Spin-orbit/Fermi contact effects

In this chapter, we use exclusively relativistically optimized or experimental geometries. Hence, we concentrate on direct relativistic effects only. They can be separated into scalar and spin-orbit/Fermi contact effects. In addition, there are, in both cases, core and valence contributions. 

Spin-Orbit/Fermi Contact Effects. While scalar relativistic effects seem to be sufficient for some systems like the metal carbonyls of Table I (even though it has been speculated (9) that spin-orbit might improve the agreement with experiment even further), there are other cases where this is not the case. We have chosen as an example the proton NMR absolute shielding in hydrogen halides HX, X = F, Cl, Br, I (7,9), Figure 1. This series has also been studied by other authors (34-38), and it may well be the most prominent example for spin-orbit effects on NMR shieldings and chemical shifts. 

Figure 1. JH absolute shielding in HX, X = F, Cl, Br, I. The figure illustrates the importance of spin-orbit/Fermi contact effects in these systems (9) scalar relativistic calculation (7) are unable to reproduce the experimental trend.

It follows from Table V that spin-orbit effects are relevant for the heavy metal shieldings and, since the spin-orbit contribution does not always have the same sign, for the relative chemical shifts. In this connection, it is interesting to note that the ZORA spin-orbit numbers are shifted as compared to their Pauli spin-orbit counterparts. This effect can be attributed, at least partly, to core contributions at the metal while scalar contributions of the core orbitals are approximately accounted for by the frozen core approximation (6,7), spin-orbit contributions of the core orbitals are neglected. Hence, the more positive (diamagnetic) shieldings from the ZORA method are due to spin-orbit/Fermi contact contributions of the s orbitals in the uranium core. 

Scalar Pauli computations and ZORA calculations with spin-orbit yield quite similar results, Table III, although the latter includes spin-orbit effects whereas the former does not. However, we note from a direct inspection of the ZORA calculations that the spin-orbit contributions are small (not shown in the table). The U-F bond is mostly po- and type and should only have very little s character at the fluorine center. Hence, Fermi contact contributions are small, and spin-orbit is not relevant here, according to the qualitative discussion above. 

Theory indicates that the J x term is composed of several contributions that characterize the effect of orbital electronic motions, the polarization of electronic spins, and the Fermi contact term. The last contribution is the most significant, and affects mainly the s valence orbitals. 

The first two terms are Zeeman terms and the third represents the hyperfine interaction of the electron and nuclear spins, / b and are the Bohr and nuclear magnetons respectively, S is a fictitious effective spin (S = 2 for a simple Kramers doublet), and / is the nuclear spin tensor. The hyperfine tensor is further split into Fermi contact, dipolar, and orbital components according to ... 

Beer and Grinter used finite perturbation theory to calculate J(Si-H), J(Si-H), (161) and J(Si-C) (143) as well as the analogous phosphorus couplings. The best results are obtained for J(Si-Q (Table XXI) for which the correlation coefficient of the best fit of calculated to observed values is 0-985. The various calculated contributions to J(Si-C) are in Table XXI. Clearly, the orbital and spin-dipolar terms, which are small and also of opposite sign, have little effect on the calculated value of J. It is concluded that the Fermi contact term is probably sufficient for the calculation of J(Si-Q and that inclusion of silicon d orbitals is not required. 

The naturally occurring isotopes of boron,, 0B (19-58%) and UB (80-42%) have spin quantum numbers of 3 and 3/2, respectively. Experimental V(C-UB) values and those calculated using an INDO-MO method, assuming only the Fermi contact mechanism, are given in Table II. (24-27) It is apparent that the calculations account for the major patterns of substituent effects on /(C-B) and that, if the orbital and dipolar mechanisms are important for these coupling constants,... 

The so-called heavy-atom chemical shift of light nuclei in nuclear magnetic resonance (NMR) had been identified as a spin-orbit effect early on by Nomura etal. (1969). The theory had been formulated by Pyykktt (1983) and Pyper (1983), and was previously treated in the framework of semi-empirical MO studies (PyykktJ et al. 1987). The basis for the interpretation of these spin-orbit effects in analogy to the Fermi contact mechanism of spin-spin coupling has been discussed by Kaupp et al. (1998b). 

The analogy with the Fermi contact interaction predicts particularly large spin-orbit effects when a large-s character of the bonding of the NMR atom to the heavy atom is present. This has been shown—e.g. for the PI cation (Kaupp et al. 1999),... 

The first defect is in ignoring or overlooking the fact that orbital effects, discussed under Electron Spin-Nuclear Spin Interaction, can make major contributions to the observed shifts and must be accounted for in order to get the correct Fermi contact contributions. Many researchers have assumed the total isotropic term is from the Fermi contact interaction. 

The Fermi contact shift term comes about from unpaired spin density transferred to the nuclei via bonding orbitals as well as screening and polarization effects. This contribution is isotropic and calculated according to Ato,... 

Let us start with the field-free SO effects. Perturbation by SO coupling mixes some triplet character into the formally closed-shell ground-state wavefunction. Therefore, electronic spin has to be dealt with as a further degree of freedom. This leads to hyperfine interactions between electronic and nuclear spins, in a BP framework expressed as Fermi-contact (FC) and spin-dipolar (SD) terms (in other quasirelativistic frameworks, the hyperfine terms may be contained in a single operator, see e.g. [34,40,39]). Thus, in addition to the first-order and second-order ct at the nonrelativistic level (eqs. 5-7), third-order contributions to nuclear shielding (8) arise, that couple the one- and two-electron SO operators (9) and (10) to the FC and SD Hamiltonians (11) and (12), respectively. Throughout this article, we will follow the notation introduced in [58,61,62], where these spin-orbit shielding contributions were denoted... 

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