Chemistry Reference
In-Depth Information
the vector potential A). In fact:
L
z
¼
i
@
@w
L
z
Y
'
m
ðWÞ¼
mY
'
m
ðWÞ
ð
10
:
76
Þ
where
Y
'
m
ðWÞ/
exp
ð
i m
wÞ
P
m
'
ð
cos
uÞ
ð
10
:
77
Þ
is a spherical harmonic in complex form (
W
stands for the solid angle
specified by
u
and
w
) and
m
¼
0
;
1
;
2
; ...; '
ð
10
:
78
Þ
is the magnetic quantum number.
Then, using RS perturbation theory up to second order in the field H:
E
1
¼hc
0
j
H
1
jc
0
i¼hc
0
jm
L
H
jc
0
i
eh
eh
ð
10
:
79
Þ
2mc
L
H
jc
0
i¼
2mc
H
hc
0
j
L
z
jc
0
i¼
0
¼hc
0
j
since m
¼
0 for a spherical ground state;
e
2
8mc
2
H
2
E
2
¼hc
0
j
H
2
jc
0
i¼
hc
0
j
x
2
þ
y
2
jc
0
i
ð
10
:
80
Þ
e
2
12mc
2
H
2
h
r
2
¼
i
00
and we obtain the Langevin contribution to the molar diamagnetic
susceptibility:
e
2
6mc
2
h
r
2
L
792
10
6
h
r
2
x
¼
N
A
i
00
¼
0
:
i
00
<
0
ð
10
:
81
Þ
a negative contribution (when r is in au, the susceptibility is given in
cgs/emu).
Table 10.3gives the calculateddiamagnetic susceptibilities for a few
simple atoms. The value for the H atom is exact. For the two-electron
atomic system, He, the hydrogenic approximation
ð
c
¼
Z
¼
2
Þ
gives
