Why do metallic elements have a very small band gap while nonmetallic elements have a large band gap? I recently was going over solid state bonding and conductivity. I was wondering why metallic elements (such as sodium, iron, etc.) have a small band gap / no band gap (as conductors that have the fermi level in the overlap between the valence and conduction band) while elements such as carbon have such large band gaps. What distinguishes carbon and sodium, for example, such that carbon is an insulator while sodium/iron is a conductor of electricity? If you could explain in terms high school to undergraduate level chemistry, that would be great.
The energy of the orbitals may be the first point to explain it at this level. You can use the example of a simple 1-D LCAO to form the bands(check page 723 of Atkins physical chemistry 8th edition). For instance, the energy difference between the bands formed by s orbitals and p orbitals will be small or large depending on the atom.