@Mazzacurati_1996]: $$P^k = \frac{1}{N_\mathrm{p}}\left[ \sum_{i=1}^{N_\mathrm{p}} (\boldsymbol{e}^k_i\cdot \boldsymbol{e}^k_i)^2\right]^{-1},$$ where $\boldsymbol{e}^k_i$ $(i=1, 2, \cdots , N_\mathrm{p})$ are the eigenvectors associated with the eigenfrequencies $\omega^k$ ($i$ is the index of the monomer particle and $N_\mathrm{p}$ is the number of monomer particles). The $\boldsymbol{e}^k_i$ represents the displacements of each monomer bead $i$ in the eigenmode $k$. It is noted that $\boldsymbol{e}^k_i$ is obtained from the diagonalization of the Hessian matrix and is orthonormalized as $\sum_{i=1}^{N_\mathrm{p}} \boldsymbol{e}^k_i\cdot \boldsymbol{e}^l_i =
\delta_{kl}$ ($\delta_{kl}$ is the Kronecker delta). The following extreme cases can occur: ${P}^k = 2/3$ for an ideal sinusoidal plane wave, ${P}^k = 1$ for an ideal mode in which all constituent particles vibrate equally, and ${P}^k = 1/N_\mathrm{p} \ll 1$ for a perfect localization, which indicates that each vibrational state is associated only with a single atom and that $e^k_i \cdot e^k_i=1$ for a single $i$, otherwise $e^k_i \cdot e^k_i=0$.

Figure \[fig7\] presents the value of $P^k$ as a function of the scaled frequency $\omega/\omega_\mathrm{D}$, for different $\varepsilon_\mathrm{bend}$. It is noted that the presented data are the binned average values. Below the BP frequency $\omega_\mathrm{BP}$, $P^k$ progressively decreases when $\omega$ decreases due to the spatially localized vibrations. The low-frequency localization below $\omega_\mathrm{BP}$ has also been 