Localization transition of instantaneous normal modes and liquid diffusion.

We analyze the structure of instantaneous normal modes (INM) associated with the Hessian matrix of a liquid. Utilizing a scaling theory developed in the framework of Anderson localization studies, we unambiguously identify the crossover point in the INM spectrum between extended and localized modes. We establish a relation between the unstable, delocalized INMs and the liquid diffusion coefficient that appears valid over a large temperature range, covering both Arrhenius and non-Arrhenius regimes of temperature dependence. These results suggest a possible route to theoretically relate dynamics to thermodynamical properties of the liquid via the tomography of the INMs.