Chemistry Reference
In-Depth Information
and electrons are delocalized, representing highly conducting materials, while for
U
1 electronic correlations dominate leading to localized states and thus
to insulating materials. Solids described by the latter case are known as Mott-
Hubbard insulators. Equation (1.7) is fully justified in the
U
/
W
>
1 limit but
when
U
becomes dominant, that is in the case of strongly correlated systems, other
approximations have to be found. In what follows we will stay within the framework
of the one-electron approximation.
/
W
One-electron approximation
A first attempt consists of assuming that each electron feels a smooth distribution
of negative charge with a charge density
ρ
c
arising from the remaining
N
e
−
1
electrons. In this case Eq. (1.4a) transforms into:
e
N
e
¯
h
2
2
m
e
∇
ρ
c
(
r
)
d
r
2
i
H
ee
=−
−
|
,
(1.9)
r
|
r
−
i
=
1
e
N
e
1e
1e
.
H
1e
where
ρ
c
(
r
)
=−
i
(
r
)
|
i
(
r
)
from Eq. (1.8) thus becomes:
i
=
1
i
¯h
2
2
m
e
∇
N
ion
H
1e
2
i
=−
+
−
V
e
−
ion
(
r
i
R
j
)
i
=
j
1
e
2
j
(
r
)
j
(
r
)
1e
1e
N
e
d
r
|
+
.
(1.10)
|
r
−
r
|
j
=
1
Equation (1.10) represents the Hartree Hamiltonian and Eq. (1.8) has to be solved
by iteration, in the sense that a guessed trial wave function
1e
i
is introduced in Eq.
(1.10) and the Schrodinger equation Eq. (1.8) solved. The resulting wave function
is again introduced in Eq. (1.10) and Eq. (1.8) is again solved until self-consistency
is achieved.
When
|
N
e
|
is expressed as the antisymmetric combination given by the Slater
determinant:
1e
1e
1e
|
1
(
r
1
)
|
1
(
r
2
)
...
|
1
(
r
N
e
)
1e
1e
1e
|
2
(
r
1
)
|
2
(
r
2
)
...
|
2
(
r
N
e
)
N
e
|
=
,
(1.11)
.
.
.
.
1e
1e
1e
|
N
e
(
r
1
)
|
N
e
(
r
2
)
...
|
N
e
(
r
N
e
)
