  We discuss the nature of the different ground states of the half-filled
Holstein model of spinless fermions in 1D. In the metallic regime we determine
the renormalised effective coupling constant and the velocity of the charge
excitations by a density-matrix renormalisation group (DMRG) finite-size
scaling approach. At low (high) phonon frequencies the Luttinger liquid is
characterised by an attractive (repulsive) effective interaction. In the
charge-density wave Peierls-distorted state the charge structure factor scales
to a finite value indicating long-range order.
