Biomedical Engineering Reference
In-Depth Information
between Dirac's and Slater's numerical coefficient seems to have been first resolved by
Gáspár (1954), and authors began to write the
local exchange potential
as
U
X
α
(
r
)
=
C
α
(
P
1
(
r
))
1/3
(20.10)
where α could take values between 2/3 and 1. This is the so-called
X
α
model
. The use
of the symbol α here is not to be confused with the use of the same symbol for a spin
eigenfunction.
Slater then had the ingenious idea of writing the atomic HF eigenvalue equation
h
F
(
r
) ψ (
r
)
=
εψ (
r
)
2
K
(
r
)
ψ (
r
)
1
h
(1)
(
r
)
+
J
(
r
)
−
=
εψ (
r
)
as
+
U
X
α
(
r
)
ψ (
r
)
h
(
1
)
(
r
)
+
J
(
r
)
=
εψ (
r
)
(20.11)
and such calculations are usually referred to as
atomic X
α
-HF
. The resulting Xα orbitals
differ from conventional HF orbitals in one major way, namely that Koopmans' theorem
is no longer valid for every orbital and so the orbital energies cannot generally be used to
estimate ionization energies. Koopmans' theorem now applies only to the highest occupied
orbital. A key difference between standard HF theory and density functional calculations
is the way we conceive the
occupation number
ν of each orbital. In molecular HF theory,
the occupation number is 2, 1 or 0 depending on whether a given spatial orbital is fully
occupied by two electrons (one of either spin), singly occupied or a virtual orbital. For a
systemcomprising verymany electrons we focus on the statistical occupation of each orbital
and the occupation number becomes a continuous variable having a value between 0 and 2.
The relationship between the electronic energy ε
el
and the occupation number of orbital
i
is
∂ε
el
∂ν
i
=
ε
i
(20.12)
so that the ionization energy from a particular orbital ψ
i
is given by
0
ε
el
(ν
i
=
0)
−
ε
el
(ν
i
=
1)
=
ε
i
dν
i
1
Notice, however, that the energy is determined by the charge density
P
(
r
). When the
calculations are performed, the resulting orbitals closely resemble those from standard HF
theory and people use them in much the same way.
At the time, such calculations of ionization energies had an edge over the traditional HF
variety because they take account of the electron relaxation that can accompany ionization.
20.2 Hohenberg-Kohn Theorems
The Thomas-Fermi and Xα approaches were constructed as approximations to the quantum
mechanical problem of calculating the electronic properties of a system of interest. The
density functional theory of Hohenberg and Kohn, to be discussed, is in principle an exact
