CHAPTER 1
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 120 to 180 ppm ("without correlation") greater than experiment. It is therefore not possible to say how large the correlation effects on ^^C shielding are from this comparison. In fact, correlation effects on shielding have been found to be small in systems where accurate calculations have been carried out, and as we can see in Table 1, the equivalent RPA/CHF/FPT do reproduce experimental values quite well, to the extent that the experimental absolute shielding values can be ascertained.
A new approach to the calculation of magnetic properties is to use the torque operator rather than the angular momentum operator in the perturbed Hamiltonian. The relative torque formalism is then used to obtain the equations for magnetic succeptibility and nuclear magnetic shielding and the conditions for gauge invariance are discussed. Very general sum rules and other quantum mechanical relations are restated in this theoretical framework. The application within the RPA approximation to the HF molecule shows that with a near-Hartree-Fock basis set the numerical results are very close in the case of 19F for different gauges (on H or on F), while they differ by about 4 ppm for 1H. Agreement of the 19F shielding tensor with experiment is excellent. The results using both formalisms are in good agreement with each other: -99.211 ppm vs. -102.760 ppm for σ[perpendicular to]p in the torque and angular momentum formalism respectively.
Shielding calculations by an approximate molecular orbital method (Xα-scattered wave) have been formulated in two ways. Both use the sum over states (SOS) uncoupled method of calculating shielding. The Xα-SW calculations fare less well compared to the others in Table 1. Some part of the discrepancy between the calculated values and experiment are due to the approximations inherent in the Xα method. However, even so, the diamagnetic terms are 9399% of ab initio values, and for 13C the paramagnetic parts are only 15 to 30 ppm poorer than the 4-31G calculations by FPT using normal gaugeless atomic orbitals. For 15N 17O and 19F the paramagnetic terms are closer to the experimental values (from spin rotation constants) than other uncoupled Hartree-Fock calculations. It is highly likely that the accuracy of the results will become comparable to the other good calculations if one of the coupled local origin approaches (rather than the common origin conventional sum over states (SOS) uncoupled approach that was used) is adopted. The advantages of the numerical Xa method (highly accurate numerical description of the radial dependence of the MOs near the nuclei and suitability for large molecular systems) may then become apparent.
Ab initio calculations of 15N, 17O, and 19F shielding in small molecules with the GIAO FPT method yield results which are in reasonably good agreement with experiment. These are compared with other calculations in Table 2. None of the methods is entirely as successful for these nuclei as they are for 13C. It should be noted in Table 2 that the calculated shielding of 17O and 15N nuclei with lone pairs give worse agreement with experiment than 13C and 10F shielding calculations. The paramagnetic term is badly underestimated by an uncoupled calculation; the Xα results in Table 2indicate this.
29Si shielding calculations have not yet reached the level of accuracy of the 13C calculations. Nevertheless, these are of particular interest especially to NMR chemists studying zeolites. The empirical patterns of shielding changes with Si-0 bond distance and with Si-O-Si bond angles are very useful for characterizing solid silicates. Unfortunately, any interpretation of the most interesting chemical shift trends would require calculations on at least (HO)3Si-O-Si(OH)3 which has 12 "heavy" atoms of which 2 are second row! At the present time such calculations with the necessary triple-zeta-plus polarization basis sets are prohibitive, particularly when several geometries have to be considered. In this review period two ab initio 29Si shielding calculations have been reported. The results do not yet bear on the pressing questions of interest but do provide a good beginning. In one calculation SiF4, SiF5-, SiF62- and two geometries of SiO4- 4 have been considered, using conventional CHF with near-Hartree-Fock quality basis sets, the biggest being [6s5p3d] on Si and [4s3pld] on F or O. Here again, more accurate results with the same basis sets could be obtained by using local origins. No absolute shielding scale is yet available for 29Si so that comparisons with experiment have to wait. Nevertheless there are some indications of the accuracy in the comparison of the relative shielding σ(SiO4-4) - σ(SiF4) = 43 ppm (calculated) and 40 ppm (experimental), σ(SiF62-) - σ(SiF4) = 114 ppm (calculated) and 75 ppm (experimental). The experimental values are from condensed phase and could be several ppm away from σe values. There are some trends worth noting in these results. (a) 29Si shielding is only very weakly dependent on Si-O or Si-F distance, suggesting that the strong empirical correlations between 29Si chemical shifts and bond distance indirectly come about from the relation between the degree of polymerization and the average Si-O bond distance. (b) Trigonal distortion of the SiO44- moiety leads to a larger value of σ along the direction of the short Si-O bond, a change which is of the same sign as the empirical correlation. (c) In the set of molecules which are silicon analogs of ethane, ethene, and ethyne, as well as also in the mixed carbo compounds H3Si-CH3, H2Si=CH2, HSi [equivalent to] =CH, the 29Si shielding is in the order
Si = < Si- < Si [equivalent to].
