QMGreensite_merged

in whichAkk¼(2/3)Skkare the principal values of the alignment tensor,
Aa¼(3/2)Azzis the axial component of the tensor, andAr¼Axx–Ayyis
the rhombic component of the tensor. If the Hamiltonian is axially
symmetric with¼0, then the above expression simplifies to

H¼^13 TrfgCuvþA^02

ffiffiffiffiffiffiffiffi

3 = 2

p

z Aa^12 3 cos^2 
 1



þ^34 Arsin^2 cos2



,

½ 2 : 324 Š

in which { (^) AP, (^) AP, (^) AP}¼{0,,} and {,} are the polar angles
describing the orientation of thez-axis of the principal axis system of the
Hamiltonian with respect to the alignment frame.
As an explicit example, the Hamiltonian for the dipole–dipole
Hamiltonian is traceless and symmetric, withdz¼
2 ð 0 = 4 Þ (^) I (^) Shr
IS^3.
Consequently, [2.322] is given by
H¼DISð 3 IzSz
ISÞ, ½ 2 : 325 Š
in which
DIS¼DmaxIS Szz^12 3 cos^2 
1

þ Sxx
Syy
 1
2 sin
 (^2) cos2
¼DmaxIS Aa^12 3 cos^2 
1

þ^34 Arsin^2 cos2
½^2 :^326 Š
is the residual dipolar coupling constant (RDC), measured in units of
Hertz, and
DmaxIS ¼

 0
I (^) Sh
4 ^2 r^3 IS
: ½ 2 : 327 Š
If the residual dipole coupling is weak, 2DIS/|!I
!S|1, then the
Hamiltonian is further truncated to
H¼ 2 DISIzSz: ½ 2 : 328 Š
This Hamiltonian has the same functional form as does the weak scalar
coupling Hamiltonian. As a consequence, if alignment occurs, then the
apparent scalar coupling constant observed experimentally is given by
JISþDIS.
The alignment of a molecule with an anisotropic magnetic
susceptibility tensor in the presence of a static magnetic field is a
simple, easily calculable example of the effects of an orienting potential.
A molecule in a magnetic field,B, has an induced magnetic dipole
moment that is proportional to the magnetic susceptibility tensor.
110 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY