Second-order Splittings

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 

ESR spectrum of the methyl radical, CH3 (note discontinuities in magnetic field axis). Simulated using hyperfine splitting from ref. 3 and eqn (2.5). 

---

For / = 0 there is obviously no first-order Zeeman splitting however, application of a magnetic field can result in second-order splitting. As such, it is necessary to evaluate the corresponding factor, which isg(j = 2 + 7,(2 + S). 

Fig. 9. Second order splittings in spin systems with two magnetically equivalent 1=1 nuclei Single crystal nitrogen ENDOR spectrum of Cu(sal)2 diluted into Ni(sal)2. (Ref. 62)...

The observed splittings of the nitrogen ENDOR lines, which vary from -230 to 430 kHz for a rotation of the crystal around a, have also to be interpreted as originating from two nonequivalent nitrogen nuclei rather than from a second order splitting related... 

From Formulas (19) and (20) it is clear that a first order effect is independent of the applied field Ih, while the second order splitting decreases as Ht increases. In this way the two effects may be experimentally distinguished. 

An example of a complex proton spectrum is that of ethyl iodide (Figure 9-32). To a first approximation, the two main groups of lines appear as equally spaced sets of three and four lines, arising from what are called first-order spin-spin interactions. Matters are further complicated by additional splitting of the three-four pattern of ethyl iodide, as also can be seen in Figure 9-32. This additional splitting is called second-order splitting. 

Figure 9-32 High-resolulion nmr spectrum of ethyl iodide, CH3CH2I, at 60 MHz relative to TMS, 0.00 ppm. The first-order splitting pattern is seen in the well-separated three-four line pattern for the CH3—CH2 resonances. The second-order splitting is the additional fine structure superimposed on the three-four pattern.

Third, the second-order splitting tends to disappear with increasing transmitter frequency. For ethyl iodide (Figure 9-32), the second-order splitting at 60 MHz is barely discernible at 100 MHz and disappears at 200 MHz. This also can be seen to occur for the three-four splitting pattern of 2-methyl-2-butanol as a function of v (Figure 9-27). 

Z.G. Sun, S.Y. Lee, H. Guo, D.H. Zhang, Comparison of second-order split operator and Chebyshev propagator in wave packet based state-to-state reactive scattering calculations, J. Chem. Phys. 130 (2009) 174102. 

A first order spectrum is observed only if the chemical shift difiereno between the coupled groups is large compared to the coupling constant, U Av/J > 7. As this ratio becomes smaller, the inner peaks of multiplets grow at the expense of outer peaks and additional or second-order, splitting ma ... 

Garbutt and Gesser (41) were able to partially resolve the second-order splitting of the center two lines of the CH radical stabilized on PVG at 77... 

K. This second-order splitting was predicted by Fessenden (42) and confirms the presence of only weak Interaction between the radical and the surface. 

Such second-order splitting was also reported for CH on silica gel, PVG, and Cabosil (43). Simulation showed the 1 2 relative intensities predicted when linewldth and separation were considered. It was also shown that the line-width varied considerably with surface coverage by the CH I. (See Table 3). 

The tumbling frequency determination must be resolved with respect to the second-order splitting. The values of the parameters used in the Klvelson equation or its equivalent must be stated if direct comparison between studies is to be possible. 

FIGURE 5.31 The relationships among the chemical shifts, line positions, and coupling constant in a two-proton spectrum that exhibits second-order splitting eifects. 

In the sections that follow, we will attempt to cover some of the most important types of benzene ring substitution. In many cases it will be necessary to examine sample spectra taken at both 60 MHz and 300 MHz. Many benzenoid rings show second-order splittings at 60 MHz but are essentially first order at 300 MHz. 

At 60 MHz, this chemical shift difference results in a complicated second-order splitting pattern for anisole (methoxybenzene), but the protons do fall clearly into two groups, the ortholpara protons and the meta protons. The 60-MHz NMR spectrum of the aromatic portion of anisole (Fig. 5.40) has a complex multiplet for the o,p protons (integrating for three protons) that is upheld from the meta protons (integrating for two protons), with a clear distinction (gap) between the two types. Aniline (aminobenzene) provides a similar spectrum, also with a 3 2 split, owing to the electronreleasing effect of the amino group. 

---