  Charge-density waves are responsible for symmetry-breaking displacements of
atoms and concomitant changes in the electronic structure. Linear response
theories, in particular density functional perturbation theory, provide a way
to study the effect of displacements on both the total energy and the
electronic structure based on a single ab-initio calculation. In downfolding
approaches, the electronic system is reduced to a smaller number of bands,
allowing for the incorporation of additional correlation and environmental
effects on these bands. However, the physical contents of this downfolded model
and its potential limitations are not always obvious. Here, we study the
potential energy landscape and electronic structure of the Su-Schrieffer-Heeger
(SSH) model, where all relevant quantities can be evaluated analytically. We
compare the exact results at arbitrary displacement with diagrammatic
perturbation theory both in the full model and in a downfolded effective
single-band model, which gives an instructive insight into the properties of
downfolding. An exact reconstruction of the potential energy landscape is
possible in a downfolded model, which requires a dynamical electron-biphonon
interaction. The dispersion of the bands upon atomic displacement is also found
correctly, where the downfolded model by construction only captures spectral
weight in the target space. In the SSH model, the electron-phonon coupling
mechanism involves exclusively hybridization between the low- and high-energy
bands and this limits the computational efficiency gain of downfolded models.
