  Constrained electronic-structure theories enable the construction of
effective low-energy models consisting of partially dressed particles. However,
the interpretation and physical content of these theories is not
straightforward. Here, we carefully explore the properties of downfolding
theories for electron-ion problems, in particular constrained
density-functional perturbation theory (cDFPT). We show that the dipole
selection rules determine whether the partially dressed phonons satisfy
Goldstone's theorem, and we prove that electronic screening always lowers the
phonon frequencies. We illustrate the theory with cDFPT calculations for
minimal example systems: the nitrogen and benzene molecule as well as graphene.
