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Internal energies, enthalpies, phonon dispersion curves, and superconductivity of atomic metallic hydrogen are calculated. The (standard) use 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 $\alpha^2 F(\omega)$ 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.

The Pseudopotential Approach within Density-Functional Theory: The Case of Atomic Metallic Hydrogen