The potential energy function is ( 24 )
W¼
1
2 0
BTB: ½ 2 : 329
Using similar derivations as used for the nuclear spin Hamiltonians
yields for the traceless symmetric component of the potential
WðÞ¼ (^) LA, (^) LA, (^) LA
B^20
0
ffiffiffi
6
p
X^2
k¼ 2
D^2 k 0 ðÞ (^) LA, (^) LA, (^) LAk 2 , ½ 2 : 330
in which^20 ¼
ffiffiffiffiffiffiffiffi
2 = 3
p
,^22 ¼ yyxx
=2,¼zz–(xxþyy)/2,
and {xx,yy,zz} are the principle values of. The isotropic component
of the potential does not contribute to the probability density, as noted
previously, and has not been included in [2.330]. Thus, a molecule in
solution has a preferential orientation with respect to the static magnetic
field. Integration of [2.312] using [2.314] and [2.330] gives
D^200
¼
B^20
15 0 kBT
,
D^20 2
¼D^202
¼
B^20 xxyy
10
ffiffiffi
6
p
0 kBT
,
½ 2 : 331
from which
Szz¼
B^20
15 0 kBT
,
SxxSyy¼
B^20 xxyy
10 0 kBT
,
½ 2 : 332
The resulting residual dipolar coupling constant is
DIS¼
(^) I (^) ShB^20
60 ^2 kBTr^3 IS
1
2
3 cos^2 1
þ
3
4
xxyy
sin^2 cos2
:
½ 2 : 333
For diamagnetic molecules, the achievable alignment is weak because
is very small. For example, a benzene molecule has ¼1.3
10 ^33 m^3 andxxyy¼0. ForB 0 ¼18.8 T (800 MHz),T¼300 K, a
C–H bond length of 0.11 nm, andDmaxCH¼45.1 kHz, a maximum value
of DCH¼0.26 Hz is obtained when¼0, and a minimum value of
DCH¼–0.13 Hz is obtained when¼/2. In this case,Szz¼5.9 10 –6,
2.8 AVERAGING OF THESPINHAMILTONIANS 111