Title: Designing Quaternary Hydrides with Potential High T$_c$ Superconductivity

Abstract: We propose three parent structures for designing quaternary hydrides of increasing complexity to optimize parameters correlated with high T$_c$ superconductivity. The first is a simple Pm$\overline{3}$m cell inspired by the FCC RH$_3$ structures (R = trivalent rare earths), which we show has moderately promising potential for high T$_c$ compounds. The second is an Fm$\overline{3}$m heterostructure inspired by our work on Lu$_8$H$_{23}$N that consistently produces metallic hydrogen sublattices, whose quantum interference with Lewis bases is designed to high DOS$_H$(E$_F$). Several examples are put forward that first-principles calculations confirm have hydrogen-dominant metal character, as well as strong network connectivity as measured with the Electron Localization Function (ELF). The third quaternary model structure allows for a more precise description of doping as well as symmetry breaking of octahedral hydrogen which improves the hydrogen network connectivity. These model structures/formulae predict compounds with high predicted T$_c$ and have enough flexibility to optimize for both T$_c$ and stability at low pressures.