in which!i is the Larmor frequency of the ith spin andJij is the
scalar coupling constant between theith and jth spins. The eigen-
functions of this Hamiltonian are used as the basis functions for the
construction of the matrix representation of the density operator. For
completeness, the effects ofstrong couplingmust be taken into account.
The product wavefunctions given by [2.138] are eigenfunctions ofH
only if 2Jij/|!i–!j| 1; this condition is known as theweak coupling
regime. If the weak coupling condition does not hold, then the spins
are said to be strongly coupled. In the strong coupling regime, the
wavefunctions in the product basis with the same total magnetic
quantum number become mixed and are no longer completely
independent. A proper basis set is obtained by taking linear combina-
tions of the subset of wavefunctions with the same value of m.
Construction of wavefunctions for strongly coupled spin systems with
N 4 2 is facilitated by use of group theoretical methods ( 8 ). Strong
coupling effects are particularly important in the analysis of coherence
transfer in isotropic mixing experiments; group theoretical analyses are
also important for treatment of identical spins (such as the three protons
in a methyl group).
To illustrate these ideas, a scalar coupled two-spin system, which
was treated in the weak coupling limit in Chapter 1, Section 1.6, will be
analyzed. The two spins will be labeledIandS. The free-precession
Hamiltonian laboratory frame for theISspin system is
H¼!IIzþ!SSzþ 2 JISIS, ½ 2 : 155 in which the scalar coupling constant isJIS. A system of two coupled
spins has the following four eigenfunctions:
1 ¼ji , 2 ¼cos
þsin
,
3 ¼cos
sin
, 4 ¼
,
½ 2 : 156 whereis known as the strong coupling parameter and is defined as
tanð 2 Þ¼
2 JIS
!I!S
½ 2 : 157 for 2in the range 0 toradians. If the spins have the same resonance
frequency, then¼/4 and the wavefunctions become
1 ¼ji , 2 ¼ 2 ^1 =^2(^) þ
(^) ,
3 ¼ 2 ^1 =^2
, 4 ¼
:
½ 2 : 158
62 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY