Theoretical and Physical Aspects of Nuclear Shielding

BY CYNTHIA J. JAMESON

1 Introduction

During this review period (June 1982 to May 1983) several papers on the relativistic theory of nuclear shielding have appeared. It is therefore appropriate that a large part of the section on general theory is devoted to this. Further developments in the shielding of conjugated systems, which yield some definitive conclusions on the physical aspects of the ring current, are described in Section 2B. Initial attempts at modelling the nuclear shielding of molecules (CO in particular) adsorbed on a surface are reported in Section 3D. A review of calculations of nuclear shielding has appeared.

2 Theoretical Aspects of Nuclear Shielding

A. General Theory. – In Volume 12 of this Series we reviewed the relativistic theory for magnetic susceptibilities and shielding formulated by Kolb, Johnson, and Shorer in terms of Dirac–Fock wavefunctions. In the solution of the coupled equations, the relativistic random phase approximation was used (which in the non-relativistic static field limit reduces to CHF theory). Applications were reported on some closed shell atoms and ions up to Z = 80. In this review period, apparently unaware of the work of Kolb et al., Pykkö and Pyper independently report formulations of relativistic theory of nuclear shielding.

We recall that in non-relativistic theory, the effective one-electron Hamiltonian in the absence of external fields and nuclear moment is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

in the local potential V and the nonlocal exchange potential VX. In a field this becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where — e is the charge electron, and A = A(r) is the combined vector potential of the external magnetic field and the nuclear magnetic moment

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

which can be written in terms of (1) and the remainder as a perturbation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

In relativistic theory the Dirac Hamiltonian is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where α and β are 4 × 4 matrices, which becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The perturbation is now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where Δ constitutes the change in the electrostatic potential due to the B0field induced changes in the charge distribution of the orbitals.

If the electronic ground state |0> and the excited states |n> are described by Dirac–Fock wavefunctions, i.e., single Slater determinants of four component molecular orbitals φi, each of which satisfy one-electron Dirac equations [equation (9)]. To calculate the shielding, we need to consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

energy terms that are linear in B0 and in μ, i.e., those terms in AB • AN coming from p • A in non-relativistic theory or from α • A in relativistic theory. In non relativistic theory such terms arise in first order from (e/m)2A2, so-called diamagnetic, and in second order from the product of matrix elements of p • AB and p • AN, so-called paramagnetic. In relativistic theory such terms arise in second order only, from the product of the matrix element of α • AB with a matrix element of α • AN. Thus, after decomposition of the relativistic current operator — eα for both diagonal and off-diagonal matrix elements, the Dirac current j = — eα has been exactly decomposed into (i) a convection current — eβ(p + eAB), which includes a magnetic field induced diamagnetic current, and (ii) an electron spin current — [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Both these currents interact with the nuclear vector potential in the usual dot product j • AN. The matrix elements of α • AB can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where K is the relativistic exchange operator. By a careful analysis of shielding in Dirac–Fock theory Pyper has shown that it predicts the nuclear shielding to consist of five terms:

(1) The diagmagnetic term σd. It differs from the non-relativistic term in that an extra factor of β is present. To lowest order in 1/c2 this extra factor of β arises from the reduction of the electron Larmor precession frequency caused by the velocity induced relativistic increase of electron mass. Although in relativistic theory there is no A2 term, such as that which gives rise to σd in non-relativistic theory, its counterpart arises from the relativistic equation when the positron-like states are summed over.

(2) The paramagnetic term σpA, which arises from the (1 + Σ) • B0 interaction. This is the exact analogue of Ramsey's σp in non-relativistic theory. It differs from the latter in only three respects: (a) in the substitution of relativistic for non-relativistic wavefunctions, (b) in the presence of factors of β (4 × 4 matrix) involving the change in the Bohr magneton caused by the relativistic increase in electron mass, and (c) in the presence of contributions from the interactions of the electron spin with both the external magnetic field, B0, and the nuclear vector potential, AN. The spin terms (c) vanish in the non-relativistic limit because a relativistic wavefunction for a closed shell system, unlike its non-relativistic analogue, is not an eigenstate of the total electron spin, although the expectation values of all components of the electron spin do vanish. Hence the spin as well as the orbital angular momentum can be partially unquenched by the B0 field, thus creating a magnetic field at the nucleus contributing to the shielding.

(3) The paramagnetic term σpE. This is purely relativistic in origin and does not have a non-relativistic counterpart. It involves expectation values over the Dirac–Fock ground state wavefunction only.

(4) The paramagnetic term σpX (the exchange term). This is also purely relativistic in origin involving orbital exchange operators.

(5) The self-consistency term σΔ which also occurs in non-relativistic theory, arises from the perturbation, Δ, constituting the change in the electrostatic potential due to the B0-field-induced changes in the charge distributions of the orbitals.

In addition to these five main terms, there are the Breit interaction terms, the Breit interaction being the leading relativistic correction to the Coulomb repulsion between a pair of electrons. The additional relativistic corrections to the nuclear shielding, which arise because the interaction between the orbital motion of the electrons and B0 causes the velocities of the electrons to change, thereby changing the Breit energy in a field-dependent way, have been derived by Pyper. The leading Breit corrections to the nuclear shielding are of the order of (1/c2) and can be interpreted as corrections to the relativistic orbit-orbit and spin-other orbit interaction. Terms up to the order of (1/c2) have been derived.

