513.001 Molecular and Solid State Physics

MIT 8.231 Physics of Solids, 2.7, sp² hybridization

a) Find the three linear combinations of s, p_x and p_y atomic orbitals necessary to form sp² orbitals in three planar bonding directions.

i) Assume the input orbitals are normalized and orthogonal. Assume the same for the hybrid orbitals.

ii) Assume the coefficients in the expansion are real.

iii) Point one of the orbitals maximally in the x direction.

iv) Make use of symmetry.

Make a sketch of the resulting orbitals assuming one is in the n = 2 shell.

b) Consider the bezene molecule. Assume the 1s states belong to the atomic cores. This accounts for 2 of the 6 electrons from each carbon. How should the 2sp² hybrid orbitals be used to form pairwise bonds to the hydrogens and between the carbon neighbors? These bonds should account for another 3 electrons per carbon.

c) Now use the remaining 2p_z states on each carbon to further bind the molecule. Make a giant molecular orbital out of a linear combination of all 6 atomic states, one centered at each carbon site. By symmetry the magnitude of each of the six coefficeints must be equal in a given molecular orbital, hence the coefficients can differ only in phase. The phase must advance uniformly as one goes from one site to the next, and must come back to the starting value when the circuit is completed.

The six allowed values of the phase are φ_k = 0, ±π/3, ±2π/3, and π. Which of these hybrid orbitals should have the lowest energy and which the highest? Which of the orbitals will be bonding and which antibonding? Which will be degenerate? Which will be filled in the bezene molecule?