Theoretical and Physical Aspects of Nuclear Shielding

BY CYNTHIA J. JAMESON

1 Theoretical Aspects of Nuclear Shielding

A. Ab Initio Calculations.- The equations-of-motion (EOM) method has been used previously in the calculation of second order magnetic properties. One well-known approximate solution of the motion equations is the random phase approximation (RPA). In this review period three methods of calculation of nuclear magnetic shielding using the framework of the RPA have been proposed. In one calculation localized molecular orbitals with local origins (LORG) are used, the others use a common origin. The equivalence of the RPA method and the Coupled Hartree-Fock (CHF) approach is demonstrated. Use of GIAOs in which each atomic orbital carries its own complex phase factor in the CHF and the Finite Perturbation Theory (FPT) have also been reported. The equivalence of the FPT and CHF methods has been previously established.

The LORG RPA method introduced by Hansen and Bouman has the following features.

(a) The occupied MOs are localized and each is associated with an origin vector relative to the magnetic nucleus.

(b) The choice of local origin distinguishes between contributions from bonds involving the magnetic nucleus for which the origin is placed at that nucleus, and distant bonds, for which the origins are identified with the corresponding molecular orbital centroids. The [??] integrals are calculated using not a common gauge origin but the origin associated with the occupied MO in the [??]/r3 integral.

(c) The "paramagnetic term" is calculated in terms of the RPA eigenvalues which can be obtained by direct Hermitian diagonalization of a matrix ([??] - [??]) including terms in the 2 electron integrals over the MOs.

(d) The "diamagnetic term" is calculated using the Hartree-Fock ground state. The diamagnetic term is not quite the same as that which would be calculated by the usual Ramsey expression. The diamagnetic shielding contribution from distant orbitals or groups are not directly included here. On the contrary, only the balance of diamagnetic and paramagnetic contributions from distant orbitals are included and this is shown to lead to an ~R-3 dependence. It is precisely this aspect of the method which reduces the long range basis set error contributions to the shielding. The combination of the RPA for the "paramagnetic term" and the Hartree-Fock ground state for the "diamagnetic term" is essential for cancellation of R-dependent contributions.

(e) The localization of the MOs and the use of local origins allow the decomposition of the total shielding into individual local bond contributions (i.e., involving only the molecular orbitals directly bonded to that nucleus) and "bond-bond" contributions involving all other bonds.

The results shown in Table 1 for 13C shielding by the LORG RPA method are very promising and are comparable to the extended basis set calculations using conventional CHF. Also shown in Table 1 are GIAO-FPT calculations using intermediate size basis sets, which are essentially triple-zeta with a set of d polarization functions on C, N, O, F atoms (a comparable basis to that used by Hansen and Bouman). The Individual Gauge Localized Orbital (IGLO)CHF method gives comparable results with a triple-zeta plus polarization basis, or larger as shown in Table 1. When a double zeta plus polarization basis set is used, the results are typically less accurate (more shielded) by about 20 ppm.

The success of the methods employing local origins in some way (an individual origin for each atomic orbital as originally introduced by Ditchfield, or an individual origin for each MO as introduced by Schindler and Kutzelnigg, or the particular local/long-range combination used in LORG by Hansen and Bouman, or even different origins for different pairs of orbitals proposed by Levy and Ridard) seems to be that when these methods are employed, the calculations leave out large terms of opposite sign which would have been exactly cancelling in the limit of a complete basis set. Thus, for a given basis set, use of local origins give an effective damping of basis set errors in long range contributions to the shielding, which leads to better agreement with experiment than a conventional CHF or FPT calculation using a common gauge origin. The results shown in Table 1 are of slightly different basis sets (as listed in the footnotes) but it is quite clear that the LORG, IGLO, and GIAO methods are each very effective and are comparable in general level of approximation and numerical accuracy. It would be interesting to have benchmark calculations on selected molecules using exactly the same basis sets and the same computer, in order to establish the comparative efficiencies of the different methods of calculation.

Incorporation of electron correlation effects in the EOM method is reputed to be more economical over CI procedures. RPA calculations of 13C shielding including double particle-hole pair interactions using a conventional common gauge origin and using a split-valence plus polarization (6-31G**) basis set are reported in Table 1. These molecules are of particular interest in that they are systems with conjugated or cumulated multiple bonds, systems which are notorious for their strong correlation in the ground state. Fortunately the 13C shielding tensors for the same set of molecules have been reported (see this chapter, previous SPR volume) and IGLO-CHF calculations with double zeta plus polarization are available as well. We compare these three in the first, third, and eighth column of numbers in Table 1. Here, the advantage of using local origins becomes obvious. The IGLO-CHF calculations give results which are closer to experiment in all systems, compared to the conventional RPA including double excitations. We have seen already that going to an improved basis set (triple zeta plus polarization) brings the IGLO result to within 1-4 ppm of the experimental 13C shielding value in CH4, HCN, HC [equivalent to] CH, CH2=CH2, CH3CH3, and H2C=O. The authors also report conventional CHF calculations using the same basis and find typically 20% difference between RPA with double excitation and conventional CHF, which they then attribute entirely to correlation effects. This is somewhat dangerous. The gauge origin problem associated with both calculations with limited basis sets lead to shieldings typically 60-100 ppm ("with correlation") and...