Nuclear Magnetic Resonance Volume 11

A Review of the Literature published between June 1980 and May 1981

By G. A. Webb

The Royal Society of Chemistry

Copyright © 1982 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-0-85186-342-9

Contents

Chapter 1 Theoretical and Physical Aspects of Nuclear Shielding By Cynthia J. Jameson, 1,
Chapter 2 Applications of Nuclear Shielding By G. E. Hawkes, 22,
Chapter 3 Theoretical Aspects of Spin–Spin Couplings By J. Kowalewski, 55,
Chapter 4 Applications of Spin–Spin Couplings By D. F. Ewing, 71,
Chapter 5 Nuclear Spin Relaxation in Fluids By A. Kratochwill, 106,
Chapter 6 Solid State N.M.R. By G. R. Hays, 128,
Chapter 7 Multiple Resonance By W. McFarlane and D. S. Rycroft, 157,
Chapter 8 Natural Macromolecules By D. B. Davies, 179,
Chapter 9 Synthetic Macromolecules By J. R. Ebdon, 205,
Chapter 10 Conformational Analysis By F. G. Riddell, 225,
Chapter 11 Oriented Molecules By C. L. Khetrapal and A. C. Kunwar, 248,
Chapter 12 Heterogeneous Systems By W. Derbyshire, 264,

CHAPTER 1

Theoretical and Physical Aspects of Nuclear Shielding

BY CYNTHIA J. JAMESON

1 Introduction

This chapter primarily reviews calculations of nuclear shielding and such experimental data as components of shielding tensors, vibrational-rotational, and intermolecular effects, which are of interest to theoreticians. Papers published in the period June 1980 to May 1981 are included. Although paramagnetic systems are not normally covered in this chapter, theoretical aspects of chemical shifts in paramagnetic systems are considered relevant and have been included this year. Isotope effects on nuclear shielding have been widely applied; theoretical aspects and such data as may be considered fundamental to its understanding are covered here. The first Section deals entirely with calculations. However, all calculations which pertain to rovibrational, isotope, and intermolecular effects are deferred until the second Section.

2 Theoretical Aspects of Nuclear Shielding

A. General Theory. — The diamagnetic shielding, σd(N), of nucleus N in a molecule with n electrons, which is represented by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

at the equilibrium configuration, has been shown to be given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where RNN are the equilibrium distances in the molecule, μ is the negative of the electronegativity of the molecule (as defined by Parr et al.),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

W is the total Born–Oppenheimer energy of the molecule as a function of the atomic numbers ZN’ and the net molecular charge Q, and e denotes the equilibrium configuration. The formula given by equation (2) is exact; it enables the calculation of σd(N) if We(ZN, ZNs Q), μ and the equilibrium geometry are known. Ray and Parr demonstrate the use of equation (2) for calculations of the diamagnetic shielding of atoms in molecules by computing σd for a series of molecules and comparing their results with those of ab initio calculations. The results are very good even when the electronegativity is neglected. Since the individual atom is a special case,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

We can in fact express the diamagnetic shielding in the molecule as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

by neglecting the derivative of the dissociation energy. Equation (5) is a commonly used method of estimating the diamagnetic shielding in molecules.

It has been shown that the second-order energy of the coupled Hartree–Fock theory can also be written in a simple sum-over-states perturbation formula. Fukui, Yoshida, and Miura have used this expression to write a formula for the nuclear shielding tensor that should be equivalent to the coupled Hartree–Fock formulae. The components of the diamagnetic term for the α component of μN and the β component of the applied field are as usual,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where r0 denotes the distance vector from the gauge origin to an electron and rN is that from the nucleus N of interest to the electron; |i> is the ith molecular orbital obtained by solving the Hartree–Fock equation. Components of the paramagnetic shielding tensor, σpαβ are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where the summation, n, runs over all excited singlet states, |n>, with excitation energies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the coefficient of the singlet excitation i ->a in the above excited state |n>. Tn and |n> are given as the nth eigenvalue and eigenvector, respectively, of the Hamiltonian matrix whose elements are defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the doubly excited singlet configuration

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The lN and l0 operators in equation (7) are defined as rN × [nabla] and r0 × [nabla], respectively. Using the SCF MO energies, εi associated with the molecular orbitals ψi, and the molecular electron repulsion integrals, Fukui et al. rewrite the matrix elements of H as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The real symmetric matrix H is diagonalized to give the diagonal matrix T by the orthogonal matrix V, whose nth column vector gives the coefficients Vi- >a,n in equation (7). Thus, HV = VT.