(d) In H3Si-X where X=BH2, CH3, NH2, the 29Si shielding decreases with increasing electronegativity of X.
The highlights of the calculations of transition metal chemical shifts which were reviewed in the previous SPR volume has been summarized by one of the authors. These ab initio FPT calculations using double zeta quality valence orbitals (MIDI-l-plus basis set for the metal and a somewhat leaner one for the ligands) are on the largest systems for which magnetic shielding has been calculated. At the other extreme are calculations of 1H shifts in OH-, H2O, H3O+, H5O2+, The latter calculations with a 6-31G** basis set using the GIAO-FPT method give results comparable to those (reviewed in the previous SPR volume) using the SOS-CI method. These results, σiso = 31.17 ppm, 24.29 ppm, and 40.21 ppm, are compared with 1H shifts in various inorganic solids containing proton environments which are essentially H2O, H3O+, and OH-, respectively: 24-27.5 ppm, 20-21 ppm, and 27-29 ppm. The calculated 31.17 ppm for H2O compares very well with the experimental value for water vapor (corrected for vibrational effects): σ0 30.052 ± 0.015. The proton in H2O and H3O+ environments in the inorganic solids are deshielded by 3 to 6 ppm owing to intermolecular effects. Thus, the H2O and H3O+ calculations are in reasonably good agreement with the experiment. The calculated OH- result is too shielded, however. Either the proton shielding in OH- ion is more strongly affected by the environment (just as electric dipole polarizabilities of negative ions are very different in going from the gas phase to the lattice), or else the calculations for a molecular ion with a net negative charge requires a biqger basis set to achieve the same accuracy as in the neutral species.
B. Semi-empirical Calculations.- Fukui et al. have proposed a separation of the shielding into diamagnetic and paramagnetic parts according to the following criteria: (a) σd depends only on an unperturbed wavefunction, (b) each of σd and σp is invariant with respect to displacement of origin, (c) each of σd and σp is written in terms of Hermitian operators. This separation is then applied to GIAO-FPT calculations with the INDO parameters of Ellis et al. augmented with multicenter integrals which have been evaluated over ST0-6G orbitals with unmodified exponents from Slaters' rules. 1H and 13C shieldings are calculated in C1 to C6 hydrocarbons using this method. Only isotropic chemical shifts relative to CH4 are reported and these agree reasonably well with experimental shifts. Other calculations of 13C chemical shifts in rigid bicyclo[m,n,o] alkanes using GIAO-FPT method within the INDO framework but with adjusted integrals (by the use of Mulliken populations) are discussed in comparison with experiment. The discrepancies observed are explained on the basis of nonbonded interactions in the highly-strained five membered rings. Estimation of different factors contributing to 13C chemical shifts in cyclopentadienyl ligands of metal complexes leads to a correlation between the observed ^3C chemical shifts and chemical properties of the complexes.
