observable and generate in-phase resonance signals. Operators of the
formI 1 z...Iði 1 ÞzIixIðiþ 1 Þz...IkzandI 1 z...Iði 1 ÞzIiyIðiþ 1 Þz...Ikzevolve into
Iix andIiy and generate antiphase resonance signals if J 1 i...,J(i–1)i,
J(iþ1)i,...,Jki are all nonzero, for kN. The resonance signals
generated by operators containingIixandIiyhave a 90 8 relative phase
difference. If the signals arising fromIixare phased to have absorptive
lineshapes, then the signals arising from Iiy will have dispersive
lineshapes.
In spin systems withN3, single-quantum coherences with 41
transverse factors in their operator representations exist and are referred
to as combination lines, combination operators, or N-spin 1
coherences (14, 15). For example, in anN¼3 spin system, the single-
quantum combination coherence
(^)
has m¼1 and is
represented by the operator I 1 I 2 Iþ 3. In the weak coupling limit,
combination operators are orthogonal to the detection operator and
consequently are not directly observable during the acquisition period.
In the strong coupling limit, the product wavefunctions are not
eigenfunctions of the Hamiltonian [2.154]; eigenfunctions are obtained
by diagonalizing the Hamiltonian as described in Section 2.5.2. The
appearance of combination lines in strongly coupled NMR spectra is
discussed by Bain ( 16 ) and references therein.
2.7.5 MULTIPLE-QUANTUMCOHERENCE
For a system of two spin-1/2 nuclei, multiple-quantum coherence
states are represented by product operators in which both spins have
transverse components. For example,
2 IxSy¼ 21 iðIþSþISÞ 21 iðIþSISþÞ: ½ 2 : 251
The first term on the right-hand side, (IþSþ–IS), is pure double-
quantum coherence (|m|¼2), whereas the second term, (IþS
ISþ), is pure zero-quantum coherence (m¼0). The multiple-
quantum coherence term 2IxSyis a superposition of both double- and
zero-quantum coherence. Multiple-quantum coherences can be prepared
by suitable combinations of pulses and free-precession periods. Such
terms have more than one transverse operator component and are not
observable directly; however, multiple-quantum coherences possess
some unique properties of considerable utility.
Multiple-quantum coherences can be expressed conveniently in
terms of Cartesian and/or shift operators. Pure double-quantum (DQ)
90 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY