QMGreensite_merged

The potential energy function is ( 24 )

W¼


1

2  0

BTB: ½ 2 : 329 Š

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 Š
in which^20 ¼
ffiffiffiffiffiffiffiffi
2 = 3
p
,^22 ¼ yy
xx

=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 xx
yy

10
ffiffiffi
6
p
 0 kBT
,
½ 2 : 331 Š
from which
Szz¼
B^20 
15  0 kBT
,
Sxx
Syy¼
B^20 xx
yy

10  0 kBT
,
½ 2 : 332 Š
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
xx
yy

sin^2 cos2

:
½ 2 : 333 Š
For diamagnetic molecules, the achievable alignment is weak because
is very small. For example, a benzene molecule has ¼
1.3 
10
^33 m^3 andxx
yy¼0. ForB 0 ¼18.8 T (800 MHz),T¼300 K, a
C–H bond length of 0.11 nm, andDmaxCH¼
45.1 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