The Raman scattering cross section is calculated using standard third order perturbation theory. The scattering is viewed as a three-stage process: an electron-hole pair is created in the metal by the incident photon; the excited electron scatters off the molecule inelastically, exciting a molecular vibration; the electron finally recombines with the hole emitting the Raman scattered photon. This is one of the mechanisms proposed by Burstein and co-workers (1). The electrons in the film are confined by infinite potential barriers at the surfaces, but are otherwise free. The resulting sinusoidal one-electron eigenfunctions are used to calculate the nonlocal conductivity tensor in the random phase approximation. Maxwell's equations are then used to detemine the vector potential inside the film, which allows the electron-photon matrix elements to be calculated. The electron-molecule interaction is deduced from experimental data on low energy inelastic electron scattering from polar molecules. Calculations are performed for the frequency dependence of the Raman cross section with two different film thicknesses.