Chemistry Reference
In-Depth Information
achieved if
a
k
=
0
;
a
k
þ
1
¼
a
k
þ
2
¼¼
0
ð
3
:
12
Þ
which implies from (3.11) the necessary condition
k
n
þ ' þ
1
¼
0
)
k
max
¼
n
'
1
ð
3
:
13
Þ
The physically acceptable radial solutions must, hence, include a
polynomial of degree
ð
n
'
1
Þ
at most.
3
Equation 3.13 determines our,
so far, unknown parameter n:
n
¼
k
þ ' þ
1
)
n
¼ ' þ
1
;'þ
2
; ...
)
n
' þ
1
ð
3
:
14
Þ
and, in this way, we obtain the well-known relation between principal
quantum number n and orbital quantum number
'
.
we see that our radial functions R(x)
will depend on quantum numbers n and
Coming back to R
ð
x
Þ¼
P
ð
x
Þ=
x
;
'
, being written in un-normalized
form as
R
n
'
ð
x
Þ¼
exp
ð
x
Þ
x
'
X
n
'
1
a
k
x
k
ð
3
:
15
Þ
k
¼
0
with
n
¼
1
;
2
;
3
;
4
; ...
' ¼
0
;
1
;
2
;
3
; ...; ð
n
1
Þ
ð
3
:
16
Þ
The functions in Equation 3.15 have
ð
n
'
1
Þ
nodes, namely those
values of x for which the function changes sign. The detailed form of the
first few radial functions can be readily obtained from (3.15) and (3.16)
using the recursion formula (3.11) for the coefficients. It is found that
3
Mathematically speaking, these are related to the associated Laguerre polynomials L
2
' þ
1
n
þ '
ð
x
Þ
(Eyring et al., 1944).
