Theoretical and Physical Aspects of Nuclear Shielding

BY CYNTHIA J. JAMESON

1 Introduction

This Chapter covers the same ground as the first Chapter of Volume 8 and the first part of the first Chapter of Volume 9, with slightly different emphasis according to the nature and distribution of original papers published during the review period June 1st 1979 to May 31st 1980. The theoretical formulation of shielding tensor components and questions of gauge invariance have been covered admirably by W. T. Raynes in past volumes of this Series and are not repeated here. Treated for the first time are the theoretical aspects of chiral shifts and the dependence of isotope shifts on structure.

2 Calculations of Nuclear Shielding

A. General Theory. — In this review period there has been a renewed interest in the current-density approach to nuclear shielding. The local shielded field Beff(r0) at the nucleus located at r0 is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where n is a unit vector in the direction of the magnetic field. The shielding field B'(r0) = – σB0 and the induced magnetic moment m are related to the current density j by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

In the semi-classical method of calculating magnetic shielding and magnetizability, the quantum-mechanical charge-density distribution of a molecule (obtained by some SCF procedure, for example) is input into the classical equations of motion of an electron density eρ(r). It has previously been shown. that the current density can be calculated from the classical equations of motion of the electron density by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where e, μ0, and me are the usual physical constants and U is the 'velocity potential' introduced by Salzer and Schmeidel. The paramagnetic current density is associated with ρ(r)[nabla]U, whereas the diamagnetic current density is associated with ρ(r × n). The equation of continuity for the currents induced by the external magnetic field

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

must be satisfied.

One might solve this equation by finding parametrized classes of functions U (variational method). Recently, Salzer developed a method of numerical solution of this partial differential equation and used the H2 molecule as a test case. His shielding tensor for H2 is in very good agreement with the results by Raynes et al. In molecules with complicated electron density functions, in which cases the adjustment of the variation function for the velocity potential U is difficult, the advantages of Salzer's procedure become evident. By use of a finer grid of points in those regions where the density function is rapidly changing, the numerical method can be applied to any charge-density distribution.

In another approach, the current density is expanded as a series in the magnetic field

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The magnetic shielding density function σN(r) is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where j(0) is zero for a diamagnetic molecule and j(1) is calculated using the coupled Hartree–Fock formulation.

The shielding density function is a surface in four-dimensional space. It is the analogue of the polarizability density function and the charge density. Upon integration these density functions yield the nuclear shielding, the polarizability, and the total number of electrons, respectively. Maps of surfaces that are plots of charge density on a particular plane in the molecule-fixed axes have been found to be useful in discussions of bonding and force constants. In the same way, shielding density maps could be used in discussions of chemical shifts. Shielding density maps have been calculated and discussed for the H atom, the F- ion, and the H and F nuclei in HF molecule. Shielding changes upon bond formation involving these species are interpreted in terms of density difference maps.

A fundamental relationship between the electronic g tensor of a molecule and the nuclear magnetic shielding tensor has been shown by Hegstrom. A theoretical formulation of the g tensor in its most general form is presented. The g tensor arises in the interaction μBS·g·B between the total electron spin of a molecule S and the external field B. Hegstrom shows that it can be written for a diatomic molecule AB as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

The first term is an isotropic correction to the free-electron g factor (ge) arising from the relativistic mass increase of the electron, the second term is a small anisotropic correction, and the third term arises from the spin-orbit coupling terms. Zn is the nuclear charge and σN is the nuclear shielding tensor. The theory is applied to the calculation of the electronic g tensors for the 2Σ ground states of the simplest paramagnetic molecule, H2, and its isotopic relatives HD+ and D2+. The components of the nuclear magnetic shielding tensor are calculated as functions of internuclear distance, tabulated, and compared with previous calculations. Finally, an interesting relationship between the second-order part (paramagnetic term) of the nuclear magnetic shielding tensor and the second-order part of the electron spin–rotation tensor is pointed out. This is exact for a one-electron molecule and may lead to useful approximations for a many-electron molecule. Note that for diamagnetic molecules the relationship between the second-order part of the nuclear magnetic shielding tensor and the nuclear spin–rotation tensor has been in wide use for many years as a means of obtaining an experimental measure of the former from accurate measurements of the latter. Such relationships between different magnetic properties are extremely useful because they provide consistency checks for experimental measurements of the different properties as well as for the corresponding theoretical calculations.

