Relaxation in supercooled liquids is formulated from the instantaneous normal modes (INM) point of view. The frequency and temperature dependence of the unstable, imaginary frequency lobe of the INM density of states, ⟨ρu(ω,T)⟩ (for simplicity we write ω instead of iω), is investigated and characterized over a broad temperature range, 10⩾T⩾0.42, in the unit density Lennard-Jones liquid. INM theories of diffusion invoke Im-ω modes descriptive of barrier crossing, but not all imaginary frequency modes fall into this category. There exists a cutoff frequency ωc such that modes with ω<ωc correspond to “shoulder potentials,” whereas the potential profiles include barrier-crossing double wells for ω>ωc. Given that only modes with ω>ωc contribute to diffusion, the barrier crossing rate, ωh, and the self diffusion constant D, are shown to be proportional to the density of states evaluated at the cutoff frequency, ⟨ρu(ωc,T)⟩. The density of states exhibits crossover behavior in its temperature dependence such that the exponential T-dependence of D(T) crosses over from Zwanzig–Bassler exp(−E2/T2) behavior at low T to Arrhenius exp(−E/T) behavior at high T; the exponential may be too weak to be observed, in which case D(T) is a power law. Based on the properties of LJ, a general INM description of strong and fragile liquids is presented, with a physical interpretation in terms of the “landscape” of the potential energy surface.