01 | Band Structure of Polyacene

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Fig. 62 Structure of polyacene.

The structure of polyacene can be simplified down to a unit cell containing only 6 atoms (4 carbon and 2 hydrogen). The unit cell is shown in Fig. 62. The unit cell repeats in the \(x\) direction to form a 1D chain. We can reasonably expect the most important frontier orbitals of this compound to be composed of \(p_z\)-orbitals of the carbon atoms. These will be the most important for bonding, reactivity, and the material’s electrical properties.

Fig. 63 Basis \(p_z\)-based molecular orbitals for the unit cell of polyacene.

The resulting band structure along the \(\Gamma - X\) path is shown below. Bands are colored by their orbital character. The \(p_z\)-orbitals are highlighted in red. If you zoom in on some of the \(s\)-based and other \(p\)-based bands you’ll also find some familiar signs of \(sp\)-mixing but by symmetry the four \(p_z\) orbitals in the unit cell are only expected to interact with each other.

We can plot the crystal orbitals as well. At gamma the crystal orbitals are generated by simply translating the mo by 1 unit cell to tile the lattice (0 nodes between repeat units). At \(X\) the crystal orbitals are generated by translating the unit cell and inverting the orbital phases with every translation (1 node between every repeat unit). All points in between are generated by interpolating between these two extremes.

Fig. 64 The \(\pi\)-based crystal orbitals of polyacene at the special points \(\Gamma\), and \(X\).