The main objection we can raise about the method is that we are trying to describe the wavefunction of the periodic solid as a combination of atomic orbitals that are eigenstates of a different Schrödinger equation with a differen potential and different boundary conditions. Moreover, the basis set is incomplete, since it lacks all the scattered wave eigenstates of the Schrödinger equation in the continuum. Although the wavefunction may be reasonably describe the states near the core of the atoms, it cannot pretebd to represent a Bloch state in the insterstitial region, where is must behave as a linear combination of free-electron plane waves.
The core wavefunctionos are appreciable only in de vicinity of the atom, and therefore the t-b approximation works reasonably well. However, in valence band states, that have higher energy than core states, the wavefunctions present more oscillations near the atomic cores. Moreover, they look more plane-wave-like in the intersticial regiono between atoms. For these states, the tight-binsing approximation does not work, and more sophisticated methods are required, as we will stee in the next section.