B. Ab Initio Calculations. — Coupled Hartree–Fock perturbation theory has been applied by Lazzeretti and Zanasi to evaluate second-order magnetic properties of the HCL, H2S, PH3, and SiH4 molecules. An efficient procedure for processing only symmetry distinct two-electron integrals in the coupled H–F algorithm is introduced, which leads to sizeable gains in computer time (roughly 3 to 12 times faster for the entire procedure than was achieved for previous calculations). This allows the use of very large basis sets, with particular attention to the polarization functions for the second row atoms. The paramagnetic contribution to nuclear shielding is dramatically dependent on the quality of the basis sets. With the improvement of basis sets the paramagnetic terms become more accurate and a higher degree of gauge dependence for the total shielding is achieved. The results for 35Cl and 31P can be compared with experiment and the agreement is good. The results for 1H show good agreement with experiment except for PH3. Keil and Ahlrichs reported coupled Hartree–Fock calculations of nuclear magnetic shielding in LiH, HF, and PH3 in its C3v and D3h structure using a 'well-polarized' basis. The average 31P shielding increases by 11.7 p.p.m. in going from the C3v to the D3h structure. Results of all-electron ab initio calculations of 31P magnetic shielding are compared in Table 1 with experimental (spin–rotation–derived) results and also with calculations using a pseudo-potential for the K and L shells. It is clear that the pseudo-potential approach leads to some error when compared with an all-electron calculation using the same set of valence shell functions. It is also clear that a flexible basis set with improved polarization functions on the heavy atom gives much better results.'

An ab initio calculation of the 31P shielding in the phosphate group has been carried out. Since the 31P nuclear shielding is widely used as a tool for the study of the backbone of nucleic acids and of the polar head of phospholipids, ab initio calculations of 31P shielding in model compounds of the biological phosphate groups are important. One of the applications of 31P n.m.r. that is of interest is the determination of the population of the gauche–gauche conformer (rotation angles about the P–O ester bonds both approximately 60°). Such an estimation is possible only if the chemical shift difference between the pure gauche–gauche (gg) and the pure gauche–trans (gt) conformations is known. While a mononucleotide with a pure gg phosphate group is known, there is no equivalent stable rigid compound with a gt phosphate group. For this reason theoretical calculations are necessary. The computations were carried out for X2PC4- anions, with one or both X = Me, H. The authors conclude that: (a) the geometry of the conformers seems to be the principal factor in the determination of the 31P chemical shielding, as it is for the energy difference between the two conformers, (b) the value of the chemical shift, σgg – σgt is most probably in the range – 3.5 to – 6.5 p.p.m., and (c) the paramagnetic component is the determining term for the calculated chemical shift variations. These calculations are able to reproduce correctly the order of magnitude of the anisotropy of the shielding tensor, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], as well as the orientation of each of its principal axes that have been measured for diester phosphates.

During this review period probably the most accurate calculations of first-order properties of the molecules N2 and CO have been reported. The results of configuration interaction calculations of the diamagnetic shielding for these molecules by Amos are compared with SCF and approximate calculations in Table 2. It has been known that the method of Flygare et al. gives very good agreement with ab initio calculations insofar as the isotropic average of σd is concerned, although it is somewhat less accurate in the prediction of the anisotropy of σd. In Table 2 we see that the average σd by all three methods are in agreement to within a few tenths of a p.p.m.

Ab initio calculations of the electric field dependence of nuclear magnetic shielding and magnetizability are reported. The components of the tensors σ(1) and σ(2) (discussed in an earlier volume) that describe the linear and quadratic electric field dependence of the nuclear magnetic shielding, were computed for H2 and HF molecules using a gauge-invariant double-finite-perturbation SCF approach.

