Biomedical Engineering Reference
In-Depth Information
∂
2
ψ
c
∂
x
c
+
h
2
8π
2
(
m
e
+
∂
2
ψ
c
∂
y
c
+
∂
2
ψ
c
∂
z
c
−
=
(ε
tot
−
ε
e
) ψ
c
(13.16)
M
)
1
r
2
r
2
∂ψ
e
∂
r
sin θ
∂ψ
e
∂θ
h
2
8π
2
μ
∂
∂
r
1
r
2
sin θ
∂
∂θ
1
r
2
sin
2
θ
∂
2
ψ
e
∂φ
2
−
+
+
+
U
ψ
e
=
ε
e
ψ
e
(13.17)
The first equation relates to the translation of the atom as a whole, and I have dealt with such
equations in earlier chapters. The second equation is usually called the
electronic equation
.
It should be clear from this discussion that in the treatment of any atomic or molecular
system the translational degree(s) of freedom may always be separated from the internal
degrees of freedom and so need not be considered in general. Also, from now on, I will
drop the subscript 'e' from the electronic equation.
In the special case of a one-electron atom, the electronic wavefunction depends only
on the coordinates of (the) one electron and so it is technically described as an
atomic
orbital
. In view of the spherical symmetry, we might expect that it would prove profitable
to consider a further separation of variables
ψ (
r
, θ , φ)
=
R
(
r
)
Y
(θ , φ)
(13.18)
This proves to be the case and
R
(
r
) is usually called the
radial function
(although I should
warn you that many authors call
P
(
r
)
rR
(
r
) the radial function; we will see why this
proves a useful choice shortly). The
Y
's turn out to be eigenfunctions of the orbital angular
momentum, and again we will see why in due course.
It turns out that most of thewavefunction solutions of Equation (13.18) are not spherically
symmetrical, and you may find this odd given that the mutual potential energy
U
depends
only on the scalar distance
r
(we call such fields
central fields
). The fact is that
U
(
r
)
being spherically symmetrical does not imply that the solutions are also spherical; the
gravitational field is also a central field, and you probably know that planetary motion is
not spherically symmetrical.
=
13.6 Radial Solutions
Before launching into a discussion of the full solutions, I want to spend a little time consid-
ering just those solutions that depend only on
r
. That is, those solutions that have a constant
angular part; this is not the same thing as finding
R
(
r
) for the general case. I will also make
the infinite nucleus approximation from this point on.
A little rearrangement and manipulation gives, for functions that depend only on
r
,
h
2
8π
2
m
e
d
2
(
rR
)
d
r
2
Ze
2
4πε
0
r
(
rR
)
−
−
=
ε (
rR
)
(13.19)
The form of this equation suggests that we make the substitution
P
rR
which explains
why some authors focus on
P
(
r
) rather than
R
(
r
) and refer to
P
(
r
) as the radial function.
This substitution gives
=
