QMGreensite_merged

TheF 2 qalso are the 2qþ1 components of an irreducible spherical tensor
of second rank. With these definitions, the nuclear spin Hamiltonian is
given by

H¼^13 TrfCguvþ

X^2

q¼
 2

ð
 1 ÞqF 2 
qAq 2 : ½ 2 : 306 Š

The advantage of writing the Hamiltonian in this form is that [2.306] is
valid in any reference frame provided that the tensors are expressed in
the suitable frame of interest. Thus, the Hamiltonian in the principal
axis frame is obtained by usingAq 2 andF 2 qexpressed in the principal
axis frame, and the Hamiltonian in the laboratory frame is obtained
by using Aq 2 and F 2 q expressed in the laboratory reference frame.
The tensorsF 2 qðlabÞare obtained in the laboratory frame from the tensors
F 2 qðPASÞin the principal axis frame by using the transformation properties
of the irreducible spherical tensors ( 20 ):

F 2 qðlabÞ¼

X^2

k¼
 2

D^2 kqðÞ (^) LP, (^) LP, (^) LPF 2 kðPASÞ, ½ 2 : 307 Š
in whichD^2 mnðÞ (^) LP, (^) LP, (^) LP are the Wigner rotation matrices, given in
Table 2.4 and { (^) LP, (^) LP, (^) LP} are the Euler angles specifying the relative
orientation of the laboratory and principal axis reference frames.
Using this relationship, the nuclear spin Hamiltonian is expressed in
the laboratory reference frame as
H¼^13 TrfgCuvþ
X^2
q¼
2
ð
1 ÞqAq 2
X^2
k¼
2
D^2 kqðÞ (^) LP, (^) LP, (^) LPF 2 kðPASÞ, ½ 2 : 308 Š
in which the vectorsuandvand the tensorsAq 2 are understood to be
expressed in the laboratory reference frame. This equation makes use
of the observation thatu,v, andAq 2 usually are much simpler to express
in the laboratory reference frame but the tensorsF 2 qhave their simplest
form in the principal axis system of the interaction.
The full form of the Hamiltonian in [2.308] is important for the
development of nuclear spin relaxation theory, and is discussed in
Chapter 5. For the consideration of first-order spectra — that is, of
resonance frequencies and intensities — the nuclear spin Hamiltonian
can be treated as a weak perturbation to the Zeeman Hamiltonian. As a
result, only the components of [2.308] that commute with the Zeeman
Hamiltonian need to be retained. This simplification is called truncation
104 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY