The unstable mode density of states (rho(u)(omega;T)) is obtained from computer simulation and is analyzed, theoretically and empirically, over a broad range of supercooled and normal liquid temperatures in the unit density Lennard-Jones liquid. The functional form of (rho(u)(omega;T)) is determined and the omega, T dependence is seen to be consistent with a theory given by us previously. The parameters in the theory are determined and are related to the topological features of the potential energy surface in the configuration space; it appears that diffusion involves a low degree of cooperativity at all but the lowest temperatures. It is shown that analysis of (rho(u)(omega;T)) yields considerable information about the energy barriers to diffusion, namely, a characteristic omega-dependent energy and the distribution of barrier heights, g(n)u(E). The improved description of (rho(u)(omega;T)) obtained in the paper is used to implement normal mode theory of the self-diffusion constant D(T) with no undetermined constants; agreement with simulation in the supercooled liquid is excellent. Use of a lower frequency cutoff on the contribution of unstable modes to diffusion, in an attempt to remove spurious contributions from anharmonicities unrelated to barrier crossing, yields the Zwanzig-Bassler temperature dependence for D(T). It is argued that the distribution of barriers plays a crucial role in determining the T dependence of the self-diffusion constant.