I know experimentally, the permittivity of a material is ##\vec D = \epsilon \vec E##, and you can also relate the Polarization to it. So it's basically a response of the material to an applied E field. But that seems like it would be a fairly complex thing to figure out theoretically. I guess one way would be to, for a given E field applied to a unit cell of the material, see how each atom gets polarized. But that seems like it would be incredibly complicated for unit cells with several atoms and a basis, and each atom having multiple electrons. So I'm wondering how you could figure out the permittivity of a general crystal, like if you were told its interatomic distances, lattice constants, band structure, etc. Is there a better way than what I described above? Thanks!