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.