Localized instantaneous normal modes in liquid Na

The instantaneous-normal-mode (INM) densities of states for liquid Na in four realistic thermodynamic states are obtained by molecular-dynamic simulation. Through the study of the size dependence for the average number of the excited particles in an INM, we show that the INMs in the high-frequency end of each lobe are localized modes. By examining the projection components of the eigenvector of a localized INM on each particle, we show that the localized INM indeed has only one excited center at the particle with the largest projection component and the spatial size of the localization extends to the first nearest neighbor shell from the central particle. We also give a scenario of how a localized INM becomes an extended INM as the frequency decreases.