN is too deshielded by about 50 ppm compared to experiment), SO2 (17O is too deshielded by 160-170 ppm), NF3 (19F is too shielded by 55 ppm), NSF (14N is too deshielded by about 700 ppm), H2CNN (end 14N is too deshielded by about 150 ppm), and pyridine-like nitrogens show discrepancies of more than 120 ppm. For 13C in olefinic systems the components perpendicular to the double bond and in the plane of the molecule are 10 to 60 ppm less shielded than experiment whereas the components perpendicular to the double bond and in the plane of the molecule are 10 to 60 ppm less shielded than experiment, whereas the components perpendicular to the double bond and the molecular plane are too shielded by about the same amount leading to a fortuitously good agreement of σ.

5. There is some indication that part of the nonrelativistic heavy atom effects, in replacement of F by Cl for example, is accounted for even without relativistic corrections. This has been qualitatively described as a nephelauxetic effect.

6. Antisymmetric parts of σ (see this chapter, Vol. 13, this series), which affects the resonance position only in second order, are small in most cases. A possible candidate for the observation of the antisymmetric part is NSF. Here (σxy - σyx = 319 ppm for 19F and (σxy - σyx) = -735 ppm for 33S. Thus, the anisotropy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is lowered by 51.3 ppm if the antisymmetric part is included. The isotropic 19F resonance in this molecule is not affected by the antisymmetric a elements in first order.

When compared with experimental chemical shifts measured in condensed phases, IGLO calculations of 33S shielding (except for SO2) are usually too shielding by an average of +25 ppm, and about the same for 29Si and 31P. On the other hand, 19F values differ by 30-60 ppm on the average, but larger variations up to 100 ppm are not rare. This does not imply that the 19F calculations are poorer. Actually, when compared with better experimental data (gas taken to the zero-pressure limit) the difference between IGLO and experiment reduces to typically below 25 ppm. (See Table 5.) Data at the zero-pressure limit have no intermolecular effects but still have rovibrational effects, whereas theory gives σe, the value appropriate to a rigid configuration at the equilibrium intemuclear separations. Such comparisons are not possible for other nuclei since there are only condensed phase data for 29Si, 33S, and 31P. Experimentally it has been shown that condensed phases are usually deshielded compared to the isolated vibrating, rotating molecule, which is itself usually deshielded compared to σe.

Comparison with experimental trends, in search of some physical insight into the universal sagging curve patterns in which the shielding is smallest for an intermediate member, have motivated calculations of shielding along series of molecules related by replacements of one atom by another. Examples are 31P in the series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and 13C in CHnF4-n. The former has not yet been observed experimentally. The latter is a well-known example of sagging curves and it is disappointing that even for molecules as small as these, the ab initio calculations do not give as good agreement with experiment as might be expected, whether by IGLO, GIAO, or conventional CHF. For CHF3 molecule the deviation from experiment in the zeropressure limit is largest not only in the sagging series of 13C shielding but also in the monotonie series of 19F shielding values in CHnF4-n molecules. Another trend of practical importance is that of 29Si shielding in silicates. Calculations show the Si-O-Si angle increase is accompanied by increased shielding on 29Si, in agreement with empirical trends.

