Paramagnetic term

The (temperature-independent) paramagnetic susceptibility tensor is given by the perturbation formula 

The angular momentum operator, expressed in terms of the position of atoms relative to the gauge origin and electrons relative to the atom, becomes 

In evaluating the first term we need the matrix elements of the linear momentum 

The second term is evaluable easily since the operator lGIa acts only on the angular momentum functions irrespective of the radial part of the atomic orbitals. For this purpose the following formulae are applicable 

The action of angular momenta operators on the basis set of real angular functions (which are linear combinations of the spherical harmonics l, m ) is summarised in Table 5.6. The matrix element (x ll/vialx ) is given by a number multiplied by the overlap integral for a shifted ket-vector 

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As indicated, the possibility of orbital decomposition is retained in the diamagnetic and paramagnetic terms in eqs.(29,30). The value for the paramagnetic shielding contribution extracted via eq.(30) does not, of course, correspond to a well-defined basis set, but is at the same level of numerical quality as the LORG calculation used for the total shielding. 

The paramagnetic term, apara, derives from the excitation of p-elec-trons by the external held, and its impact is opposite to that of diamagnetic shielding. The term,, derives from the effect of neighboring groups, which can increase or decrease the held at the nucleus, a can also be affected by intermolecular effects, in most cases deriving from interaction of the solvent. 

It is not our concern here to delve into the theory of chemical shifts, since most studies mentioned are essentially empirical in their approach and have aimed to correlate the measured shifts with electronic factors. The main fact, however, is that unlike proton shifts, the shifts for other nuclei are governed by the Van Vleck second-order paramagnetic term, which is not easy to handle since its magnitude depends not only on the ground electronic state but also on the manifold of excited states. 

A general comment on the use of the empirical correlation between Si and Sn NMR (and likewise on C/ Si or Sn/ Pb NMR) chemical shifts is in order. The basis for this correlation is that the paramagnetic term Op dominates the chemical shift. According to Ramsay s theory, Op is proportional to the reciprocal energy difference h.E between the magnetically active orbitals and proportional to the expectation value for the electron radii (r )np- Thus, a linear correlation between the 5 Si and 8 Sn implies that the ratio of both determining factors of Op is constant for the all compounds of interest. In particular, it is not clear, however, if the ratio for tetravalent silicon and tin compounds is the same as for trivalent silicon and tin compounds. Therefore, the extension of a correlation based exclusively on the... 

Ramsey obtained ay, by first-order and op by second-order perturbation theory (76) variational treatments give similar results (14, 71, 73). The term wp is sometimes called the second-order paramagnetic term and sometimes the high-frequency term (14), because of the dependence of the (temperature-independent) paramagnetism in molecules on the high-frequency matrix elements of the orbital moments (91). 

The dependence imposes a periodicity on the variation of the paramagnetic term with atomic number Z of the nucleus if the molecular environments are similar. Figure 3 shows the variation with Z of... 

This phenomenon of antiparamagnetic paramagnetic terms clearly needs a name and is called here the Cornwell effect (ideally the Cornwell-Santry effect). Positive contributions to op (which may or may not be positive overall) are expected in heteronuclear diatomics if they have a IT state this excludes, e.g., HF, InF, and TIF. In homonuclear diatomics, the IT -> a excitation is symmetry-forbidden. The possibility has been mentioned for XeF (34), although, from the chemical shift and calculated values of aa, the resultant Op ( F) is negative in XeFg and KrFj (cf. Fig. 7). Another candidate is FC DH, from the evidence of the fluorine chemical shift and spin-rotation interaction (96). According to this interpretation there should be a substantial upheld shift of the... 

Table II shows that the observed 29) shifts 8 (taken as positive downfield) show a degree of linear additivity in the methyl-substituted methanes, with a shortfall for neopentane. Table II shows also the division of the shielding into atom-plus-ligand diamagnetic and paramagnetic parts, relative to methane for which era is 295 ppm. The diamagnetic terms were calculated by Flygare s method 25). With each substitution of hydrogen by carbon, aa for the central carbon increases by 28 ppm but o-p increases by about 37 ppm, and the line moves 9 ppm downfield. Analogous relationships have been demonstrated for shielding in methyl-substituted NH3 and NH4+ 30). Table II shows that the shortfall at neopentane is in the paramagnetic term.

The most striking feature of Figs. 7 and 8, particularly as compared with the hydride plot, is the plunge downfield of the resultant shielding, following the paramagnetic term, across the row of the central atom for the typical elements, and also for the early transition metals (Fig. 8), with ready circulation of fluorine 2p electrons into empty t g orbitals in the complexes. Fluorine is h hly shielded however in the d molecules and ions, and this was discussed in Section III, B as a possible Cornwell effect (62). 

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