Downfolding approaches to electron-ion coupling: Constrained density-functional perturbation theory for molecules

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.