Instantaneous normal modes (INM) are the harmonic approximation to liquid dynamics. This is an extension of the phonon description of lattice dynamics, in which case Bloch\u27s theorem shows that all modes are extended. Long-range order is destroyed in liquids and glasses, and the INM spectrum has contributions from both extended and localized modes. We use the soft-potential mode to describe localized modes. This model is a high-temperature extension of the standard two-level-system model for glasses. The equilibrium position of any atom in the liquid has only temporary character, and relaxation processes in the liquid are associated with particles hopping over potential energy barriers. Barrier tops have negative curvature so that an INM spectrum has an imaginary frequency (unstable) lobe in addition to the conventional stable mode contributions; conversely the unstable modes carry information about diffusion. We derive analytic expressions for the frequency and temperature dependence of the unstable lobe that are in agreement with results from computer simulations. Self-diffusion of particles in the liquid is governed by the fraction of unstable modes originating from double-well potentials. For the diffusion constant, we find a crossover behavior from Arrhenius temperature dependence to Zwanzig-Bässler dependence. We find an explicit expression for the distribution of barrier heights. In agreement with Stillinger\u27s inherent structure approach to glass-forming liquids, this distribution is uniform, or Gaussian, for high and low temperatures, respectively
