7.2 Electron Current Density Induced by Magnetic Fields
and Nuclear Magnetic Dipoles
The expression for the quantum mechanical current density induced in the electron
cloud by a homogeneous, static magnetic field B ¼ $ Â A
B , where
A
B
¼
1
2
B Â r
ð7:7Þ
is the vector potential in the Coulomb gauge, is obtained as a sum of diamagnetic
and paramagnetic contributions by theoretical procedures illustrated by McWeeny
[59] and employed in previous Refs. [3, 15, 60], whose notation is used in the
following. The diamagnetic contribution,
J
B
d ðrÞ ¼ À
e
2
m e
A
B
ðrÞc
ð0Þ
ðrÞ;
ð7:8Þ
is related to the probability density of the unperturbed molecule [59],
c
ð0Þ
ðrÞ ¼ n
Z
dx 2 . . .dx n W
ð0Þ
a ðr; x 2 ; . . .x n ÞW
ð0ÞÃ
a ðr; x 2 . . .x n Þ;
ð7:9Þ
and the paramagnetic contribution, parallel to the canonical momentum vector ^ p, is
given by
J
B
p ðrÞ ¼ À
ne
m e
Z
dx 2 . . .dx n
 B Á W
BÃ
a ðr; x 2 ; . . .x n Þ^ pW
ð0Þ
a ðr; x 2 ; . . .x n Þ
h
þ W
ð0ÞÃ
a ðr; x 2 . . .x n Þ^ pB Á W
B
a ðr; x 2 ; . . .x n Þ
i
:
ð7:10Þ
The total current density is obtained by summing,
J
B
¼ J
B
d þ J
B
p :
ð7:11Þ
Analogous expressions are found for the current density induced by a nuclear
magnetic dipole m I via the Ramsey nuclear spin/electron orbit interaction [12, 13,
15, 16, 51, 52, 61–63]. The diamagnetic contribution is connected to the probability
density of the unperturbed system [59], Eq. (7.9),
J
m I
d ðrÞ ¼ À
e
2
m e
A
m I ðr À R I Þc
ð0Þ
ðrÞ;
ð7:12Þ
156
P. Lazzeretti