QMGreensite_merged

coherence is represented by suitable combinations of bilinear product
operators,

1

2 ðI

þSþþI
S
Þ¼ 1

2 ð^2 IxSx
^2 IySyÞ¼DQx, ½^2 :^252 Š

1

2 iðI

þSþ
I
S
Þ¼ 1

2 ð^2 IxSyþ^2 IySxÞ¼DQy: ½^2 :^253 Š

Pure double-quantum coherence precesses at thesumof the two chemical
shifts involved, e.g., during a delay,t,

DQx )

(^) IIztþ (^) SSzt
DQxcos½ŠþðÞ (^) Iþ (^) St DQysin½ŠðÞ (^) Iþ (^) St, ½ 2 : 254 Š
DQy^ IIztþ^ SSzt)DQycos½Š
ðÞ (^) Iþ (^) St DQxsin½ŠðÞ (^) Iþ (^) St: ½ 2 : 255 Š
Similarly, pure zero-quantum (ZQ) coherence is represented by
1
2 ðI
þS
þI
SþÞ¼ 1
2 ð^2 IxSxþ^2 IySyÞ¼ZQx, ½^2 :^256 Š
1
2 iðI
þS

I
SþÞ¼ 1
2 ð^2 IySx
^2 IxSyÞ¼ZQy, ½^2 :^257 Š
and evolution occurs at thedifferenceof the chemical shifts of the spins
involved,
ZQx^ IIztþ^ SSzt)ZQxcos½ŠþðÞ (^) I
(^) St ZQysin½ŠðÞ (^) I
(^) St, ½ 2 : 258 Š
ZQy )
(^) IIztþ (^) SSzt
ZQycos½Š
ðÞ (^) I
(^) St ZQxsin½ŠðÞ (^) I
(^) St: ½ 2 : 259 Š
Two-spin multiple-quantum coherence such as that noted previously,
does not evolve under the influence of the scalar coupling of the two spins
involved in the coherence (the active coupling). This principle can be
rationalized using [1.57] and the energy level diagram of Fig. 1.7b.
Double-quantum coherence connects the states ji and j i. The
difference in energy between these two states does not depend on the
active scalar coupling constant. Similar considerations are relevant for
zero-quantum coherence that connects statesj iandj i. However,
multiple-quantum coherence can evolve under the influence of a scalar
coupling to a thirdpassivespin. For example, consider the three-spin
system I, S, R, where the couplings present are JIS,JIR, and JSR.
A multiple-quantum coherence term can be identified by the appearance
of more than one transverse Cartesian operator in the product;
therefore, the operator, 4IySxRz, is a multiple-quantum coherence with
2.7 PRODUCTOPERATORFORMALISM 91