Deformation-potential theorem in metals and in dielectrics

Using linear response within the density-functional theory, I provide an expression of the matrix elements for electron scattering by long-wavelength phonons, both optic and acoustic. In the latter case, I prove that the matrix elements are exactly related to the strain-induced shifts of the electronic levels by the identity known as the deformation-potential theorem, and which had been demonstrated so far only for metals within the rigid- or deformable-ion models. The case of dielectrics shows additional complications due to the long-range nature of the Coulomb interaction; a generalized form of the theorem is proved in this case, where the macroscopic effects are embedded in a bulk fourth-rank tensor. The relationship to recent results on the ``absolute'' deformation potentials for homogeneous strain is discussed.