Bands from molecular orbitals

From molecular orbitals to bands

In the Hückel model each atom contributes one π orbital, and neighbouring atoms couple through the hopping integral β. For a chain of N atoms the orbital energies are , j = 1 … N.

Read j as a wavevector and every finite molecule's levels lie on one curve, the band of the infinite chain, . More atoms just sample that curve more finely, so discrete levels become a continuous band.

Bond alternation δ

Real conjugated chains don't have equal bonds: polyacetylene alternates short double bonds and long single bonds. The model captures this with two hopping integrals, , alternating along the chain. The unit cell now holds two atoms, so the band splits in two:

with a gap of at k = π/2. At half filling that gap sits exactly at the Fermi level, so the chain turns from a metal into a semiconductor.

• The lowest orbital has no sign changes (bonding everywhere); the highest changes sign at every bond.
• Each level holds two electrons. With one electron per atom the band is half full.
• The density of states piles up at the band edges, where E(k) is flat.