Chemistry Reference
In-Depth Information
"
#
ð'
m
Þ=
X
2
ð'
m
1
Þ=
X
2
m
=
2
Q
'
m
ð
x
Þ¼ð
1
x
2
a
2k
x
2k
a
2k
þ
1
x
2k
þ
1
Þ
þ
ð
3
:
38
Þ
k
¼
0
k
¼
0
where the first term in brackets is the even polynomial and the second
term is the odd polynomial, whose degree is at most k
max
¼ '
m
ð
0
Þ
for both.
Using (3.38) together with the recursion formula (3.29) with
l
¼ 'ð' þ
1
Þ
, we obtain the following for the first few angular solutions
ð
x
¼
cos
uÞ
:
<
' ¼
0
;
m
¼
0
'
m
¼
0
Q
¼
a
0
00
' ¼
1
;
m
¼
0
'
m
¼
1
Q
¼
xa
1
/
cos
u
10
Q
20
¼
a
0
þ
a
2
x
2
¼ð
1
3x
2
' ¼
2
;
m
¼
0
'
m
¼
2
Þ
a
0
1
=
2
¼ð
1
x
2
' ¼
1
;
m
¼
1
'
m
¼
0
Q
Þ
a
0
/
sin
u
11
:
1
=
2
Q
21
¼ð
1
x
2
' ¼
2
;
m
¼
1
'
m
¼
1
Þ
xa
1
/
sin
u
cos
u
Þ
a
0
/
sin
2
Q
22
¼ð
1
x
2
' ¼
2
;
m
¼
2
'
m
¼
0
u
Þ
xa
1
/
sin
2
¼ð
1
x
2
' ¼
3
;
m
¼
2
'
m
¼
1
Q
u
cos
u
32
ð
3
:
39
Þ
where a
0
and a
1
are normalization factors for the even polynomials and
odd polynomials respectively. It is seen that the angular solutions (3.39)
are combinations of simple trigonometric functions proportional to the
associated Legendre polynomials P
m
'
ð
x
Þ
, well known in mathematics in
potential theory (Abramowitz and Stegun, 1965; Hobson, 1965) with a
proportionality factor
m
þ½ð' þ
m
Þ=
2
P
m
'
ð
x
Þ¼ð
1
Þ
Q
'
m
ð
x
Þ
ð
3
:
40
Þ
where
½
stands for 'integer part of', a factor anyway irrelevant fromthe
standpoint of the differential equation. The skilled reader can verify the
solutions (3.39) by direct substitution in the differential equation (3.25)
with
l
¼ 'ð' þ
1
Þ
.
