  We compare the acoustic Su-Schrieffer-Heeger (SSH) model with two of its
optical variants where the phonons are defined on either on the sites or bonds
of the system. First, we discuss how to make fair comparisons between these
models in any dimension by ensuring their dimensionless coupling $\lambda$ and
relevant phonon energies are the same. We then use determinant quantum Monte
Carlo to perform non-perturbative and sign-problem-free simulations of all
three models on one-dimensional chains at and away from half-filling. By
comparing the results obtained from each model, we demonstrate that the optical
and acoustic models produce near identical results within error bars for
suitably chosen phonon energies and $\lambda$ at half-filling. In contrast, the
bond model has quantitatively different behavior due to its coupling to the $q
= 0$ phonon mode. These differences also manifest in the total length of the
chain, which shrinks for the bond model but not for the acoustic and optical
models when $\lambda \neq 0$. Our results have important implications for
quantum Monte Carlo modeling of SSH-like interactions, where these models are
sometimes regarded as being interchangeable

Strongly Correlated Electrons