Biomedical Engineering Reference
In-Depth Information
For this equation to hold for all values of the variable
r
, the coefficients of all powers of
r
must separately vanish; thus
β
2
2ε
α
2
=
+
0
β
2
2ε
α
1
+
+
2 (
Z
−
2β) α
2
=
0
β
2
2ε
α
0
+
+
2 (
Z
−
β) α
1
+
2α
2
=
0
2
Z
α
0
=
0
It follows that
β
2
2
=−
ε
Z
2
β
=
Z
α
1
+
2α
2
=
0
α
0
=
0
The first two equations give an energy of
−
Z
2
/8 and because ε is actually the energy divided
by an atomic unit of energy
E
h
we have
Z
2
8
E
h
=−
ε
Z
2
2
2
E
h
The third and fourth equations determine all the coefficients but one, which we are free to
choose. If we take α
1
=
1
2
=−
1 then α
2
=−
Z
/2 and we have
Z
2
r
2
Φ (
r
)
=−
+
r
(13.25)
13.7 Atomic Orbitals
We must now examine the general case where the wavefunction depends on three variables.
Substitution of
ψ (
r
, θ , φ)
=
R
(
r
)
Y
(θ , φ)
(13.26)
into the electronic Schrödinger equation (Equation (13.17)) gives
r
2
∂
R
∂
r
sin θ
∂
Y
∂θ
1
R
∂
∂
r
8π
2
m
e
h
2
1
Y
sin θ
∂
∂θ
1
Y
sin
2
θ
∂
2
Y
∂φ
2
(13.27)
+
(ε
−
U
)
r
2
=−
−
By our usual argument, both sides of this equation must be equal to a constant that we will
call λ. We then have two equations
