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Suppose we have a square lattice with periodic condition in both $x$ and $y$ direction with four atoms per unit cell, the configuration of the four atoms has $C_4$ symmetry. What will be a general wavefunction in this lattice look like? How will it different from the usual Bloch wavefunction?

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What do you mean different from the "usual Bloch wavefunction"? Are you asking about the effect of the $C_4$ symmetry? Or are you confused about having multiple atoms in a tight binding description? Or some combination thereof? –  BebopButUnsteady May 10 '13 at 0:18

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