Chemistry Reference
In-Depth Information
2
L
Y
¼
lY
ð
3
:
4
Þ
l
0 is a first separation constant.
1
Equation 3.3 is the differential
equation determining the radial part of the HAOs and Equation 3.4 is the
eigenvalue equation for the square of the angular momentum operator
L
where
2
determining the angular part of the orbitals. The latter equation is
found in general in the study of potential theory, the eigenfunctions
Y
ðu; wÞ
in complex form being known in mathematics as spherical
harmonics. At variance with the radial eigenfunctions R(r), which
are peculiar to the hydrogen-like system, the Y
ðu; wÞ
areusefulingeneral
for atoms.
3.2.2 Solution of the Radial Equation
The radial equation (3.3) has different solutions according to the value of
the parameter E, the eigenvalue of Equation 3.1.
E
>
0 corresponds to the continuous spectrum of the ionized atom, its
eigenfunctions being oscillatory solutions describing plane waves. It is of
no interest to us here, except for completing the spectrumof theHermitian
operator H.
E
<
0 corresponds to the electron bound to the nucleus, with a discrete
spectrum of eigenvalues, the energy levels of the hydrogen-like atom as
observed from atomic spectra.
It is customary to pose R
ð
r
Þ¼
P
ð
r
Þ=
r and
Z
2
2n
2
E
¼
ð
3
:
5
Þ
where n is a real integer positive parameter to be determined,
2
and to
change the variable to
Z
n
r
x
¼
ð
3
:
6
Þ
1
We shall see later in this section that
'
0 is the orbital quantum number.
2
It is seen that (3.5) is nothing but the result in atomic units of Bohr's theory for the hydrogenic
system of nuclear charge
þ
Z.
l
¼ 'ð' þ
1
Þ
, where