Earlier unpublished work on this subject has been described in abstracts of a symposium. A partial relativistic theory in which only some of the lowest order (to the order of 1/c2) relativistic corrections are included, was formulated earlier, by carrying Ramsey's treatment to third order perturbation theory, involving products of matrix elements of spin-orbit coupling, the interaction of the orbital angular momentum with the field B0 and the electron spin–nuclear spin coupling in the field-free molecular states. Even earlier was a partial relativistic theory of nuclear magnetic shielding in which only the larger two of the four component wavefunction are used. The application of this to a H-like atom gives a relativistic correction that is(– 1/15) times the non-relativistic shielding for the ground state. The relativistic theory of nuclear shielding for the H+2 molecule ion has been previously reviewed here.

The nuclear magnetic shielding tensor is represented, in general, by an asymmetric second rank tensor with nine independent components, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], whereas [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. These two expressions are not necessarily equivalent, as was pointed out by Buckingham and Pople in 1963 and by Buckingham and Malm in 1971. By inspection of Ramsey's formulae it becomes obvious that the diamagnetic term can be represented by a symmetric second-rank tensor with a maximum of six components, with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. However, the paramagnetic part is not necessarily symmetric. Following Buckingham's suggestion, the o tensor can be written as the sum of an isotropic part, a traceless symmetric part, and a traceless antisymmetric part [equation (11)]. The isotropic part is observed as the

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

scalar shift in liquids. If the resonance frequency is measured as a function of rotation angles about three mutually orthogonal axes fixed in a single crystal, the observed angular dependence of the frequency shift [equation (12)] yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

A, B, and C, which are, in general, quadratic functions of the shielding tensor components. It has been shown that if only terms which are linear in σij are included in the analysis of the angular variation of the resonance frequency, relying on the argument that σij [much less than] 1 and σ2ij [much less than] σij, then only the six components (the isotropic plus the traceless symmetric part) influence the results. This is due to the fact that the antisymmetric part of the shielding tensor affects the resonance position only in the second order. Conversely, information about the antisymmetric part is lost. It has been suggested that some spin–lattice relaxation time measurements may yield the sum of the squared antisymmetric components. The spin–lattice relaxation equation due to chemical shielding interactions is expressed in equation (13).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In order for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to be extracted from T1 measurements, the sum of (σa)2 terms must not be negligible compared to the (Δσ)2 terms. A recent calculation of the antisymmetric components gives the following results. Since the diamagnetic shielding is a symmetric tensor, the antisymmetric part σa can be obtained from equation (14), where [??]p is the transpose of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

σp. The results are shown in Table 1. The ratios of the (σa)2 terms to the (Δσ)2 terms are 2.2 × 10-2, 5.6 × 10-4, and 4.2 × 10-4 for 13C in CHFClCH3, CH3CHO, and NH2CHO, respectively. This calculation seems to indicate that the antisymmetric part of the shielding tensor may be too small to measure even for the heavy nuclei.

The effects of electron correlation on nuclear shielding can be calculated using a theory at the configuration interaction (CI) level. This was presented by Dabom and Handy, (reported in Volume 12 of this Series) and more recently by Fukui. For the F nucleus in HF, electron correlation has a shielding effect (+ 30 p.p.m.), and for the H nucleus in the same molecule, a deshielding effect (– 0.7 p.p.m.).

B. Ab Initio Calculations. – Ab initio calculations of nuclear shielding for molecules larger than benzene have previously only been possible with Ditchfield's GIAO method. The IGLO (individual gauge for localized orbitals) method (reviewed in Volume 12 of this Series) has been applied to the calculation of nuclear shielding (and magnetic susceptibilities) in hydrocarbons and other organic molecules. In this method, the gauge origin for the position vector r of an electron in a field-dependent molecular orbital is chosen at the centre of gravity Rk of the localized orbital to which it is related by the factor eiλk. That is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The local paramagnetic contributions calculated with these individual gauge origins are small since the localized molecular orbitals are nearly spherical, thus the errors in them have only a small effect on the final σ, which is obtained as the sum of a local diamagnetic term, o and a local paramagnetic term, σp. The results of IGLO calculations of 1H and 13C shieldings in about a hundred compounds are reported. Only chemical shifts relative to CH4 in the gas phase, rather than absolute shielding values, are shown. In most cases, the agreement with experiment appears to be better than that provided by other calculations. A particularly promising result of this paper is the apparent transferability of orbital contributions. Since the nuclear shielding is obtained by the IGLO method directly as sums over contributions of the localized orbitals, it is possible to assign specific contributions from various parts of the molecule and develop an ab initio incremental system for shielding. For 1H, the most important contribution is from the bonding orbital to the carbon to which the proton is attached. One order of magnitude smaller are the contributions of those bonds that directly involve this carbon; all other contributions are smaller by two orders of magnitude, except in cyclic systems. Of course, empirical incremental systems have been in use for some time, but this paper shows that increments with a definite physical meaning can be defined. The IGLO method can be very useful in gaining insight into the shielding tensor by looking at the individual contributions from localized orbitals in selected molecules, more so than in reproducing large numbers of experimental data. For example, it would be instructive to see the ab initio increments which make up the components of the shielding tensors in the H3C–CH3, H2C=CH2, and HC [equivalent to] CH series and the CHnF4-n and C2HnF4-n series.