What is the relation between electronic band structure that is calculated by using free electron approximation (FEA) and tight binding approximation (TBA)?
In FEA, it seems we start with a parabolic dispersion relation (the relation between the energy of an electronic state and the crystal momentum of the state). Then we adjust the shape of the curve near the boundaries of Brillouin zones because of the perturbation of the periodic potential. At last, we just "project" the resulting dispersion relation to those paths in the first Brillouin zone.
In TBA, we don't calculate the general band structure directly, instead, we specifically calculate the bands formed by some atomic orbital. For example, for graphene, we can specifically calculate the energy band of $2P_z$ orbital of $C$ atoms.
In FEA, for each calculated energy band, how do we know it is formed by which atomic orbital? On the other hand, when using TBA, how do we know that we have exhausted every possible combination of atomic orbitals?
Is there a relation between the energy bands calculated by the two different methods?