of the Hamiltonian, with the result
H¼^13 TrfgCuvþA^02 F 20 ðlabÞ¼^13 TrfgCuvþA^02X^2
k¼ 2D^2 k 0 ðÞ (^) LP, (^) LP,0F 2 kðPASÞ
¼^13 TrfgCuvþA^02
ffiffiffi
3
2
r
dz
n
D^200 ðÞ0, (^) LP,0:
þ
1
ffiffiffi
6
p D^220 ðÞþ (^) LP, (^) LP,0 D^220 ðÞ (^) LP, (^) LP,0
o
: ½ 2 : 309
The third Euler angle is unnecessary for determining the truncated
Hamiltonian, and has been arbitrarily set to zero, because only
D^2 m 0 ðÞ , , are required to obtainF 20 ðlabÞusing [2.307]. This simplifica-
tion results because the truncated Hamiltonian commutes with the
Zeeman Hamiltonian and consequently is unaffected by a rotation
around thez-axis of the laboratory reference frame. Also,
D^2 m 0 ðÞ¼ , ,0 Ym 2 ðÞ , ,
D^20 mðÞ¼0, , Ym 2 ðÞ , ,
½ 2 : 310
in whichYm 2 ðÞ, are the modified spherical harmonic functions used in
Chapter 5 ( 20 ). If operators of the formuþv–andu–vþcommute with the
Zeeman Hamiltonian, thenA^02 is given in [2.303]. If operators of the form
uþv–andu–vþdo not commute with the Zeeman Hamiltonian, then these
are truncated as well and
A^02 ¼
ffiffiffiffiffiffiffiffi
2 = 3
p
uzvz: ½ 2 : 311
For example, this simplified expression forA^02 is obtained ifu¼Iand
v¼Srefer to different heteronuclear spins.
In solution NMR spectroscopy, the Hamiltonians given in [2.308]
and [2.309] must be averaged over the rotational distribution of
molecules in solution. The angular dependence of the Hamiltonian is
expressed by the angular dependence of the Wigner rotation matrices.
This means that theD^2 mnðÞ (^) LP, (^) LP, (^) LP in [2.308] and [2.309] must be
replaced by average valueshD^2 kqð (^) LP, (^) LP, (^) LPÞi. The rotational average
ofDlmnðÞ , , is defined in general by
DlmnðÞ , ,
¼
Z
Dmnl ðÞ , , pðÞ , , sin d d d , ½ 2 : 312
106 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY