Coupling constant second-order effects

Our analysis thus far has assumed that solution of the spin Hamiltonian to first order in perturbation theory will suffice. This is often adequate, especially for spectra of organic radicals, but when coupling constants are large (greater than about 20 gauss) or when line widths are small (so that line positions can be very accurately measured) second-order effects become important. As we see from... 

Here, A is the nearly isotropic nuclear coupling constant, I is the nuclear spin (Iun = I), and m is the particular nuclear spin state. It may be observed that the zero field splitting term D has a second-order effect which must be considered at magnetic fields near 3,000 G (X-band). In addition to this complication nuclear transitions for which Am = 1 and 2 must also be considered. The analysis by Barry and Lay (171) of the Mn2+ spectrum in a CsX zeolite is shown in Fig. 35. From such spectra these authors have proposed that manganese is found in five different sites, depending upon the type of zeolite, the primary cation, and the extent of dehydration. 

The value of P—H coupling constant determined from the spectrum is slightly different from the true value of V(P H) because of second-order effects (see later) and, hence, is designated as the apparent P - H coupling constant, iJ (P—H). 

Before we leave the topic of second-order effects, we revisit the three-spin system previously introduced in Example 9.15. There we established that if the chemical shifts of the three nuclei in a structure such as 9-3 are sufficiently different (i.e., an AMX system), the spectrum will consist of 12 lines 3 (first-order) doublets of doublets, each one centered at the appropriate chemical shift and exhibiting line spacings equal to the appropriate coupling constants. 

As the ratio of A 8v (the difference in chemical shift between two coupled nuclei) to J decreases, the relative intensities of the lines in a multiplet deviate further from first-order (e.g., Pascal triangle) ratios. Inner lines (those facing the coupled multiplet) increase in intensity, while outer lines lose intensity. This slanting of the multiplets is one type of second-order effect. At very small values of A Sv/J, not only may extra lines appear in the multiplets but also apparent line positions and spacings may not equate with true chemical shifts and coupling constants (e.g., deceptive simplicity and virtual coupling). 

In Section 9.9 we discussed how the appearance of a spin-coupled NMR spectrum is determined by the ratio of Av (the difference in chemical shifts between the coupling nuclei) to J (the coupling constant they share). For the spectrum to exhibit first-order multiplet intensities (Pascal s triangle Section 8.5), the value of Av//has to be at least 10. Smaller values of Av/J lead to progressively greater complications due to second-order effects. 

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