Theoretical and Physical Aspects of Nuclear Shielding

BY CYNTHIA J. JAMESON

1 Introduction

This chapter reviews the papers published during the period June 1st 1981 to May 31st 1982. The first part deals with general theory and calculations of the components of nuclear magnetic shielding. The second part reviews experimental data on the anisotropy of the shielding tensor, effects of rotation and vibration, isotope effects, intermolecular effects, and the theoretical calculations in support of these physical aspects. Several new developments in the theory of nuclear shielding are discussed in Section 2A (gauge invariant techniques, relativistic calculations, and inclusion of correlation effects on σ by perturbed configuration interaction methods). All examples given are ab initio calculations which are also discussed in this section. Other ab initio calculations using standard methods are reviewed in Section 2B. The theoretical treatment of the variation of nuclear magnetic shielding with internal co-ordinates (bond lengths and bond angles) is discussed in Section 3B since this is intimately connected with the rotational and vibrational averaging which gives rise to the temperature dependence of nuclear magnetic shielding in the isolated molecule and the mass dependence of this rovibrational average (isotope shifts).

2 Theoretical Aspects of Nuclear Shielding

A. General Theory. — It has been shown that it is possible to obtain very accurate values of the nuclear magnetic shielding provided that the computation is carried out using a very large AO basis set. Some examples of these were reviewed in Chapter 1 of Volume 11 of this series. In principle, σ is gauge invariant, that is, its value is independent of the choice of origin if Ramsey's theory is carried out with a complete basis set, and if the energies used in the perturbation expansion are obtained by diagonalizing exactly the exact electronic Hamiltonian with zero external fields (i.e., if the wavefunctions used are exact Hartree-Fock functions). In practice, of course, a truncated basis set is used and the energies are obtained in some approximate way. The approximations to σ, obtained with finite LCAO bases, converge very slowly when these bases are enlarged. Therefore, calculations carried out in the standard manner (by the coupled Hartree–Fock method) lead to different results for different choices of gauge origin and these differences (gauge dependence) are especially pronounced when the basis set used is small. It has been found that with a judicious choice of gauge origin, for example the centre of electronic charge, the results are in better agreement with experiment. Tests for gauge dependence have been devised to determine a priori the 'goodness' of a shielding calculation, that is, the adequacy of the basis set used, without resorting to comparison with experiment. For example, the sum rules [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where N is the number of electrons, or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], etc. can be applied.

A different approach proposed by Ditchfield avoided the choice of gauge origin altogether by using field-dependent atomic orbitals (GIAO). It now becomes clear that this approach is only one of several possible approaches to a gauge independent calculation. These are gauge-independent in the sense that (a) the gauge origins are predetermined, and (b) not a single origin is chosen but several origins. Three different approaches have been used:

(1) different gauge origins for different atomic orbitals – GIAO, by Ditchfield;

(2) different gauge origins for different localized molecular orbitals – IGLO, by Schindler and Kutzelnigg;

(3) different gauge origins for different pairs of orbitals (AOs or MOs), by Levy and Ridard.

The first is well known by its many applications to various small molecules. The last two have been reported during this review period.

The physical basis for the improvement in these methods over the standard CHF method has not been demonstrated. The improvements probably result because an appropriate gauge origin choice reduces both the diamagnetic and paramagnetic contributions to shielding, thereby minimizing the paramagnetic part which is more difficult to calculate accurately. Thus, by choosing in each instance (for each AO or for each localized MO) an appropriate origin, the total error can be reduced. In these approximations to Ramsey's theory the nuclear magnetic shielding is calculated using a single configuration (Slater determinant) for the unperturbed wavefunction.

In the method of Schindler and Kutzelnigg,individual gauge for localized orbitals (IGLO), calculations are carried out with a new molecular orbital ψk that is related to (φk (molecular orbitals which are expanded in powers of field strength, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) by the following relationship: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where Λk is some local multiplicative operator proportional to the field strength.

For comparison we show here the general form of the expression for the nuclear magnetic shielding in Schindler and Kutzelnigg's method and the standard CHF method. In the standard CHF method the shielding of a nucleus with magnetic moment μ at the position ρ is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

φk0 are the occupied molecular orbitals of the ground electronic state of the 2n electron system and are eigenfunctions of the Fock operator with eigenvalues εk. φk1 are the molecular orbitals [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with Cpk(1) being the coefficients obtained by solving the simultaneous equations (the coupled Hartree–Fock equations):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

and (qi|pj) are the two electron repulsion integrals. Note that the common gauge origin is at R. Schindler and Kutzelnigg's analogous expression for the shielding is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

ψk0 = φk0 are the occupied molecular orbitals but are not necessarily eigenfunctions of the Fock operator, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the solutions to a set of simultaneous equations which may be considered as analogues of equation (4) with operators transformed by eiΛk and matrix elements constructed in ψk. Note that the gauge origin is at Rk, a different origin for each molecular orbital φk. The advantage of switching from the φk to the ψk orbitals becomes obvious when Λk is chosen so that one has the optimum gauge origin for ψk. With the choice

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

in which r is the usual electron position vector in the molecular co-ordinate system, and Rk is taken as the position vector of the centroid of charge of the molecular orbital ψk0 = φk0, one has a much better gauge origin for this orbital than a common origin for all orbitals. Consequently, one gets 'local' diamagnetic and paramagnetic contributions, of which the latter are rather small. In the standard CHF calculation, with a common gauge origin for all the orbitals, large diamagnetic and paramagnetic contributions are obtained which cancel each other to a large extent. The advantage of having 'local' terms with small errors in Schindler and Kutzelnigg's theory comes at the expense of having to calculate many additional terms. Expressions for these 'exchange corrections' and 'resonance corrections' are given and further approximations are made in their calculations.

