the spin state of one nucleus changing while the spin state of the other
nucleus remains constant. The spectrum can be conveniently labeled
with the spin states of the coupled spins as shown in Fig. 2.1. Similar
considerations apply to the three-spin system, although the appearance
of the spectrum is a little more complex. The three spins are denotedI,R,
andSand have three scalar coupling constants,JIS,JIR, andJSR. The
wavefunctions for the scalar coupled three-spin system are denoted
jimI,mS,mR in the product basis, and the energies of theeightlevels can
be calculated by generalizing [1.57] or by direct application of [2.154]
and [2.7]. The energy level diagram for a three-spin system is shown
in Fig. 2.2a. The single-quantum transitions that connect pairs of
eigenstates in which the spin state of one of the three nuclei changes are
represented as solid or dashed arrows. Each of the indicated transitions
hasm¼–1 and, just as in the two-spin case, is associated with a
resonance line of a specific multiplet (in this case the multiplets are
quartets) in the one-dimensional NMR spectrum. Schematic NMR
spectra are shown in Fig. 2.2b,c. As seen by comparing b and c in
Fig. 2.2, the appearance of the spectrum depends on the relative
chemical shifts of theI,S, andRspins and on the relative sizes ofJIS,
JIR, andJSR; however, Fig. 2.2a is sufficient for illustrative purposes.
The two transitions 1–2 and 2–4 share a common eigenstate (2 in this
case); consequently, these two transitions are referred to asconnected
transitions. The spin state of one of the three spins remains unchanged
across connected transitions (e.g., theIspin state isji for the connected
aabbabba1234FIGURE2.1 Multiple-quantum transitions forISspin system. Shown are the
zero-quantum flip-flop transitions between statesj iandj iand the double-
quantum flip-flip transitions between statesj iandj i.
76 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY