The recently developed Enskog theory for binary friction for fluids with continuous potentials, such as the Lennard-Jones, has been extended to calculate the frequency dependence of this friction, zeta E(omega). This zeta E(omega) is then applied to study vibrational energy relaxation of low-frequency modes via the Landau-Teller expression. The agreement with simulation results is found to be satisfactory. In the present approach we provide an exact prescription for the binary friction and thus remove a lacuna in this area. Zeta E(omega) shows an interesting structure with a hump at low frequency, the signature of which has already been seen in many simulation studies.