Chemistry Reference
In-Depth Information
As we have seen before, the quantization of the solution comes about when we impose
the boundary conditions; because the electron is confined to move around the nucleus, it
can only occupy particular states: those with an integer
m
l
quantum number.
Equation (A9.17) provides the functional form for
), but we usually work with
normalized wavefunctions. Since the overall wavefunction is a product of angular and
radial functions, normalizing each term individually will ensure a normalized function
when they are brought together.
We can obtain the normalization factor
N
φ
m
l
(
φ
m
l
(
φ
for
) from the integral
π
π
∗
d
φ
=
N
2
φ
exp(
−
i
m
l
φ
)exp(i
m
l
φ
)d
φ
=
1
(A9.18)
−
π
−
π
where the limits are chosen to give a complete revolution around the
Z
-axis. This is a
straightforward integral to do because the product of exponentials must give 1 for any
value of
m
l
.Sowehave
π
1
√
2
N
2
φ
d
φ
=
1
which gives
N
φ
=
(A9.19)
π
−
π
The normalized solution can then be written as
1
√
2
m
l
=
exp( i
m
l
φ
)
(A9.20)
π
The direct solution of Equation (A9.15), for
), is much more involved and we leave
that to texts dedicated to quantum mechanics, some of which are referenced in the Further
Reading section of this appendix. The normalized solutions are
lm
l
(
θ
2
l
(
l
P
|
+
1
− |
m
l
|
)!
|
m
l
lm
l
(
θ
)
=
( cos
θ
)
(A9.21)
l
2
(
l
+ |
m
l
|
)!
Here,
P
|
l
are the associated Legendre polynomials, which can be obtained from the
sources in the Further Reading section of this appendix. We go this far to note that
m
l
lm
l
is only defined for
l
being an integer and
|
m
l
| ≤
l
, since, while 0! is taken to be 1, the
factorial of a negative number is not defined.
The combination of
m
l
to give the angular part of the wavefunction defines
the spherical harmonic functions,
Y
lm
l
=
lm
l
m
l
, which are the solutions to the full angu-
lar equation. The first six spherical harmonics are sufficient for us to develop H atom
wavefunctions up to d-orbitals, and these are listed in Table A9.1 along with the real
combinations that are in common usage, which are described later. Notice that all these
functions conform to both of the angular boundary conditions given in Equation (A9.16),
and so the spherical harmonics give continuous functions in
lm
l
and
θ
and
φ
.
A general property of the spherical harmonic functions is that
2
θφ
−
Y
lm
l
=
l
(
l
+
1
)
Y
lm
l
(A9.22)
