[Mel22]

Can anyone help me finding out why Si has a smaller band gap than Diamond?

[Dan]

When it comes to band gaps, I find it useful to think of the radial extension of the overlapping orbitals. The more diffuse the orbitals, the smaller the band gap.

Think about an MO diagram for I2 vs F2. The orbitals on I2 are more diffuse, overlap less effectively, and so the bonding-antibonding gap is smaller than in F2. In the context of solids like diamond and silicon, the bonding orbital is your valence band, the antibonding is your conduction band - Si orbitals are more diffuse, and so the band gap is less.

Another point to think about is band width. A large band width can also be attributed to diffuse orbitals, as larger orbitals will interact with more distant neighbours, which leads to the creation of more combinations of atomic orbitals, and thus a wider spread of MO energies.

Im not sure how well I am explaining this, but a VERY good, readable book for this sort of thing is "the electronic structure and chemistry of solids" - P.A. Cox. Definately check it out