related to the operator
^
M
i
I ¼
l 0
4p
r i À R I
r i À R I
j
j
3
 ^ p i ¼
l 0
4p
^ l i ðR I Þ
r i À R I
j
j
3
:
ð7:21Þ
The operator ^ l i ðR I Þ represents the angular momentum of electron i with respect
to the origin at nucleus I with position R I [3].
The spin-dipolar and the Fermi contact terms of the Ramsey theory [12, 13] are
obtained from the general expression for the current density
J ðr Þ ¼ r  MðrÞ; MðrÞ ¼ À
e
m e
Qðr Þ;
ð7:22Þ
due to the magnetization density M associated to the electron spin density defined
via the spin density matrix [59, 62],
Q ðr; r
0
Þ ¼
Z
s¼s 0
^ scðx; x
0
Þds; QðrÞ ¼ Qðr; rÞ;
ð7:23Þ
denoting by ^ s the spin operator. The corresponding contributions to the current
density are [4, 61, 62]
J
m I
SD ðrÞ ¼ À
en
m e
Z
dx 2 . . .dx n
 $  m I Á W
SDÃ
a ðr; x 2 ; . . .x n Þ^ sW
ð0Þ
a ðr; x 2 ; . . .x n Þ
n
þ W
ð0ÞÃ
a ðr; x 2 ; . . .x n Þ^ sW
SD
a ðr; x 2 ; . . .x n Þ Á m I
o
;
ð7:24Þ
J
m I
FC ðrÞ ¼ À
en
m e
Z
dx 2 . . .dx n
 $  m I Á W
FCÃ
a ðr; x 2 ; . . .x n Þ^ sW
ð0Þ
a ðr; x 2 ; . . .x n Þ
n
þ W
ð0ÞÃ
a ðr; x 2 ; . . .x n Þ^ sW
FC
a ðr; x 2 ; . . .x n Þ Á m I
o
;
ð7:25Þ
In these cases the current field is parallel to $ Â ^ s.
Definitions of the second-order energy alternative to the RSPT’s, Eq. (7.4), are
obtained in terms of J
B and J
m I , via relationships of classical electromagnetism
easily applicable to calculate magnetizability, nuclear magnetic shielding, and
reduced spin-spin coupling tensors, that is, via the expression
W
BB
¼ À
1
2
Z
A
B
Á J
B d
3 r;
ð7:26Þ
158
P. Lazzeretti