QMGreensite_merged

For example, using the strong coupling eigenbasis [2.158], the matrix
element½R
x^1 ðފ 12 is calculated as

R
x^1 ðÞ

   

12 ¼hj
^1

N

j¼ 1

Ecos^

2

þisin^

2

Tj



ji
 2

¼hj

N

j¼ 1 Ecos

2 þisin

2 Tj



cos

þsin



¼hj Ecos^2

2

þicos^

2

sin^

2

T 1 þicos^

2

sin^

2

T 2 
sin^2

2

T 1 T 2



 cos

þsin



¼icos^

2

sin^

2

sinþicos^

2

sin^

2

cos

¼icos^2 sin^2 ðÞcosþsin:

½ 2 : 174 Š

This result is calculated using the property that the inversion operatorTj
changes the spin state of spinjfrom to andvice versa. As another
example, R
x^1 ðÞ

   

14 is given by

R
x^1 ðÞ

   

14 ¼hj
^1

N

j¼ 1

Ecos^

2

þisin^

2

Tj



ji
 4

¼hj

N

j¼ 1

Ecos

2 þisin

2 Tj



¼hj Ecos^2

2 þicos

2 sin

2 T^1 þicos

2 sin

2 T^2 
sin

2

2 T^1 T^2



¼
sin^2

2 :

½ 2 : 175 Š

Repeating these calculations for every element of the matrix representa-
tion of the pulse operator yields

R
x^1 ðÞ¼

c^2 icsu icsv 
s^2

icsu 1 
s^2 u^2 
s^2 uv icsu

icsv 
s^2 uv 1 
s^2 v^2 icsv


s^2 icsu icsv c^2

2

6

(^64)
3
7
(^75) , ½ 2 : 176 Š
where c¼cos( /2), s¼sin( /2), u¼cosþsin, and v¼cos– sin.
Because the rotation operators are unitary, Rx( ) is the adjoint
66 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY