In this section we will approach the problem of generating the band structure of regulartrans-polyacetylene in a different way – by usingfragment orbitalsto generate the Bloch orbitals. When trying to understand the electronic structure of a complex molecule it is useful to decompose the system into different fragments for which the electronic structure is easily derived. It is then possible to analyse the interaction between the different fragment orbitals that lead to the orbitals of the molecule. Using this process, a clear understanding of the main electronic factors governing the stability of the molecule is usually reached. For instance, the carbon–carbon bonding of the ethylene molecule may be studied by considering the interaction of two CH2fragments. In the
Fig. 5.11
A view of thetrans-polyacetylene distortion based on Lewis structures.
same vein we can use the orbitals of a –CH2–CH2– fragment to generate the crystal orbitals oftrans-polyacetylene. To this end, let us consider theπ-type normalised molecular orbitals of the –CH2–CH2– fragment, i.e. theπ andπ∗ orbitals represented in Fig. 5.12.
Fig. 5.12
π-type orbitals of the C2H2fragment, with only the upper lobes of thepy atomic orbitals represented.
We can generate a base of Bloch orbitals(BO)π(k)and(BO)π∗(k) using the two fragment orbitalsπandπ∗.
BOπ(k) = 1
√n
n
m=−n+1
exp(i2πmka)(π)m (5.1)
BOπ∗(k) = 1
√n
n
m=−n+1
exp(i2πmka)(π∗)m (5.2)
Since both orbitals are antisymmetric with respect to thex O zplane, theπ-type crystal orbitals of the system result from the interaction of these two Bloch orbitals:
(i) a mathematical resolution of the secular equations, leading to the analyt- ical expression of the crystal orbitals and their energies, according to the H¨uckel approach;
(ii) a more qualitative approach based on the analysis of the interaction term between the Bloch orbitals(BO)π(k)and(BO)π∗(k).
5.4.1 Calculation of the band structure by means of the H¨ uckel approach
The E(k) energies associated with the crystal orbitals may be obtained by imposing that the secular determinant be nil using this new basis:
hπ,π(k) −E(k)S π,π(k) hπ,π∗(k) −E(k)S π,π∗(k) hπ∗,π(k) −E(k)S π∗,π(k) hπ∗,π∗(k) −E(k)S π∗,π∗(k)
=0 (5.4)
where
hπ,π(k) =
BOπ(k)|ˆ h|BOπ(k) , hπ∗,π∗(k) =
BOπ∗(k)|ˆ h|BOπ∗(k) , hπ∗,π(k) =
BOπ∗(k)|ˆ h|BOπ(k)
=
hπ,π∗(k)∗ Sπ,π(k) =
BOπ(k)|BO π(k) , Sπ∗,π∗(k) =
BOπ∗(k)|BO π∗(k) Sπ∗,π(k) =
BOπ∗(k)|BO π(k)
=
Sπ,π∗(k)∗
The different terms may be calculated using the procedure developed in Sec- tion 3.2.2 and the H¨uckel approach:
hπ,π(k) =α+2βcos2(πka) ; hπ∗,π∗(k) =α−2βcos2(πka) (5.5a) hπ,π∗(k) =iβsin(2πka) ; Sπ,π∗(k) =0 ; Sπ,π(k) =Sπ∗,π∗(k) =1
(5.5b) Let us note that hπ,π(k) and hπ∗,π∗(k) represent the energies of the Bloch orbitals(BO)π(k) and(BO)π∗(k), respectively, whereas h π,π∗(k) describes the interaction between the two orbitals.
The secular equation (eqn (5.4)) may be written in the form:
hπ,π(k)−E(k) hπ∗,π∗(k)−E(k)
−hπ,π∗(k)hπ∗,π(k)=0
Developing the different terms by means of eqn (5.5) we obtain the following equation for the allowed energies of the system:
E±(k) =α±2βcos(πka) with ka∈
−1 2,1
2
(5.6) Now let us compare the energy of the crystal orbitals (eqn (5.6)) with those of the Bloch orbitals(BO)π(k)and(BO)π∗(k) (eqn (5.5a)). The similarity of expressions (5.6) and (5.5a) is striking: they only differ in the square power of the cosine function. Figure 5.13 is a representation of the variation as a function ofkaof the energy of the CO±(k) crystal orbitals (continuous lines) and the(BO)π(k) and(BO)π∗(k) Bloch orbitals (dotted lines).
Fig. 5.13
Energy of the Bloch orbitals
{(BO)π(k), (BO)π∗(k)}generated from theπandπ∗fragment orbitals (dotted lines, see eqn (5.5a)). The energy of the crystal orbitals of the system are also shown (continuous lines, see eqn (5.6)).
There is a strong similarity between the lines representing the energy of the crystal orbitals and those of the Bloch orbitals(BO)π(k)and(BO)π∗(k)as a function ofka. This means that the band structure fortrans-polyacetylene may be obtained qualitatively assuming that the Bloch orbitals interact weakly, in other words assuming that thehπ,π∗(k)term can be neglected.
5.4.2 Qualitative determination of the band structure
In this section we will try to generate the band structure for regular trans- polyacetylene without explicitly solving the secular determinant. We will only consider the shape of the dotted lines in Fig. 5.13, which represent the energy of the Bloch orbitals (BO)π(k) and (BO)π∗(k) as a function of ka. The interaction between these orbitals is stronger when their energy difference is smaller (i.e. near the X point) and when the interaction term (hπ,π∗(k), see eqn (5.5b)) is large (i.e. near the point associated with ka equal to 1/4). As a consequence, the mixing between the two Bloch orbitals near is small:
the CO+(k)and CO−(k)crystal orbitals scarcely differ from the (BO)π(k) and(BO)π∗(k)Bloch orbitals and thecπ+(k)/c+π∗(k)andc−π(k)/c−π∗(k)ratios are, in absolute value, considerably larger than one. In contrast, for points not near, an interaction between(BO)π(k) and(BO)π∗(k) develops and leads to a crystal orbital CO+(k) more stable than(BO)π(k) and a crystal orbital CO−(k) less stable than(BO)π∗(k), as shown in Fig. 5.14.
It may then be concluded that the band structure of regular trans- polyacetylene may be generated by ‘pushing out’ from each other the curves representing the energy dependence of the Bloch orbitals generated by theπ andπ∗ fragments. The ‘repulsion’ between Bloch orbitals is at a minimum around the center of the Brillouin zone () and a maximum around the ka=1/4 point (see Fig. 5.15).
At X the interaction term is nil, so that the Bloch orbitals BOπ(X) and BO∗π(X)form a basis for the degenerate level. Near X, although the interaction term is small, the mixing between Bloch orbitals is noticeable because the two Bloch orbitals are quasi-degenerate. This explains the differences between the curves associated with the crystal orbitals and Bloch orbitals around X:
whereas the tangent is horizontal for the latter it is non-nil for the former.
Again, this means that the Bloch orbital curves ‘repel’ each other to give the curve associated with the crystal orbitals.
Fig. 5.14
Schematic diagram showing how the (BO)π(k)and(BO)π∗(k)Bloch orbitals interact and lead to the crystal orbitals of the system.
Fig. 5.15
Consequences of the interaction between the Bloch orbitals: the band structure may be obtained by ‘repelling’ the dotted lines representing the energy of the (BO)π(k)and(BO)π∗(k)Bloch orbitals.
This example illustrates how convenient it can be to develop the crystal orbitals in terms of a basis of Bloch orbitals associated with appropriate frag- ment orbitals. In favourable cases it allows the determination and qualitative analysis of the band structure of the system. Often the crystal orbitals atand X may be identified with one of the Bloch orbitals generated by appropriate fragment orbitals, thus greatly simplifying the analysis. In the following chap- ters we will use this idea as much as possible since it gives useful hints on the nature of the band structure of many compounds without having to do any detailed calculations.