Chemistry Reference
In-Depth Information
3.2
The linear chain of atoms in problem 3.1 undergoes a distortion so
that the even-numbered atoms move to the right, doubling the unit
cell size to 2
L
, and leading to an interaction of strength
V
+
V
between the 2
n
th and
(
2
n
+
1
)
th site, and an interaction of strength
−
(
−
)
V
th site. By writing the wave-
function within a given unit cell as a linear combination of the two
orbitals in that cell, and then applying Bloch's theorem, show that
the band structure in the distorted lattice is given by
V
between the 2
n
th and
2
n
1
2
V
2
cos
2
2
sin
2
E
sq
=
E
s
±
(
qL
)
+
(
V
)
(
qL
)
(3.51)
and hence show that the distortion can be regarded as opening an
energy gap of magnitude 4
2
L
in the band structure of
problem 3.1. Estimate the average band-structure energy gained per
atom when this distortion, referred to as a Peierls distortion, occurs
in a linear chain with one electron per atom on the chain.
(
V
)
at
q
=
π/
3.3
Show for the K-P potential of eq. (3.31) that the magnitude of the
n
th energy gap at
q
=±
π/
L
is given in the NFE model by
sin
(
2
V
0
2
n
−
1
)π
b
E
g
=
(3.39b)
(
−
)π
2
n
1
L
while the
n
th gap at
q
=
0 is given by
V
0
n
E
g
=
sin
(
2
n
π
b
/
L
)
(3.39c)
π
3.4
Show for the K-P potential of eq. (3.31) that the NFE wavefunction,
ψ
(
x
)
, of the lower state at the
n
th gap at
q
=±
π/
L
is given by
n
,
l
2
L
sin
(
2
n
+
1
)π
x
ψ
=
n
,
l
L
while that of the upper state,
ψ
(
x
)
, is given by
n
,
u
2
L
cos
(
2
n
+
1
)π
x
ψ
=
n
,
u
L
Hence justify why the NFE method provides a better estimate of the
energy, to larger values of
b
, for the lower state than for the upper
state.
3.5 We can improve the accuracy of the NFE method by including more
basis states in the NFE calculation of the K-P band structure. We
see from the previous question that the four lowest basis states at
