Chemistry Reference
In-Depth Information
where
F
½
n
a
;
n
b
¼
min
!ð
n
b
Þ
ð
T
½þ
V
ee
½Þ
½
6
n
a
;
¼
½
n
a
;
n
b
þ
½
n
a
;
n
b
½
7
T
V
ee
is a universal functional independent of
v
ext
s
ð
r
Þ
.
Minimizing the total energy density functional, Eq. [5], for both spin
densities by taking the functional derivative
d
=
d
n
s
, and using the Euler-
Lagrange multiplier technique leads to
d
F
½
n
a
;
n
b
þ
v
ext
s
ð
r
Þ¼
m
½
8
d
n
s
where
is the chemical potential of the system.
Next we imagine a system of noninteracting electrons with the same spin
densities. Applying the HK theorem to this noninteracting system, the poten-
tials,
v
S
s
, that give densities
n
s
ð
r
Þ
m
as the ground-state spin densities for this
system are unique. This is the fictitious Kohn-Sham (KS) system,
2
and the fully
interacting problem is mapped to a noninteracting one that gives the exact
same density. Solving the KS equations, which is computationally simple (at
least compared to the fully interacting problem, which becomes intractable
for large particle numbers), yields the ground-state density. The KS equations
are
1
2
2
r
þ
v
S
s
ð
r
Þ
f
j
s
ð
r
Þ¼
E
j
s
f
j
s
ð
r
Þ
½
9
with spin densities
X
N
s
2
n
s
ð
r
Þ¼
1
j
f
j
s
ð
r
Þj
½
10
j
¼
where
v
S
a
,
v
S
b
are the KS potentials and
N
s
is the number of spin
s
electrons,
(
N
a
þ
N
).
In the top panel of Figure 2, we plot the exact density for the He atom
from a highly accurate wave function calculation, and below that we plot the
exact
KS potential for this system. One can see that the KS potential is very
different from the external potential. This is due to the fact that the KS single
effective potential for the noninteracting system must give the correct interact-
ing electron density. Because the Coulomb repulsion between the electrons
shields the nucleus, and makes the charge density decay less rapidly than
e
4
r
, the KS potential is more shallow than
v
ext
N
b
¼
ð
r
Þ
.
