QMGreensite_merged

in whichp( , , ) is the probability distribution for the set of Euler
angles { , , }. In isotropic solution, all molecular orientations are
equally likely. Consequently, p( , , )¼1/(8^2 ) andDlmnðÞ , ,
¼0.
Therefore, the second term in [2.308] and [2.309], depending on the
traceless tensorC(2), is zero as a result of averaging over the random
distribution of molecular orientations. The first term is a scalar product
and is invariant to rotation. The rotationally averaged nuclear spin
Hamiltonian in isotropic phase is given by

H¼^13 TrfgCuv: ½ 2 : 313 Š

This is the form of the nuclear spin Hamiltonians considered thus far
in this text. For example, the chemical shift Hamiltonian [2.98] has the
form of [2.313] withuEv¼ B 0 Izand^13 Trfg¼ris the isotropic chemical
shielding. The strong scalar coupling Hamiltonian [2.154] has the form
of [2.313] withu¼(Ix,Iy,Iz)T; v¼(Sx,Sy,Sz)T;C¼J, whereJis the
scalar coupling tensor; and^13 TrfJg¼ 2 JIS. The dipole–dipole and
quadrupole tensors are traceless and consequently do not contribute to
the rotationally averaged nuclear spin Hamiltonian in isotropic phase.
The isotropy of a solution of molecules is destroyed if the molecules
are subject to a potential of mean force,W( , , ), that depends on the
orientation of a molecular fixed frame, relative to the laboratory
reference frame. The time-dependent Euler angles { , , } describe the
relative orientation of these two frames of reference. The probability
distribution is given by the Boltzmann equation:

pðÞ¼ , ,

R exp½Š
WðÞ^ ,^ ,^ =kBT

exp½Š
WðÞ , , =kBT sin d d d



1

8 ^2



ðÞ 1 
WðÞ , , =kBT: ½ 2 : 314 Š

The second equality is obtained by assuming that the potential of mean
force is weak and that
Z
WðÞ , , sin d d d ¼ 0 : ½ 2 : 315 Š

Because the probabilityp( , , ) is unaffected by adding a constant to
W( , , ), the zero of potential energy always can be chosen to satisfy
this constraint. The Wigner rotation matrices form a complete set;
therefore, the probability density also can be expressed as ( 20 )

pðÞ¼ , ,

1

8 ^2

X

l

ðÞ 2 lþ 1

Xl

m,n¼
l

DlmnðÞ , , 

DlmnðÞ , , : ½ 2 : 316 Š

2.8 AVERAGING OF THESPINHAMILTONIANS 107