QMGreensite_merged

does not affect the energy levels of the nuclear spin systems and
consequently does not contribute to the observed resonance frequencies.
This component of the nuclear spin Hamiltonian will not be considered
further in this text. The Hamiltonian can then be written as

H¼^13 TrfCguvþuTCð^2 Þv: ½ 2 : 300 Š
In the principal axis reference frame of the tensor,C(2)is diagonal
with elementsdk¼Ckk
Tr{C}/3, in whichCkkfork¼{x,y,z} are the
principal values ofC. In this frame,

uTCð^2 Þv¼dxuxvxþdyuyvyþdzuzvz: ½ 2 : 301 Š

Equation [2.301] is expressed in terms of the Cartesian components ofu,
v, andC(2), which facilitates a physical interpretation of spin interac-
tions. However, the effects of rotation are more easily considered by
expressing the Hamiltonian using spherical, rather than Cartesian,
tensors. Thus, [2.301] can be reformulated as

uTCð^2 Þv¼

ffiffiffiffiffiffiffiffi

3 = 2

p

dzA^02 þ^12 dzðA^22 þA
 22 Þ, ½ 2 : 302 Š

in which ¼ðdx
dyÞ=dz, Aq 2 are the 2q þ1 components of the
irreducible spherical tensor operator of second rank:

A^02 ¼p^1 ffiffi 6 ð 3 uzvz
uvÞ,

A^21 ¼^12 ðu^ vzþuzv^ Þ,

A^22 ¼^12 u^ v^ ,

½ 2 : 303 Š

u^ ¼ux iuyandv^ ¼vx ivy, anduandvare expressed in the principal
axis frame. In obtaining [2.302], the relationshipdz¼
(dxþdy) has been
invoked because the tensorC(2)is traceless. For the chemical shielding
tensor, dz¼^23  [1.49]. The expression [2.302] can be written
equivalently in the form

uTCð^2 Þv¼

X^2

q¼
 2

ð
 1 ÞqF 2 
qAq 2 ½ 2 : 304 Š

by making the identifications

F 20 ¼

ffiffiffiffiffiffiffiffi

3 = 2

p

dz,

F 21 ¼0,

F 22 ¼^12 dz¼^12 dx
dy



:

½ 2 : 305 Š

2.8 AVERAGING OF THESPINHAMILTONIANS 103