j i$j itransition is obtained by inverting the phase of the final
y-pulse. The propagatorU¼exp(iI^ Sx) is obtained by shifting the
phases of all pulses by/2.
2.8 Averagingof the Spin Hamiltonians and Residual Interactions
The presentations in the preceding sections of this chapter have
focused on evolution of the density operator under the isotropic
components of the nuclear spin Hamiltonian. At this point, a more
formal analysis is presented of the nuclear spin Hamiltonian and the
effects of isotropic and nonisotropic averaging in solution. The nuclear
spin Hamiltonians important in NMR spectroscopy of diamagnetic
molecules are described most generally in the form
H¼uTCv, ½ 2 : 297 in whichuandvare vectors, the superscript T indicates the transpose,
andCis a general second-rank Cartesian tensor. The Cartesian tensorC
is represented by a 33 matrix and can be decomposed into the sum of
irreducible tensors of rank 0, 1, and 2:
C¼Cð^0 ÞþCð^1 ÞþCð^2 Þ, ½ 2 : 298 in which Cð^0 Þ¼^13 TrfCgE,Cð^1 Þ¼ðCCTÞ=2 is traceless and anti-
symmetric, and C(2)¼(CþCT)/2 –C(0) is traceless and symmetric.
The vectoru normally will be an angular momentum operator, the
vectorvwill be an angular momentum operator or a magnetic field
vector, andCwill depend on the particular magnetic spin interaction
being considered. For example, the chemical shielding Hamiltonian
(introduced in Chapter 1, Section 1.5) is described, in the laboratory
reference frame, byuT¼(Ix,Iy,Iz),v¼(0, 0, B 0 )T, andC¼r, in which
r¼ 11 12 13
21 22 23
31 32 332
43(^5) ½ 2 : 299
is the Cartesian nuclear shielding tensor for theIspin (which should not
be confused with the density operator in this context).
The observation that the nuclear spin Hamiltonian must be invariant
to rotation has profound consequences for NMR spectroscopy because
this constraint limits the types of interactions that can couple to the
nuclear spin angular momentum operators. The antisymmetric tensor
102 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY