Biomedical Engineering Reference
In-Depth Information
In Hartree-Fock (HF) theory we write the wavefunction as a Slater determinant and this
gives an additional exchange term in the electronic energy
M
ψ
R
(
r
1
)
h
(
r
1
) ψ
R
(
r
1
) dτ
1
ε
el
=
2
R
=
1
2
ψ
R
(
r
1
) ψ
R
(
r
1
)
M
M
+
g
(
r
1
,
r
2
)ψ
S
(
r
2
) ψ
S
(
r
2
) dτ
1
dτ
2
ˆ
(20.2)
R
=
1
S
=
1
ψ
R
(
r
1
)
ψ
S
(
r
1
)
M
M
−
g
(
r
1
,
r
2
)
ψ
R
(
r
2
)
ψ
S
(
r
2
)
d
τ
1
d
τ
2
ˆ
R
=
S
=
1
1
These expressions are general to Hartree and HF theory in that they do not depend on the
LCAO approximation. In fact the Hartrees produced highly accurate numerical tables of
radial functions with no LCAO coefficients in sight. The atomic problem is quite different
from the molecular one because of the high symmetry of atoms and the HF limit can be
easily reached by numerical integration. It is not necessary to invoke the LCAO approach.
There has been a resurgence of interest in atomic HF calculations because astrophysicists
want to study highly ionized species in the interstellar medium. They look to theory for
their energy-level data because the species are hard to prepare and study in the laboratory.
I have also described the LCAO versions, where the energy expressions are written in
terms of the 'charges and bond orders' matrix
P
together with the matrices
h
(1)
,
J
and
K
:
1
2
Tr
(
PJ
)
ε
el,H
=
Tr
(
Ph
1
)
+
(20.3)
for the Hartree model and
1
2
Tr (
PJ
)
1
4
Tr (
PK
)
ε
el,HF
=
Tr (
Ph
1
)
+
−
(20.4)
for the HF version. The
J
and
K
matrices depend on
P
in a complicated way, but the
Hartree and HF electronic energies are completely determined from knowledge of the
electron density
P
.
There was initially a great deal of confusion about the extra term in the HF energy
compared to the Hartree energy. The name 'exchange term' was coined and some authors
tried to describe it in terms of a mysterious force called the 'exchange potential'.
In themeantime, solid-state physics had been developing along a quite different direction.
Wigner and Seitz (1934a,b) suggested what is now called the 'cellular method' for handling
the problemof computing crystal orbitals. These orbitals have energies that formcontinuous
bands and such models are therefore known as
energy band theories
.
20.1 Pauli and Thomas-Fermi Models
There is nothing sinister about the exchange term; it arises because of the fermion nature
of electrons. Nevertheless, it caused considerable confusion among early workers in the
field of molecular structure theory and much effort was spent in finding effective model
potentials that could mimic electron exchange. I discussed the earliest models of metallic
