We present a comprehensive first-principles investigation of a hypothetical cubic $Pm\overline{3}m$ phase of the ternary hydride ${\mathrm{NaAlH}}_3$, focusing on its lattice dynamics, electronic structure, and electron-phonon-mediated superconducting properties at ambient pressure. Using density functional theory and the Migdal-Eliashberg formalism, we find an exceptionally strong electron-phonon coupling ($\lambda =2.23$), resulting in a superconducting critical temperature of up to 73.7 K for a Coulomb pseudopotential ${\mu }^{*}=0.1$. Phonon dispersion calculations, complemented by ab initio molecular dynamics simulations, indicate dynamic and thermal stability within the adopted theoretical framework. The electronic structure exhibits a metallic character with substantial contributions from Al- and Na-derived states at the Fermi level. The resulting superconducting gap ratio ($2\mathrm{\Delta }\left(0\right)/k_B{T}_c\approx 4.8$) and specific heat jump ($\mathrm{\Delta }C/\gamma T_c\approx 2.2$) significantly exceed BCS weak-coupling predictions, highlighting the strong-coupling nature of superconductivity in this hypothetical phase.

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• Electron-phonon coupling
• Electronic structure
• Superconductivity
• High-temperature superconductors
• Hydrides
• Density functional calculations
• First-principles calculations