Chemistry Reference
In-Depth Information
V
(
x
) can also be expressed as a Fourier series:
N
V
nG
e
i
nGx
V
(
x
)
=
,
(1.41)
n
=−
N
where
V
nG
are the Fourier coefficients. Introducing Eqs. (1.40) and (1.41) in the
Schrodinger equation one obtains the Bethe equation (Bethe, 1928):
¯h
2
k
2
2
m
e
−
E
c
k
+
N
V
nG
c
k
−
nG
=
0
,
(1.42)
n
=−
N
which leads to the equation system:
¯h
2
k
2
2
m
e
−
V
0
c
k
+
E
+
V
G
c
k
−
G
=
0
,
(1.43a)
¯
h
2
(
k
c
k
−
G
+
G
)
2
−
−
E
+
V
0
V
G
c
k
=
0
.
(1.43b)
2
m
e
The energy
E
is obtained by setting to zero the determinant from Eq. (1.43):
¯
h
2
k
2
2
m
e
−
E
+
V
0
V
G
=
0
,
(1.44)
¯
h
2
(
k
G
)
2
−
V
G
−
E
+
V
0
2
m
e
which results in:
2
G
2
¯h
2
2
m
e
2
1
¯h
2
2
m
e
2
G
2
2
V
G
,
E
=
V
0
+
η
+
±
η
+
(1.45)
1
using
η
=
k
−
2
G
.At
k
=
π/
a
,
η
=
0 and Eq. (1.45) reduces to:
1
2
G
2
¯
h
2
2
m
e
E
=
V
0
+
±
V
G
.
(1.46)
Equation (1.46) shows that a gap
E
zb
a
induced by the weak
periodic potential. Following the same arguments given above, doubling
a
to 2
a
opens a gap at
=
2
V
G
opens at
k
=
π/
2
a
. We thus see that both the chemical and physical approaches
lead to the same conclusions as expected.
Let us now consider the ideal case of the BFS (see Fig. 1.18). In the absence
of anions and neglecting sulfur- or selenium-induced lateral molecule-molecule
interactions, the organic molecules can be modelled as forming an ideal 1D chain
with
p
π
orbitals aligned in the direction of the chain with a lattice constant
a
given
π/
