Chemistry Reference
In-Depth Information
equation. The Hohenberg-Kohn (HK) theorem
1
of 1964 states that for a given
nondegenerate ground-state density
n
ð
r
Þ
of fermions with a given interaction,
the external potential
v
ext
ð
r
Þ
that produced that density is unique (up to an addi-
tive constant). Hence if the density is known,
v
ext
ð
r
Þ
is then known and so
H
, the
Hamiltonian, is known. From this and the number of particles (determined by
the integral of the density), all properties of the system may be determined. In
particular, the ground-state energy of the system
E
would be known. This is
what we mean when we say these properties are functionals of the density,
e.g.,
E
. It was later shown that the HK theorem holds even for degenerate
ground states,
4
and modern DFT calculations use an analogous theorem applied
to the spin densities,
n
a
ð
r
Þ;
½
n
1
2
, respectively.
The total energy for
N
electrons consists of three parts: the kinetic energy
n
b
ð
r
Þ
, where
a
;
b
¼
½
, the electron-electron interaction
V
ee
[
], and the external potential
T
energy
V
ext
½
, each of which is defined in Eqs. [1], [2], and [3], respectively:
*
+
2
X
N
i
r
1
2
i
T
½¼ j
j
½
1
*
+
2
X
N
X
N
1
1
j
r
i
r
j
j
j
V
ee
½¼ j
½
2
i
j
6¼
i
*
+
X
N
V
ext
½¼ j
v
ext
ð
r
i
Þj
½
3
i
By the Rayleigh-Ritz principle, we find
hj
H
E
¼
min
ji
¼
min
ð
T
½þ
V
ee
½þ
V
ext
½Þ
½
4
If we rewrite the minimization as a two-step process,
155
we find
E
¼
min
n
a
;
min
!ð
n
b
Þ
ð
T
½þ
V
ee
½þ
V
ext
½Þ
n
b
n
a
;
where the inner search is over all interacting wave functions yielding spin den-
sities
n
a
;
n
b
. We may pull the last term out of the inner minimization to give
!
ð
d
3
rv
ext
s
ð
r
Þ
X
E
¼
min
n
a
;
F
½
n
a
;
n
b
þ
n
s
ð
r
Þ
n
b
s
¼
min
n
a
;
n
b
ð
E
½
n
a
;
n
b
Þ
½
5
