Analysis of eigenvalues and eigenvectors of polymer particles: random normal modes

We investigate the density of vibrational states g(ω) for 6000 atom polymer particles involving all 18,000 degrees of freedom. The particles are efficiently generated using a molecular dynamics-based computational algorithm and a molecular mechanics method. The density of states spectrum g(ω) clearly shows two distinguishable frequency regions in the polymer system: high $\text{(760<ω<1240}\text{cm}{}^{\mathrm{-1}}\text{)}$ and low $\text{(0<ω<}\text{350}\text{cm}{}^{\mathrm{-1}}\text{)}$ frequency modes. By calculating the level-spacing distributions, we find the distribution of the low eigenfrequency corresponds to that of a Wigner distribution. In contrast, Poisson behavior is found for the high frequency region. The eigenvectors for the two regions are analyzed by using a random walk method and Stewart's perturbation theory, both indicate random character for the eigenvectors of the low frequency modes. The random character of the eigenvectors should have ramifications to most types of spectroscopy since transformations of the transition operator to random normal coordinates will cause a widespread mixing, i.e., no selection rules.