C. Semi-empirical Calculations. — In this review period we have seen a number of intermediate neglect of differential overlap calculations of nuclear shielding. Dobosh, Ellis, and Chou have extended their earlier INDO models used to calculate chemical shifts for carbon, to proton shifts, the major change being the inclusion of all three-centre integrals in the calculation of the shielding tensors. The parametrization had to be changed once again, using 13C chemical shifts for a representative set of molecules (methane, ethane, ethylene, acetylene). The proton calculations appear to hold promise for examining small effects such as conformational and substituent effects on proton shielding. MINDO/3 calculations of 19F and 11B shielding in a variety of compounds show highly overestimated 11B results and a standard deviation of 69.7 p.p.m. in the comparison with experimental 19F chemical shifts over a range of 700 p.p.m. It is to be concluded that the MINDO/3 method is not suitable for shielding calculations. CNDO/S and INDO/S methods are specifically parametrized to account for excited states and are expected to have a better likelihood of success in calculating second-order properties. For 13C in several compounds containing cumulative double bonds, these two methods give standard deviations of 23 and 22 p.p.m. in comparison with experimental chemical shifts. Since the range of chemical shifts studied was 187 p.p.m., the correlation is really not better in a relative sense than the aforementioned 19F MINDO/3 calculations, although the absolute deviations are much smaller here. Semi-empirical calculations of nuclear shielding using magnetically active excitation energies from ultraviolet spectroscopy and photoelectron spectroscopy can be used to interpret 13C shielding differences. By regrouping the terms in the diamagnetic and paramagnetic contributions to shielding, one could consider (a) 'local' terms, which are large, and (b) distant contributions, which effectively cancel. It has been suggested that instead of lumping together all terms other than those involving orbitals centred on the nucleus in question (A):

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one could simply assume that nearest neighbours (L) matter but all others give cancelling contributions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

This is the 'atom-plus-ligand' approach. The first term is defined as having been calculated by using Flygare's method, including only nearest neighbours. When this term is subtracted from absolute shieldings defined by the 13CO spin-rotation measurement, one can get an empirical measure of the second term. It is the second term that is presumed to correlate inversely with excitation energies for families of compounds. There are indeed some correlations found in series of alkenes and fluoroalkenes, but as would normally be expected, the interpretations are not so simple.

As has been mentioned above, semi-empirical methods involve parametrization of some part or other of the calculation. In calculations involving the sum-overstates approach as opposed to the coupled Hartree–Fock approach, there is always the problem of having to use a limited number of states or use an average excitation energy. However, there is a model system that can be used to mimic real molecules, and which provides an exactly solvable quantum-mechanical system in an external magnetic field. The anisotropic harmonic oscillator is such a model system. The environment of a nucleus in a diatomic or linear polyatomic molecule is simulated by placing the nucleus along the long axis of the ellipsoid. The environment of a nucleus in benzene is simulated by placing the nucleus in the plane perpendicular to the short axis. Harris and Swope show that the magnetic shielding tensor of a nucleus in the presence of a charged anisotropic harmonic oscillator may be reduced to quadrature. The sum over states in the paramagnetic terms collapse to one term for the harmonic oscillator because when lx or ly operates on the ground state the resulting state is proportional to one of the excited states of the oscillator. Thus one may remove the energy denominator of that state and use completeness. In other words, the use of an average energy is exact for the harmonic oscillator. The nucleus is put where it is to donate a spin but not to interact coulombically with the electron. Thus the absolute magnitudes of shielding calculated with this model cannot be expected to be accurate. However, the ratios of the components to one another such as σxx/σzz, or (σin-plane/σzz are quite good for both H2 and benzene. Even though the model is purely geometric, it exemplifies the diamagnetic character of the shift in fields perpendicular to the plane of the molecule. The physical effect is easily understood in terms of angular momentum conservation in this model system, and there is no necessity to invoke aromaticity or other attributes.