Non-linear self-consistent screening of a proton by a high density electron gas has been used to find effective electron–proton potentials for metallic hydrogen for a number of densities and for both face-centered cubic and body-centered cubic structures. The resulting proton–proton potentials have been employed to calculate the phonons in the self-consistent harmonic approximation, following which the effective distributions α2F(ω) were evaluated in the plane wave, spherical approximation. From these, the superconducting transition temperatures Te and functional derivatives were found.Non-linear effects are seen to be important. For both structures, dynamical instabilities occur for rs ≥ 1.0, indicating densities higher than those predicted by linear theory are required. In addition, for the fcc case, Te is enhanced.Te is found to depend sensitively on the structure assumed; for the bcc case, it is very small.For fcc H. McMillan's equation overestimates Te by about 40%, even when λ = 0.5. Leavens' formula agrees with solutions of the Eliashberg gap equations to within about 10%.