Understanding the vibrational density of states of liquids using instantaneous normal mode theory

Liquid dynamics play crucial roles in chemical and physical processes, ranging from biological systems to engineering applications. Yet, the vibrational properties of liquids are poorly understood when compared to the case of crystalline solids. Here, we report experimental neutron-scattering measurements of the vibrational density of states (VDOS) of water and liquid Fomblin in a wide range of temperatures. In the liquid phase, we observe a universal low-energy linear scaling of the experimental VDOS as a function of the frequency, which persists at all temperatures. The low-frequency scaling of the VDOS exhibits a sharp jump at the melting point of water, below which the standard Debye's law is recovered. On the contrary, in Fomblin, we observe a continuous transition reflecting its glassy dynamics, which is confirmed by structure measurements. More importantly, in both systems, we find that the slope of this linear behavior grows with temperature following an exponential Arrhenius-like form, as predicted by instantaneous normal mode (INM) theory. The microscopic origin of this exponential behavior lies in the thermally activated hopping across the energy barriers in the liquid potential landscape. We confirm this experimental trend using molecular dynamics simulations and show that the predictions of INM theory for the shape of the VDOS in the liquid phase are in qualitative agreement with the experimental and simulation data. On the other hand, from a more quantitative perspective, the predictions from the normal modes analysis under-estimate the energy scale entering in the exponential temperature behavior of the VDOS slope by a factor of ≈ 2-3. Anharmonic effects, that are not entirely captured by the INM analysis, are probably the origin of this discrepancy.