There are a sufficient number of calculations of shielding of nuclei involved in multiple bonds to present a clearer partitioning of the relative contributions of sigma and pi components of the double bond in the IGLO method where separate contributions from the localized MOs are obtained. The general observation is that the sigma component of a double bond contributes significantly more than the pi component to the paramagnetic part of the shielding, its magnitude and its anisotropy being larger than the pi and often larger than that of a lone pair contribution. This was found to be the case in benzene early on, and was the contention of Musher's arguments against the physical reality of ring currents. To a good approximation, the total shielding anisotropy of a nucleus is the stun of the anisotropies of its adjacent sigma bonds. Hence the largest sigma bond anisotropy will govern that of the whole tensor. The contribution to σP (the paramagnetic shielding) of a localized sigma bond can be resolved, locally, into components parallel and perpendicular to the bond axis. The local angular momentum operator l[parallel] gives no significant contribution to σP whereas l[perpendicular to] does. The relative importance of the various sigma bonds around a nucleus is governed by the difference in the MO energies and the availability of matching virtual orbitals which are related to the ground MOs by a rotation operation. Viewed in this way, the anisotropy of the shielding tensor of a nucleus X in H2X=XH2 will be largely determined by the geometry of the molecule, so that in ethene-type structures X will have principal axes along the same directions, and the relative magnitudes of the components will be similar to 13C in ethene. That is, the most shielded (smallest ap) component being perpendicular to the double bond and in the plane of the molecule, the intermediate component lying along the double bond. Table 8 shows the calculated components for X = 13C and 29Si. 31P shielding makes an interesting case because of the existence of a lone pair rather than a third sigma bond around P. This turns the two in-plane axes (of intermediate and largest of ct values) so that the intermediate one lies more or less along the lone pair axis (rather than along the double bond). Table 8 shows the calculated components for three P=P bonds in geometries of the type trans AP=PA, where A=H, CH3, or aryl. They have the same ordering and therefore the sign of the anisotropy is the same for all. This is most exaggerated for P=P, of course, where each P has two lone pairs. In going from a single X-X to a double bond, X=X there is a substantial change in the contribution of the XX bond but the changes in contributions of the lone pairs and of the XH bonds are of the same order of magnitude. This has been found for X=C, N, Si, and P. It is most pronounced in PP.

Comparison of observed 13C shielding in carbocations with calculations offer some evidence for classical or nonclassical types. The cations where intramolecular charge delocalization is possible to a large extent, that is, aromatic "nonclassical" and allylic cations, exhibit good agreement between theory and experiment. The cations with localized charge tend to have strong interactions with solvent molecules and counter ions. Not unexpectedly, for these ions carbon nuclei are more shielded than calculated by ab initio (IGLO) method. For cations with unknown geometry, comparison of experiment with the calculated 13C shielding for several proposed geometries can rule out certain ones. For example C4H7+ is assumed to be either a bicyclobutonium or a bisected cyclopropylcarbinyl ion. The latter is found inconsistent with observed 13C shielding. The former type of structure is also found for the methyl-substituted C4H7+, namely C4H6CH3+. 2-norbornyl cation 13C spectra are in good agreement with those computed for the nonclassical (symmetrically bridged) structure (1) and very different (by over 100 ppm in all cases) from classical structures. This appears to be the final word on the structure of the 2-norbomyl cation. Similarly, the 13C shieldings of 2-bicyclo[2.1.1]hexyl cation (2) is consistent with a bridged nonclassical structure, whereas the 2-methylbicyclo[2.1.1]hexyl cation is consistent with a classical structure and the l,2-dimethylbicyclo[2.1.1]hexyl cation is consistent with an equilibrium between two equivalent classical structures. Likewise IGLO calculations of 13C shielding is consistent with a nonclassical structure for the 1,3-dehydro-5,7-adamantanediyl dication (3).

"Strain increases shielding" is an empirical rule of thumb which has been found in hydrocarbons. IGLO calculations appear to be consistent with this. One example is P4 in which the PP bonds are strained, has the most shielded 31P among the many molecules studied theoretically during this review period. The 31P shielding in 1,2,3-triphenylphosphirene (4) is also high and has a very large anisotropy, Δσ = 694 ppm. Dimethyldioxirane (5) is a strained cyclic peroxide for which the 13C shielding is not too different from dioxane. IGLO calculations are said to agree with experiment.

At this stage of development of shielding computations, calculated shielding values should be reported as such rather them as a chemical shift from some calculated value for a reference molecule (which value the authors invariably fail to report). Such practices could be tolerated in the past when shielding calculations were so poor that only trends in the calculated values could be discussed. At the present stage it is no longer appropriate to suppress the actual results of the shielding calculations by reporting only shifts from some (usually) unreported number. The comparison of reported values with other calculations then become impossible.