Biomedical Engineering Reference
In-Depth Information
r
2
d
R
d
r
8π
2
m
e
h
2
R
1
r
2
d
d
r
λ
r
2
+
(ε
−
U
)
−
=
0
(13.28)
sin θ
∂
Y
∂θ
1
sin θ
∂
∂θ
1
sin
2
θ
∂
2
Y
∂φ
2
+
+
λ
Y
=
0
I have discussed the second equation inAppendixAwhen considering angular momentum;
the allowed solutions are the spherical harmonics
Y
l
,
m
l
(
θ , φ
)
, where λ
1) and
l
and
m
l
are integers. Introducing the value for λ into the first equation of (13.28) and expanding
the first term gives
=
l
(
l
+
8π
2
m
e
h
2
R
d
2
R
d
r
2
2
r
d
R
d
r
+
l
(
l
+
1)
+
(ε
−
U
)
−
=
0
(13.29)
r
2
or, in terms of
P
(
r
) introduced above (where
P
(
r
)
=
rR
(
r
))
8π
2
m
e
h
2
P
+
d
2
P
d
r
2
l
(
l
1)
+
−
−
=
(ε
U
)
0
(13.30)
r
2
The term in
l
(
l
1) is called the
centrifugal potential
; it adds to the Coulomb term to give
an
effective potential
.
The radial equation (13.30) is more complicated than Equation (13.20) because of the
+
l
(
l
1)/
r
2
term but in essence the same method of solution can be used. The details are
given in standard traditional quantum chemistry texts such as Eyring, Walter and Kimball
(EWK) (Eyring
et al.
1944). The radial solutions are a set of functions from mathematical
physics called the
associated Laguerre polynomials
(the Laguerre polynomial
L
α
(
x
) of
degree α in
x
is defined as
+
d
α
d
x
α
L
α
(
x
)
=
exp (
x
)
(
x
α
exp (
−
x
))
and the βth derivative of
L
α
(
x
) is called an associated Laguerre polynomial).
13.7.1
l
=
0 (s-orbitals)
The atomic orbitals are named depending on the three quantum numbers. The first three
s-orbitals are given in Table 13.2; they are conventionally written in terms of the variable
ρ
=
Zr
/
a
0
(where
Z
=
1 for hydrogen).
Table 13.2
The first few s-orbitals
n
,
l
,
m
Symbol Normalized wavefunction
Z
a
0
3
/
2
1
√
π
1, 0, 0
1s
exp
(
−
ρ)
Z
a
0
3
/
2
−
ρ)
exp
2
1
4
√
2
π
ρ
2, 0, 0
2s
(
2
−
Z
a
0
3
/
2
2
ρ
2
exp
3
27
2
81
√
3
π
ρ
3, 0, 0
3s
−
18
ρ
+
−
