Chemistry Reference
In-Depth Information
where
U
½
n
is the Hartree energy, given by
ð
d
3
rd
3
r
0
2
X
ss
0
s
0
ð
r
0
Þ
j
r
r
0
j
1
n
s
ð
r
Þ
n
U
½
n
¼
½
13
and the exchange-correlation (XC) energy is defined by Eqs. [7] and [12]:
E
XC
½
n
a
;
n
b
¼
T
½
n
a
;
n
b
T
S
½
n
a
;
n
b
þ
V
ee
½
n
a
;
n
b
U
½
n
½
14
Inserting Eq. [14] into Eq. [8] and comparing that to Eq. [11] gives a definition
of the KS potential:
v
S
s
ð
r
Þ¼
v
ext
ð
r
Þþ
v
H
ð
r
Þþ
v
XC
s
ð
r
Þ
½
15
where the Hartree potential is the functional derivative of
U
½
n
:
ð
d
3
r
0
X
s
0
ð
r
0
Þ
j
r
r
0
j
ð
r
Þ¼
d
U
½
n
n
v
H
ð
r
Þ
¼
½
16
d
n
s
0
while the XC potential is given by the functional derivative of the XC energy:
v
XC
s
ð
r
Þ¼
d
E
XC
½
n
a
;
n
b
½
17
d
n
s
ð
r
Þ
This derivation then closes the relationship between the KS system and
the original physical problem. Once
E
XC
½
n
a
;
n
b
is known exactly or is
approximated,
v
XC
s
ð
r
Þ
is determined by differentiation. The KS equations
can be solved self-consistently for the spin densities and orbitals, and the total
energy found by inserting those densities and orbitals into the total energy
functional
E
is not known
exactly and must be approximated. There exists in the scientific community a
functional soup
of many different approximations of varying accuracy and
computational cost, many of which are discussed in the following section.
For the He atom, when the KS equation is solved with the exact potential
as shown in Figure 2, the highest occupied molecular orbital (HOMO) level is
at
¼
T
S
þ
U
þ
E
XC
þ
V
ext
. Unfortunately
E
XC
½
n
a
;
n
b
592 eV, the negative of the ionization energy of helium. In exact DFT,
Koopmans theorem, which states
I
24
:
¼
E
HOMO
, is exactly true.
157
In ground-
state DFT, this is the only energy level of the fictitious KS system that has an
immediate physical interpretation.
Before leaving our tutorial on ground-state formalism, we mention the
optimized effective potential (OEP) method
158,159
in which the XC functional
is written as a functional of the KS orbitals (which in turn are functionals of
