Chemistry Reference
In-Depth Information
Figure 1.32. Band structure of a linear chain of period
a
at half filling.
Orbital mixing
Let us now study the effect of including a perturbation
H
per
to the zero-order
Hamiltonian
H
0
, so that the new Hamilton operator
H
will be given by
H
=
H
per
. We further proceed with our example of the linear chain of period
a
with
N
at
H
0
+
1, for the sake of simplicity, and continue to extract fundamental
information. Figure 1.32 shows the band dispersion of this half-filled system.
Let us consider a wave vector
k
of an occupied state close to
k
F
and its equivalent
=
−
k
2
k
F
as indicated in Fig. 1.32. The Bloch functions of both states will be given
by Eq. (1.31):
N
ion
1
√
N
ion
1e
(
k
e
i
kna
at
(
x
|
φ
,
x
)
=
|
ψ
−
(
n
−
1)
a
)
,
(1.51a)
n
=
1
N
ion
1
√
N
ion
1e
(
k
e
i(
k
−
2
k
F
)
na
at
(
x
|
φ
−
2
k
F
,
x
)
=
|
ψ
−
(
n
−
1)
a
)
.
(1.51b)
n
=
1
The contribution of the perturbation to the total energy of the system is such
that
1e
(
k
1e
(
k
φ
−
2
k
F
,
x
)
|
H
per
|
φ
,
x
)
=
0
,
