Some spin-coupling effects second-order

8 SOME SPIN-COUPLING EFFECTS (SECOND-ORDER) 

In discussing second-order effects, which arise jointly from two perturbations, two forms of the perturbation theory of Section 11.4 are open to us. It will often be convenient, as in Sections 11.5 and 11.6, to use the form in which one perturbation is applied first and the second is used as a probe to study its effect. In this way, it is possible to gain valuable insight into the origin of various types of coupling. At the same time, the approach leads naturally towards the theory of linear response, taken up in the next chapter. Again, we consider one-by-one a few typical and important examples. 

The separation of levels that differ only in nuclear spins is due primarily to the nuclear 2 eman terms, given in (11.3.17), namely 

To verify this simple picture, we look for two perturbations among those listed in Section 11.3 that can lead to cross-terms in (11.4.14) bilinear in field components and nuclear-spin components the obvious candidates are (using Hi, H2 in place of AH /iH ) 

Here J(r) is the current density at point r (in the perturbed state 5i), with components given in (11.6.18) r = r-il is the position vector of this 

---

The pulse sequence, as a variant of the spin echo experiment, also refocuses the spread of frequencies caused by field inhomogeneity, so that some improvement in resolution is obtained. The inset at the lower right of Figure 6-18 shows the normal ID spectra of H-4 and H-5 at the top (Figure 6-18c and e) and the unrotated projection of the 2D J-resolved spectra at the bottom [Figure 6-18d and f, extracted from the projected spectrum (Figure 6-18a) at the top of the 2D display]. The much higher resolution of the 2D resonances is clearly evident. Thus, the procedure is an effective way to measure J accurately, particularly when J is poorly resolved in the ID spectrum. The experiment fails for closely coupled nuclei (second-order spectra). 

Equation (2.3) describes line positions correctly for spectra with small hyperfine coupling to two or more nuclei provided that the nuclei are not magnetically equivalent. When two or more nuclei are completely equivalent, i.e., both instantaneously equivalent and equivalent over a time average, then the nuclear spins should be described in terms of the total nuclear spin quantum numbers I and mT rather than the individual /, and mn. In this coupled representation , the degeneracies of some multiplet lines are lifted when second-order shifts are included. This can lead to extra lines and/or asymmetric line shapes. The effect was first observed in the spectrum of the methyl radical, CH3, produced by... 

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... 

The obvious effects associated with second-order spectra (for instance, extra lines, distorted intensity patterns, and unequal spacings) generally preclude any injudicious attempts to analyze such systems by first-order rules. However, in some cases second-order spectra have features that are qualitatively indistinguishable from some features of first-order spectra, and so are often misinterpreted. It must be understood that the three cases discussed below are not physical phenomena—they are simply the result of certain combinations of the chemical-shift and spin-coupling parameters. 

---