Internal energies, enthalpies, phonon dispersion curves, and superconductivity of atomic metallic hydrogen are calculated. The standard use of pseudopotentials in density-functional theory are compared with full Coulomb-potential, all-electron linear muffin-tin orbital calculations. Quantitatively similar results are found as far as internal energies are concerned. Larger differences are found for phase-transition pressures; significant enough to affect the phase diagram. Electron–phonon spectral functions ${\alpha }^{2}F\left(\omega \right)$ also show significant differences. Against expectation, the estimated superconducting critical-temperature T${}_c$ of the first atomic metallic phase I4${}_1$/amd (Cs-IV) at 500 GPa is actually higher.