Biomedical Engineering Reference
In-Depth Information
20.3 Kohn-Sham (KS-LCAO) Equations
The Kohn-Sham paper gives us a practical solution to the problem, based on HF-LCAO
theory; again, it is such an important paper that you should have sight of the abstract that
in any case is self-explanatory (Kohn and Sham 1965).
From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous
system of interacting electrons are developed. These methods are exact for systems of slowly
varying or high density. For the ground state, they lead to self-consistent equations analogous
to the Hartree and Hartree-Fock equations, respectively. In these equations the exchange and
correlation portions of the chemical potential of a uniform electron gas appears as additional
effective potentials. (The exchange portion of our effective potential differs from the due to
Slater by a factor of 2/3.) Electronic systems at finite temperatures and in magnetic fields are
also treated by similar methods. An appendix deals with a further correction for systems with
short-wavelength density oscillations.
The Kohn-Sham equations are modifications of the standard HF equations and we write
+
U
XC
(
r
)
ψ (
r
)
h
(1)
(
r
)
+
J
(
r
)
=
εψ (
r
)
(20.15)
where
U
XC
(
r
) is the local exchange-correlation term that accounts for the exchange phe-
nomenon and the dynamic correlation in themotions of the individual electrons.We speak of
the KS-LCAO procedure. It is usual to split
U
XC
(
r
) into an exchange term and a correlation
term and treat each separately.
+
U
C
(
r
)
ψ (
r
)
h
(1)
(
r
)
+
J
(
r
)
+
U
X
(
r
)
=
εψ (
r
)
(20.16)
The electronic energy is usually written in DFT applications as
ε
el
[
P
]
=
ε
1
[
P
]
+
ε
J
[
P
]
+
ε
X
[
P
]
+
ε
C
[
P
]
(20.17)
where the square brackets denote a functional of the one-electron density
P
(
r
). The first
term on the right-hand side gives the one-electron energy, the second term is the Coulomb
contribution, the third term the exchange and the fourth term gives the correlation energy.
We proceed along the usual HF-LCAO route; we choose a basis set and then all that is needed
in principle is knowledge of the functional forms of
U
X
(
r
) and
U
C
(
r
). It is then a simple
matter in principle to modify an existing standard HF-LCAO computer code to include
these additional terms. Either component can be of two distinct types:
local functionals
that
depend only on the electron density at a point in space and
gradient-corrected functionals
that depend both on the electron density and its gradient at that point.
In their most general form, the exchange and correlation functionals will depend on the
density of the α and of the β electrons, and the gradients of these densities. Each energy
term ε
C
and ε
X
will therefore be given by a volume integral
f
X
P
α
,
P
β
, grad
P
α
, grad
P
β
dτ
ε
X
=
f
C
P
α
,
P
β
, grad
P
α
, grad
P
β
dτ
ε
C
=
(20.18)
where
f
is an energy density.
