eigenstates,ji ,
(^)
,
(^)
, and
(^)
, where the first symbol in each
represents the state of theI spin and the second symbol represents
theSspin. The single-element operator basis set contains four so-called
population terms:
ji hj ¼I^ S^ ,
(^)
¼I^ S^ ,
(^)
¼I^ S^ ,
(^)
¼I^ S^ :
½ 2 : 214
The basis set contains eight terms representing the single-quantum
transitionsassociated with the two spins (remembering the definitions of
[2.209]):
ji
(^)
¼I^ Sþ,
(^)
hj ¼I^ S,
(^)
(^)
¼I^ Sþ,
(^)
(^)
¼I^ S,
ji
(^)
¼IþS^ ,
hj ¼IS^ ,
(^)
¼IþS^ ,
(^)
¼IS^ :
½ 2 : 215
In these cases, one spin remains ‘‘untouched’’ and the transition involves
only the change in spin state of the other spin. These operators describe
the single-quantum coherences associated with the single-quantum
transitions. The basis set contains four terms representing transitions
in which both spins change their spin state simultaneously. These
coherences are classified as double-quantum coherence if both spins
change spin states in the same sense,
ji
(^)
¼IþSþ,
hj ¼IS, ½ 2 : 216
or zero-quantum coherence if both spins change spin states in the
opposite sense,
(^)
¼IþS,
(^)
¼ISþ: ½ 2 : 217
Each of these product operators has a simple interpretation in terms of
energy levels and transitions shown in Fig. 2.4. ForNscalar coupled
spins-1/2, a full operator set contains 4Nelements.
The relationship between the Cartesian and single-element operators
[2.210] can be seen as, for example,
Iz¼ 2 ðIzÞ^12 E
¼^12 I^ I^
S^ þS^
,
Sx¼ 212 E
ðÞ¼Sx^12 I^ þI^
SþþS
,
2 IzSy¼ 2 ðÞIz Sy
¼ 21 i I^ I^
SþS
:
½ 2 : 218
82 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY