Chemistry Reference
In-Depth Information
insulators because of electron localization (small
v
values). This effect has been
termed chemical pressure.
Each atomic orbital admits two electrons (spins up and down), so that for an
N
ion
-system each band would be able to allocate 2
N
ion
electrons. In our example,
if each atomic orbital has only one electron, the band would have
N
ion
electrons,
which means it is half-filled, while if there were two electrons per atomic orbital
the band would be completely filled. Half-filled bands correspond to the metal-
lic state, and indeed there are 1D materials that are metals, such as TTF-TCNQ
and some Bechgaard salts as discussed in Section 1.5 (see also the discussion on
page 281). An illustrative example of the formation of bands from MOs has been
experimentally obtained for
n
-phenylene linear molecules (
n
6) (Seki
et al.
, 1984) and
n
-alkanes C
n
H
2
n
+
2
(Grobman & Koch, 1979), where ultraviolet
photoemission spectroscopy (UPS) has been systematically performed in the gas
phase for increasing
n
-values. When
n
increases the MO bands broaden due to the
increasing contribution of the discrete MOs. In the case of
n
-alkanes the compar-
ison of gas phase and solid phase spectra reveals that
n
=
1
,...,
=
13 already provides a
convenient finite model for the band structure of an infinite linear polymer. This
comparison also shows that the influence of intermolecular interactions on the band
structure is rather small, the most significant difference between spectra from the
two phases being a slight increase in peak widths in the solid phase.
Within the limit of negligible electronic correlations the filling of the band
N
at
and
k
F
are related through:
N
at
2
a
=
4
k
F
.
(1.34)
For
N
at
=
1, corresponding to a half-filled band,
k
F
=
π/
2
a
and for
N
at
=
2, cor-
responding to a filled band,
k
F
=
π/
.
Let us consider briefly the rather simple polyacetylene molecule, -(CH)
n
,in
which each carbon is
a
(see Fig. 1.28). In fact for CTSs
N
at
=
σ
bonded to only two neighbouring carbons and one hydrogen
atomwith one
electron on each carbon (the
p
π
orbital pointing perpendicularly to
the chain direction). If the carbon bond lengths were equal, with one
π
π
electron per
formula unit, it would imply a metallic state (
E
β
<
0) as discussed above. However,
neutral polyacetylene is a semiconductor with an energy gap of approximately
1.5 eV. The reason for this discrepancy is discussed next.
6
Let us imagine now that for some reason, either intrinsic or extrinsic, the lin-
ear chain from Fig. 1.28(a) dimerizes but preserves its 1D structure. The new
distribution is illustrated in Fig. 1.29(a). The first consequence is that the period
6
The bandwidth of linear polymers is of the order of 10 eV, thus sensibly larger than the typical values found for
MOMs, which lie in the
.
π
0
5 eV range. This difference is due to the fact that the
-overlap is intramolecular
and intermolecular for polymers and MOMs, respectively.
