Past exam of the chemistry A course of the University of Cambridge / 2022 / Paper 2 / A9 / a / i / Solution

In the orthogonal nearest-neighbour tight-binding model,The on-site integral ${\alpha }_s$ sets the band centre, while the hopping integral ${\beta }_s$ describes coupling between neighbouring orbitals. With a consistent phase convention, the bonding combination at the zone centre has ${\beta }_s<0$.

For p orbitals directed along the chain, useHere every positive $p$ lobe points in the same spatial direction. Neighbouring equal-sign coefficients then bring opposite-sign facing lobes together and make the zone-centre combination antibonding; the alternating zone-boundary combination is bonding. Thus the usual axial $p$ hopping has the opposite sign to $s$ hopping. Orbital phase and the sign of one-dimensional hopping must be treated consistently; replacing only ${\alpha }_s$ by ${\alpha }_p$ while silently keeping the same physical hopping sign would give the wrong band ordering.