Using particle swarm optimization with density functional theory, we identify the positions of hydrogen in a hypothetical Mg-end-member of phase egg (MgSiO 4 H 2) and predict the most stable crystal structures with MgSiO 4 H 2 stoichiometry at pressures between 0 and 300 GPa. The particle swarm optimization method consistently and systematically identifies phase H as the energetically most stable structure in the pressure range 10-300 GPa at 0 K. Phase Mg-egg has a slightly higher energy compared to phase H at all relevant pressures, such that the energy difference nearly plateaus at high pressures; however, the combined effects of temperature and chemical substitutions may decrease or even reverse the energy difference between the two structures. We find a new MgSiO 4 H 2 phase with the P4 3 2 1 2 space group that has topological similarities to phase Mg-egg and is energetically preferred to phase H at 0-10 GPa and 0 K. We compute the free energies for phase Mg-egg, phase P4 3 2 1 2, and phase H at 0-30 GPa within the quasi-harmonic approximation and find that the effect of temperature is relatively small. At 1800 K, the stability field of phase P4 3 2 1 2 relative to the other polymorphs increases to 0-14 GPa, while pure phase Mg-egg remains energetically unfavorable at all pressures. Simulated X-ray diffraction patterns and Raman spectra are provided for the three phases. Additionally, the crystallographic information for two metastable polymorphs with the P1 space group is provided. Our results have implications for the deep hydrogen cycle in that we identify two novel potential carrier phases for hydrogen in the mantles of terrestrial planets and assess their stability relative to phase H. We determine that further experimental and computational investigation of an extended compositional space remains necessary to establish the most stable dense hydrated silicate phases.