  We study the one-dimensional Holstein model of spinless fermions interacting
with dispersion-less phonons by using a recently developed projector-based
renormalization method (PRM). At half-filling the system shows a
metal-insulator transition to a Peierls distorted state at a critical
electron-phonon coupling where both phases are described within the same
theoretical framework. The transition is accompanied by a phonon softening at
the Brillouin zone boundary and a gap in the electronic spectrum. For different
filling, the phonon softening appears away from the Brillouin zone boundary and
thus reflects a different type of broken symmetry state.
