color bar. Data are shown as the average values over bins in the frequency domain of $\left[ \omega - \Delta\omega/2, \omega + \Delta\omega/2\right]$ with $\Delta \omega \simeq 0.06$. The vertical line indicates the position of $\omega_\mathrm{BP}/\omega_\mathrm{D}$ averaged over the examined systems with varied $\varepsilon_\mathrm{bend}$. ](fig7.pdf){width="48.00000%"}

Low-frequency vibrational spectra
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### Reduced vDOS

Finally, the spectra of vibrational eigenmodes in polymer glasses are studied. The vibrational mode analysis is performed on the configuration of the polymeric system at $T=0$, which corresponds to the inherent structure [@Kittel_1996; @Ashcroft]. The Hessian matrix is diagonalized to obtain the eigenfrequencies $\omega^k$ that corresponds to the square root of the eigenvalues $\lambda^k$, i.e., $\omega^k = \sqrt{\lambda^k}$ ($k=1,2,...,3N_\mathrm{p}$). The specific expression of the Hessian matrix is given in Supplementary Material [^1]. The statistics of the eigenfrequency provide the vDOS, $g(\omega)$. Figure \[fig4\] presents the reduced version of the vDOS, $g(\omega)/\omega^2$, when changing the rigidity $\varepsilon_\mathrm{bend}$ and for $L=50$ in (a) and $L=3$ in (b). The reduced vDOS, $g(\omega)/\omega^2$, of the Debye theory is the so-called Debye level $A_\mathrm{D}$ [@Kittel_1996; @Ashcroft]. $A_\mathrm{D}$ is calculated from the elastic moduli, $K$ and $G$, as follows: $A_\mathrm{D} = 3/\omega_\mathrm{D}^3$, where $\omega_\mathrm{D}$ is the Debye frequency defined as $\omega_\mathrm{D}
= \left[ 18 