Biomedical Engineering Reference
In-Depth Information
The electronic Schrödinger equation for electron 1 is
ψ
B
r
12
dτ
B
ψ
A
=
h
2
8π
2
m
e
∇
Ze
2
4πε
0
r
1
+
e
2
4πε
0
2
−
−
ε
A
ψ
A
(14.25)
That gives us ψ
A
. We now focus on electron 2, for which
ψ
A
r
12
dτ
A
ψ
B
=
h
2
8π
2
m
e
∇
Ze
2
4πε
0
r
2
+
e
2
4πε
0
2
−
−
ε
B
ψ
B
(14.26)
and calculate ψ
B
then back to electron 1 so on. The calculation is an iterative one and
we stop once the change between iterations is sufficiently small. Each electron experi-
ences a field due to the remaining electrons, and at the end of the calculation the average
electron density derived from the field must be the same as the field and so Hartree
(1927) coined the phrase
self-consistent field
(SCF for short). William Hartree and his
son Douglas did much of the early work and so we speak of the
Hartree self-consistent
field method
.
The theory of atomic structure is dominated by angular momentum considerations, since
the square of the orbital angular momentum operator and its
z
component commute both
with each other and the electronic Hamiltonian. This simplifies the problem considerably
and the Hartrees wrote each atomic orbital as
1
r
P
(
nl
;
r
)
Y
l
,
m
l
(θ , φ)
ψ (
r
)
=
(14.27)
where
P
(
nl
;
r
) is a radial function and
Y
lm
a spherical harmonic. Their notation for
P
should
be clear; each shell has a different radial function. From now on I am going to make the
notation more consistent with previous chapters and write
1
r
P
nl
(
r
)
Y
l
,
m
l
(
θ , φ
)
Thus for fluorine we would expect three different radial functions,
P
1s
(
r
),
P
2s
(
r
) and
P
2p
(
r
).
Because of the spherical symmetry of atoms, all the 2p solutions have the same radial part.
Details of the method are given in the topic
The Calculation of Atomic Structures
(Hartree
1957) and essentially the radial functions are determined from the variation principle. That
is to say, they are chosen so as to minimize the variational energy
Φ
∗
H
Φ dτ
Φ
∗
Φ dτ
ψ
(
r
)
=
14.10 Hartree-Fock Model
Hartree's calculations were done numerically. It soon became apparent that these early
calculations gave energies that were in poor agreement with experiment; Fock (1930a,b)
pointed out that Hartree had not included the Pauli principle in his method. Essentially, the
Hartree model
considered a simple orbital product such as
Ψ
Hartree
=
ψ
A
(
r
1
) α (
s
1
) ψ
A
(
r
2
) β (
s
2
) ψ
B
(
r
3
) α (
s
3
) ψ
B
(
r
4
) β (
s
4
)
(14.28)
