  The Peierls instability in one-dimensional electron-phonon systems is known
to be qualitatively well described by the Mean-Field theory, however the
related self-consistent problem so far has only been able to predict a partial
suppression of the transition even with proper account of classical lattice
fluctuations. Here the Hartree-Fock approximation scheme is extended to the
full quantum regime, by mapping the momentum-frequency spectrum of
order-parameter fluctuations onto a continuous two-parameter space. For the
one-dimensional half-filled Su-Schrieffer-Heeger model the ratio $d=\Omega/2\pi
T_c^0$, where $\Omega$ is the characteristic phonon frequency and $2\pi T_c^0$
the lowest finite phonon Matsubara frequency at the mean-field critical point
$T_c^0$, provides a natural measure of the adiabaticity of lattice
fluctuations. By integrating out finite-frequency phonons, it is found that a
variation of $d$ from the classical regime $d=0$ continuously connects $T_c^0$
to a zero-temperature charge-density-wave transition setting up at a finite
crossover $d=d_c$. This finite crossover decreases within the range $0\leq d
\approx 1$ as the electron-phonon coupling strength increases but remaining
small enough for weak-coupling considerations to still hold. Implications of
$T_c$ supression on the Ginzburg criterion is discussed, and evidence is given
of a possible coherent description of the charge-density-wave problem within
the framework of a renormalized Mean-Field theory encompassing several aspects
of the transition including its thermodynamics close to the quantum critical
point.
