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...