should be considered as well [@Monaco_2006; @Niss_2007; @Hong_2008; @Mizuno2_2013].

In addition, the length scale of collective vibrational modes in the BP region is discussed. For atomic LJ glasses, the length scale was evaluated as $\xi_\mathrm{BP}=2 \pi c_\mathrm{T} /\omega_\mathrm{BP}$, which corresponds to the size of approximately $23$ particle [@Leonforte_2005]. This length scale diverges near the isostatic point or the marginally stable point, theoretically [@Wyart_2005; @Wyart_2006; @Wyart_2010; @DeGiuli_2014] as well as numerically [@Silbert_2005; @Lerner_2014; @Karimi_2015; @Shimada_2018; @Mizuno_2018]. The present study evaluates the length scale of collective vibrational modes in polymeric glasses as $\xi_\mathrm{BP}=2 \pi c_\mathrm{T}
/\omega_\mathrm{BP} \approx 12$, which corresponds to half of that for LJ glasses. The vibrational modes in the BP region are more localized nature due to the polymerization. Moreover, the value of $\xi_\mathrm{BP}$ is independent of the bending rigidity $\varepsilon_\mathrm{bend}$ because of $\omega_\mathrm{BP}
\propto \omega_\mathrm{D} \propto c_\mathrm{T}$. In other words, the bending rigidity does not affect the length scale of the collective vibrational motions in the BP region.

Discussion
==========

The glass transition temperature, elastic properties, and the low-frequency vibrational spectra were studied in polymeric glasses. In particular, the bending energy scale was highly varied for long chains ($L=50$) and short chains ($L=3$). As the system becomes rigid 