Chemistry Reference
In-Depth Information
_
+
+
z
= 0
= 1
.
_
_
_
+
+
+
+
_
z
_
z
+
= 2
= 3
Figure 3.2 Schematic polar diagrams of the angular part of s, p, d, and f AOs with
m
¼
0
Q
'
0
with
7
The polar diagrams of the angular functions
' ¼
0
;
1
;
2
;
3 are
sketched in Figure 3.2.
3.2.4 Some Properties of the Hydrogen-like Atomic Orbitals
The HAOs in complex form
c
n
'
m
ð
r
; u; wÞ¼
R
n
'
ð
r
Þ
Y
'
m
ðu; wÞ
ð
3
:
41
Þ
satisfy simultaneously the following three eigenvalue equations:
<
:
H
c
n
'
m
¼
E
n
c
n
'
m
n
¼
1
;
2
;
3
; ...
2
L
ð
3
:
42
Þ
c
n
'
m
¼ 'ð' þ
1
Þc
n
'
m
' ¼
0
;
1
;
2
;
3
; ...; ð
n
1
Þ
L
z
c
n
'
m
¼
m
c
n
'
m
m
¼
0
;
1
;
2
;
3
; ...; '
where H is given by (3.1) and E
n
by (3.5). Each eigenvalue is character-
ized by one quantum number. So, the principal quantum number n
7
It is seen that
'
equals the number of nodal planes of the functions.