B. The general concept of a density function for a molecular electronic property was considered and the origin dependence of some electric and magnetic property densities investigated, the nuclear magnetic shielding density function being used as a typical example, by Jameson and Buckingham. Density difference functions have been defined and calculated for some simple cases. For example, the shielding density difference function due to molecular bond formation is the function resulting by taking differences at each point in space of the shielding density for a nucleus in the molecule and the nucleus in the free atom. This function, when plotted, provides a way of viewing the polarizations and distortions in the shielding density upon molecule formation, and provides a basis for discussion of contributions to the nuclear shielding change from various parts of the molecule. When integrated over all space this difference function results in a single number: the shielding difference between the molecule and the free atom. Density difference maps are also shown for the HF molecule upon bond extension, that is for HF at Re + 0.1 a.u. minus that for HF at the equilibrium distance, for both 1H and 19F shielding. This difference map provides a physical interpretation of the derivative [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

B. Ab Initio Calculations. — The most accurate calculations (to date) of the nuclear magnetic shielding constants for the hydrides of the elements of the first and second rows of the periodic table have been recently carried out. These are coupled Hartree–Fock calculations using basis sets that are satisfactory with respect to gauge invariance. Some of the results are summarized in Tables 1 and 2. Closure relations for choosing an optimum gauge origin for nuclear shielding were tested. For the heavy-atom shielding, the origin of the best gauge is situated on the heavy atom and for 1H it is found to be very close to the heavy atom. If a reasonable gauge origin (as given by the closure relations) is chosen, much smaller basis sets can be used. The nuclear magnetic shielding tensors for the simplest hydrocarbons are reported. The results are shown in Table 3 and compared with experimental data and other theoretical calculations. The z axis is taken to be the unique axis. In the case of ethylene z is along the C — C bond and the x axis is perpendicular to the plane of the molecule. The magnetic susceptibility tensors for some hydrides and hydrocarbons are also reported. There has been some ambiguity about the magnitude and sign of the anisotropy of the magnetic susceptibility tensor, which has been used for empirical calculations of neighbour magnetic anisotropy contributions to nuclear shielding. The results shown in Table 3 for the components of the magnetic susceptibility tensor are probably the most accurate calculations available for these quantities and better than the available experimental values.

The semi-classical approach for calculation of the nuclear magnetic shielding and magnetic susceptibility tensors presented by Schmiedel for which a finite element method of numerical solution has been demonstrated (reviewed in Volume 10), has been applied to polyatomic molecules. The procedure is a gauge-invariant calculation, which only requires a knowledge of the unperturbed electron density. The numerical solution is based on a partitioning of the electron density into localized parts. Since the basis set for the unperturbed electron densities used for the first attempt to apply this semi-classical method to polyatomic molecules is a relatively limited one, the results cannot be expected to be very good. Small differences in unperturbed electron densities can lead to significant differences in the magnetic susceptibility and shielding tensors calculated from them by this semi-classical approach. Only the results for proton shielding are included in Table 3. The 13C shielding calculations do not give results that are comparable to the quality of the other theoretical values shown in Table 3. The relationship of the semi-classical approach to uncoupled Hartree–Fock perturbation theory and to the variational method of Rebane is also shown.

Ab initio calculations of the one-electron properties, including the diamagnetic shielding for all nuclei, have been carried out for the COF2, SO2F2, and SOF2molecules. For COF2 the paramagnetic shielding for F has been calculated previously from the spin–rotation constant obtained from hyperfine structure in the rotational transitions measured in a molecular beam maser spectrometer. This result combined with the 19F diamagnetic shielding yields the following components of the 19F shielding tensor in COF2: σzz = 178.62, σyy = 281.88, and σxx = 230.46 p.p.m., where z is parallel to the C=0 direction and x is perpendicular to the plane of the molecule. The error limits on σ are determined by the precision of the spin–rotation data, of course, which translates to ±10 p.p.m. in the nulear shielding components. The estimates of σd, obtained by using the Flygare method, can be compared with these recent values: σxx = 637.64 p.p.m., ab initio (633.5 p.p.m., Flygare), σyy = 543.59 (542.4), σzz = 584.91 (582.3). We see that the approximate method of Flygare, which is easily applied, given only the equilibrium geometry of the molecule, continues to hold up as a very good approximation.

C. Semi-empirical Calculations. — The relationship of the paramagnetic term in the shielding (with the gauge origin at the nucleus of interest) to the nuclear quadrupole coupling constant (which is a measure of the electric-field gradient at the nucleus) was expressed in 1954. Both are essentially dependent upon the ‘fraction of the unbalanced p electrons on the atom in question’. It is not surprising then that Na chemical shifts for each of the (Na+)LnL’4-n species, where L are various ligands, are directly proportional to the square root of the observed linewidths normalized to unit viscosity. In this case linewidths depend on the electric-field gradient, which shares a common origin with σp: namely injection of electrons donated by the ligands into 3p orbitals on the Na+ ion, according to a theoretical interpretation of the 23Na chemical shifts.