The results of Schindler and Kutzelnigg are very encouraging; some of them are given in Tables 1 and 2 for comparison with other theoretical calculations reported during this review period. In addition, these authors carried out calculations for the following systems: BH, BH3, B2H6, CH4, H2CO, NH3, and F2 all at the equilibrium geometry; (H2)2 in the D2h configuration, as a function of distance between H2 bond centers, CH3+ in the C3v configuration as a function of angle and bond length.

In the third method, different gauge origins are used for different pairs of orbitals (AOs or MOs) by Levy and Ridard. The second order energy is written in terms of pair contributions. In the diamagnetic term there are only two orbital indices involved so that the pairs are chosen uniquely. In the paramagnetic term there are four orbital indices and an arbitrary choice of pairs is made [equation (10)]. The

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

different pair contributions are not physical observables so that the theory is not gauge invariant. If each pair contribution is calculated with a specific gauge origin, Gij, using a truncated basis set, then the value of the pair contribution obtained using the specific gauge origin Gij, Eij(Gij), is related to the value Eij obtained using a common gauge origin, by Eij = Eij(Gij) + ρij, with a non- vanishing correction ρij for each pair. Levy and Ridard then give a recipe for computing the sum of the pair corrections. In practice, they compute the second order energy in some basis in a standard (CHF) way and then subtract the value of the correction [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] obtained by use of the orbital pair model in this basis. It is hoped that doing this subtraction will correct to a large extent the effect of the truncated basis and thus will give a result nearly equivalent to that obtained by using a very large basis set. The numerical test was carried out on PH3. From previous calculations (reported in Volume 10 of this Series) the 31P and 1H shielding values from 78- and 83-function basis sets provide a standard large basis calculation. The present calculation uses 37 basis functions. The individual gauge origins Gij are chosen at the midpoint between the centres of the main components φi, and φj. The results show that the agreement between the calculated values using 37 functions with pair correction and those using 78 and 83 functions is good. Three different calculations with the 37 basis functions using a common origin in each calculation are carried out. Results obtained for 1H are 88.0, 25.0, and 3.1 p.p.m. with the origin at H, at P and at 1.8 a0 above P along the symmetry axis respectively, using standard CHF theory. With the orbital pair theory the values obtained are 31.6, 22.4, and 19.1 respectively. The experimental value is 28.3 p.p.m. We note that this model does not make an adequate correction; the corrected values still show some gauge dependence.

In all the methods discussed above, one is solving directly either by iterative techniques or by matrix solution of simultaneous equations, for that part of an electronic energy eigenfunction 'P (in terms of MOs φk1) which is first order in the uniform magnetic field B. A different approach is suggested by Parker: find a way of transforming the equation for Ψ(1) into an equation for a new unknown function f which, unlike Ψ(1), is independent of the choice of gauge. The transformation depends on the physical assumption that when B is applied parallel to one of the principal axes of the dipole magnetizability tensor (χ), the induced current density tends to be perpendicular to the magnetic field. This method is applied to a calculation of χ in the context of the independent electron approximation. The equation to be solved in the function f is in terms of the normalized zero-field probability density obtained from a non-degenerate real zero-field eigenfunction Ψ0 a 3-dimensional density P0 (x, y, z):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where [nabla]t is the transverse gradient, that part of [nabla] which is perpendicular to B, the magnetic field along principal axis α. For calculations of nuclear shielding, this equation is transformed further to an equivalent form which is not too sensitive to errors in fα near the shielded nucleus at r = 0. The 3-dimensional density P0 (x, y, z) is replaced by a series of 2-dimensional densities R(xs, y, z) characteristic of slices through a molecule perpendicular to the axis α at positions xs = x1, x2, ..., so that R(xs, y, z) dydz is the conditional probability of finding the electron in dydz given that it is in the slice at xs. The results for H2, using a 2-term function and a 5-term function, give shielding values which are in very good agreement with those of a 48-function calculation (see Table 1). The advantage of this method appears to be that the shielding calculation carried out with a P0 value taken from a very high quality SCF (or better) wavefunction, is no more complicated than one with a P0 value from a small basis set. Byproducts of the calculation (from solving the equation for fα in slices) are contour maps of the modulus and the direction of current densities in the molecule for a magnetic field along a principal axis.

The CHF technique has been generalized to treat atoms and ions of high nuclear charge where the relativistic effects as well as correlations are expected to be important. The method is the relativistic random phase approximation (RRPA) which in the non-relativistic limit reduces to the random phase approximation with exchange. The latter theory describes the linear response of an atom to a dynamic external field, and reduces to the CHF theory in a static uniform field. The magnetic shielding for closed shell atoms Xe, Pd, Kr, Ar, Ne, He as well as their closed shell ions with the same number of electrons: 46, 36, 18, 10, 2 [for example Xe(0), Xe(+8), Xe(+18), Xe(+44) and Xe(+52)] have been reported together with the entire Ne ioselectronic sequence, from Ne(0) (σ = 558.6 × 10-10) to U(+82) (σ = 1.744 × 10-2). The non-relativistic approximation to σ is also shown for these systems. For Ne(0) and U(+82) they are 553.4 × 10-6 and 0·8225 × 10-2, respectively. We note that the relativistic calculations become necessary for highly charged ions of moderate atomic number Z as well as atoms of high Z.