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Among the mechanisms for lattice structural deformation, the electron-phononinteraction mediated Peierls charge-density-wave (CDW) instability in singleband low-dimensional systems is perhaps the most ubiquitous. The standardmean-field picture predicts that the CDW transition occurs at all fillings andall values of the electron-phonon coupling $g$ and the adiabaticity parameter$t/\omega_0$. Here, we correct the mean-field expression for the Peierlsinstability condition by showing that the non-interacting staticsusceptibility, at twice the Fermi momentum, should be replaced by the dynamicone. We derive the Luttinger liquid (LL) to CDW transition condition, {\itexact to second order in a novel blocked perturbative approach}, for thespinless one-dimensional Holstein model in the adiabatic regime. The smallparameter is the ratio $g \omega_0/t$. We present the phase diagram atnon-half-filling by obtaining the surprising result that the CDW occurs in amore restrictive region of a two parameter ($g^2 \omega_0/t$ and $t/\omega_0$)space than at half-filling.Comment: Made changes in the appendices and also in notatio

Phase transition and phase diagram at a general filling in the spinless one-dimensional Holstein model