In the past several years a variety of carbocations in solution have been prepared and characterized. Differentiation between carbocations with classical or non-classical structures by a range of physical methods may or may not lead to unequivocal conclusions. For a pair of degenerate equilibrating classical carbonium ions, the observed nuclear shieldings are averages of the shielding for various geometries from one extreme static structure to the other. For non-classical ions, however, the observed shielding is merely averaged over the very small displacements undergone by any rotating-vibrating molecule. N.m.r. allows the unambiguous determination of the cation structure provided its spectrum corresponds to an asymmetrical classical structure. Some cations, however, have spectra characteristic of a symmetrical structure down to the lowest temperature of observation, so that either a non-classical symmetrical structure or rearrangements of classical structures with a frequency exceeding the frequency difference of the averaged n.m.r. lines are possible interpretations. It could be concluded that an ion has a non-classical spectrum if its spectrum differs from the average spectra of classical structures. The problem in applying this criterion is that the n.m.r. spectra of the limiting unobserved classical structures are not known and the nature of the averaging motion has to be assumed. Thus, calculations of nuclear shielding will allow, in principle, the differentiation to be made. One calculation for the non-classical structure, calculations for the limiting classical structures and for an assumed character of the motion, and calculations for the structures through which one limiting classical structure passes into the other one need to be carried out. Cheremisin and Schastnev have carried out semi-empirical (INDO) calculations of 13C nuclear shielding in some typical carbonium ions with this strategy as the ideal approach. In practice, only the average between the 13C shieldings in the two limiting structures is reported. Since there are usually several 13C nuclear sites whose shieldings relative to CH4 can be compared with experiment, there is some possibility of success. In some cases they arrive at definite conclusions that agree with evidence from other physical methods. In others no definite assignment is made. The classes of cations considered include the 2-norbornyl, 7-norbornenyl, 7-norbornadienyl, bicyclohexyl, and cyclopropyl cations, as well as some simpler cations. The validity of their conclusions may be affected in part by solvent effects, which are unaccounted for, and the improper averaging of just the two limiting structures in the cations undergoing rearrangement. In addition, the bond lengths and angles of the non-classical cations may be incorrectly estimated and the method of summation over states in the INDO approximation may be inadequate for the sometimes subtle distinctions in 13C shieldings being made.

In approximate calculations of nuclear magnetic shielding it is customary to consider the total shielding as being partitioned into separate contributions, e.g., ring currents, electric field, and magnetic anisotropy effects from various parts of the molecule. Calculations of only one or two contributions are then undertaken for a series of related compounds, in which all other contributions, if this is to be believed, remain the same. The approach is to try to find a dominant varying contribution that is independent of the varying structural details of the molecules in the series. This seems to work when the series of compounds studied are sufficiently similar and when the shielding changes are indeed largely due to the contribution upon which attention is being fixed. This seems to be a sensible approach for long-range effects. In this review period several calculations of this nature have been reported. Cheremisin and Schastnev have introduced a method of calculating the contributions to the nuclear shielding from spin-orbit interactions by third-order perturbation theory. Allowing for the perturbation terms in L•S, L•B, and S•B [interaction between spin and orbital angular momentum, between orbital (or spin) angular momentum and the magnetic field] in the molecular Hamiltonian, the wavefunction of the ground state ψ(B) depends on the external magnetic field and owing to the spin-orbital interaction the singlet ground state has a small admixture of triplet components. The hyperfine contribution to the energy of the system can be obtained by averaging the hyperfine interaction operator (contact and dipolar terms in the interaction of electron spin with nuclear spin angular momentum) over the wavefunction ψ'(B). In the case of the halogen hydrides, HBr and HI, the spin-orbital contributions to the 1H shielding have been shown to be comparable with the diamagnetic and paramagnetic contributions. In the case of 13C shielding in halogen-substituted methanes, CHnX4-n (X = Cl, Br, I), the above method, applied in the INDO approximation using gauge-dependent atomic orbitals, leads to contributions which are not additive; for example, 6.2, 15.2, 35.6, and 76.3 p.p.m. in going from CH3Br to CBr4. These contributions constitute a large fraction of the total 13C shielding relative to CH4 in the cases of Br and I substitution. The results obtained describe quantitatively 13C shielding changes in the halomethanes of the type CHnX4-n as n is varied. It would have been interesting to see if the data on the mixed halomethanes (CXnY4-n; X, Y = F, Cl, Br, I) could also be reproduced. Perhaps the authors are already working on these. Nearly all the spin–orbit contribution comes from the contact or isotropic term rather than the dipolar one.

(Continues…)Excerpted from Nuclear Magnetic Resonance Volume 11 by G. A. Webb. Copyright © 1982 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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