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...