  We use the Variational Exact Diagonalization to investigate the single
polaron properties for four different dual models, combining a short-range
off-diagonal (Peierls) plus a longer-range diagonal (Holstein or
breathing-mode) coupling. This allows us to investigate the sensitivity of
various polaron properties both to the range of the diagonal coupling and to
the specific diagonal coupling chosen. We find strong sensitivity to the range
for all duals models as the adiabatic limit is approached, however considerable
sensitivity is observed for some quantities even for large phonon frequencies.
Also, the strong dependence of the results on the specific form of the diagonal
coupling is observed everywhere in the parameter space. Taken together, these
results suggest that careful consideration must be given to the specific
coupling and its proper range when quantitative comparisons with experiments
are sought.
