In order to perform the averaging over the probability distribution,
the principal axis system of the spin Hamiltonian must first be
transformed to the molecule-fixed frame, which will be referred to as
the alignment frame, and then the alignment frame must be transformed
to the laboratory reference frame. The principal axis system is oriented
with time-independent Euler angles { (^) AP, (^) AP, (^) AP} with respect to the
alignment frame. The alignment frame is oriented with time-dependent
Euler angles { (^) LA, (^) LA, (^) LA} with respect to the laboratory frame. The
relationships between the three frames are illustrated in Fig. 2.8. Thus,
[2.307] is generalized to
F 2 qðlabÞ¼
X^2
j¼ 2
X^2
k¼ 2
D^2 jqðÞ (^) LA, (^) LA, (^) LAD^2 kjðÞ (^) AP, (^) AP, (^) APF 2 kðPASÞ ½ 2 : 317
and the rotationally averaged truncated nuclear spin Hamiltonian
becomes, by extension of [2.309] ( 21 ),
H¼^13 TrfgCuvþA^02
X^2
j¼ 2
X^2
k¼ 2
hD^2 j 0 ðÞi (^) LA, (^) LA,0 D^2 kjðÞ (^) AP, (^) AP, (^) APF 2 kðPASÞ
¼^13 TrfgCuvþA^02
X^2
j¼ 2
X^2
k¼ 2
SjD^2 kjðÞ (^) AP, (^) AP, (^) APF 2 kðPASÞ,
½ 2 : 318
aAP
bAP
gAP
yPAS
A → LAB
zLAB
xLAB
yLAB
{aLA, bLA, gLA}
xPAS
zPAS
zA
xA yA
FIGURE2.8 Coordinate transformations. The principal axis system (PAS) of the
nuclear spin interaction is oriented with fixed angles {, } with respect to the
molecule-fixed alignment reference frame,A. The alignment frame is oriented
with respect to the laboratory (LAB) reference frame by time-dependent Euler
angles { (^) LA, (^) LA, (^) LA}.
108 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY