  For the one-dimensional Holstein model, we show that the relations among the
scaling exponents of various correlation functions of the Tomonaga Luttinger
liquid (LL), while valid in the thermodynamic limit, are significantly modified
by finite size corrections. We obtain analytical expressions for these
corrections and find that they decrease very slowly with increasing system
size. The interpretation of numerical data on finite size lattices in terms of
LL theory must therefore take these corrections into account. As an important
example, we re-examine the proposed metallic phase of the zero-temperature,
half-filled one-dimensional Holstein model without employing the LL relations.
In particular, using quantum Monte Carlo calculations, we study the competition
between the singlet pairing and charge ordering. Our results do not support the
existence of a dominant singlet pairing state.
