Consider the outer-most s orbitals for Mg or Ca solid. They will form one band. The 2 electrons in the outer-most shell fill this band. Shouldn't the next p orbital band be above the fermi level and make Mg and Ca insulators?
One can also think about free electron model where the electron-ion interaction is treated as perturbation to the free electrons Hamiltonian. In that case, the first order perturbation theory will lift the degeneracy at the boundary of the first brillouin zone. The number of states in the first band (depending on the size of supercell) should exactly allow twice the number electron to occupy the band full, and make the crystal an insulator.
The above arguments are obvious wrong/incomplete due to the fact that Mg and Ca are metals in reality. However, I want to know what went wrong with my argument. What makes the two metals?